1. Introduction

Advances in nanoscale technologies and integrated optics have led to an aggressive interest in the development of effective means for the exploitation of optical micromanipulation (“trapping”) techniques, which are widely utilized ranging from stretching DNA molecules [1, 2] and engineering optically-driven pumps [3, 4] to optically-patterning the new photonic materials [5] and creating model systems for many-body physics studies [6, 7]. The techniques used to manipulate the particles have evolved in parallel with techniques used to enhance the visibility of particles with smaller sizes or in low contrast applications. For example, sensitive fluorescent techniques are often used to enhance the visibility of the particles at the smaller size and the low contrast applications [8]. To our best knowledge, the microscope is the only tool presently used to image trapped particles or assemblies. This approach has been successful when the previous research emphasis has largely remained upon manipulation of either individual particles or a very limited number of particles. However, large assemblies of submicron particles have attracted increasing attentions for their broad applications in electronics and photonics [9, 10], and the recent progress in optical manipulation has led to extensive research of manipulating smaller particles over a larger area [11], as well as the dynamic systems [12]. The fact that a high power microscope objective lens normally has a small field of view makes it conflictive to obtain the direct image of smaller particles/lattice structures over a larger area simultaneously. Although imaging a larger scale of assembly was previously possible by taking multiple pictures via translating the microscope objective lens, the processing and analyzing of multiple digital images becomes substantially challenging even for modern computers when the number of particles increases. Consequently, such approaches are not typically efficient when small particles are manipulated over a large area. Furthermore, the extensive computing duty makes it nearly impossible to perform in situ analysis of the macroscopic (statistic) characters of a dynamic system consisting large number of particles, which evolve at a time scale where Brownian motion dominants. Therefore, an alternative approach to image and analyze the statistic behavior of large amount of particles over a wider area is needed.

Motivated by X-ray crystallography, we report an alternative method to analyze large, optical-assembled arrays of particles based on the light diffracted by the assembly itself. As an analogy to the X-ray measurements in the crystallography, where the X-ray wavelength is in the same order of the crystalline lattice constant, the size of the trapped microscopic particles and their spacings, are comparable with the wavelength of the typical lasers. Hence the trapped particles diffract any probe beams incident on the assembly. The diffraction patterns result from the interferences of the diffracted light from all particles illuminated within which the source is coherent, which can be large for a laser. The relative intensities and the profile of the diffracted beams contain much valuable information, such as the lattice constant, crystal orientation, and (with further analysis) defect properties of the assembly. Furthermore, the light used to trap the particles is too diffracted by the particles simultaneously while the assembly is organized. Significantly different from the conventional microscope-based imaging technique, we do not observe the direct image of the individual manipulated particles. Instead, we observe the diffraction pattern associated with the assembly as a whole, in situ, without the need to translate an external microscope imaging setup, as required for real-space imaging. Statistical properties of the assembly can be derived from the diffraction pattern itself.

2. Experimental Instrumentation

In this work, we utilized the diffraction approach to study a dynamic optically-bound assembly of spherical colloidal particles. Our spheres were optically assembled and manipulated by two coherent laser beams, derived from a single source. A one-dimensional periodic optical potential was generated from the interference of the two beams. The spacing of the interference maxima is given by $d=\frac{\lambda }{2\sin (\theta ⁄2)}$, where d is the period, λ is the wavelength of the incident laser, and θ is the angle between the two beams. This interference provided strong gradients of the electric field along the direction of the standing wave. Weaker, two-dimensional gradients were generated simultaneously by slightly focusing the input beam, characterized by the typical Gaussian intensity profile. We applied this light field to dielectric colloids — here, polystyrene spheres suspended in water. The polarizability of the spheres implies a position-dependent energy within the optical lattice. The strong gradient force organized the spheres onto periodic lines and the weak gradient force in the normal direction prevented the particles from diffusing out along each line. Details concerning the various adaptation of such setup for the optical-assembly experiments are reported elsewhere [12, 13]. Our technique allows us to assemble large numbers of polystyrene spheres into two-dimensional line-like arrays. The particles trapped in this manner act as a planer grating albeit with various imperfections to be discussed. Since our sample was assembled as a single layer, we have two-dimensional diffraction, rather than the three dimensional Bragg diffraction encountered in crystallography. For any incident angle, the diffraction light is expected at an angle given by: $\sin {\theta }_{\mathit{out}}-\sin {\theta }_{\mathit{in}}=m\frac{\lambda }{d}$, where m is the order of the diffraction, λ is the wavelength of the laser beam, d is the lattice constant of the crystalline particle assemblies, θ_in and θ_out are the incident angle and the diffracted angle, respectively. The diffraction patterns, as reported in our earlier work[14], were projected on a flat screen and then recorded by a video camera.

It is most convenient if one of the input beams, which are used for assembling the spheres, is taken to be the incident beam. This gives us a way to perform the measurement in situ and independent of the observation of microscopic particles. We note in passing that one could always filter out the beams used to trap the particles and illuminate the array with a third beam having a different wavelength which is incident, say, in a direction normal to the array.

 

Fig. 1. Polystyrene spheres assemblies. (a) an assembly trapped on a template of square lattices. (b) an assembly trapped on a template of hexagonal lattices.

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Fig. 2. Diffraction patterns of the polystyrene spheres assemblies. (a) the assembly trapped on a template of square lattices. (b) the assembly trapped on a template of hexagonal lattices.

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Examples of assemblies trapped on templates of a square lattice and a hexagonal lattice were studied in this work. The direct images, as those in the conventional microscope based imaging setup, were taken with a 50x microscope objective lens, as shown in the Figure 1. Although significant amount of point defect were observed in the colloidal lattices, as shown in the figure, it is clear that the over-all symmetry of the lattices were demonstrated in the corresponding diffraction patterns, which were taken at the far zone without microscope objective lenses, as shown in Fig. 2. The spacing between the nearest neighbors was read out directly from the images as 7.28 µm and 3.09 µm, for Fig 1(a) and (b) respectively. It is clear that the diffraction patterns represent nearly identical assemblies in reciprocal space. The lattice constant of the interference maxima calculated from the (imperfect) diffraction patterns using the 2D diffraction equation above were 7.13 µm and 3.33 µm, respectively, which were consistent with the spacing calculated from the direct images. Comparing the measurements from both approaches, we found an uncertainty of 2% between the diffraction measurements and the direct measurement for the square lattice and a larger uncertainty of 8% for the hexagonal lattice. The discrepancy arises from a “mapping distortion” of the diffraction pattern. In this work, the diffraction patterns were projected onto a flat screen in front of the camera. Since the various orders of the diffracted beams spaced uniformly in the angular space, rather than in a linear space, mapping the diffraction pattern onto a flat screen introduced a “pincushion” distortion to the pattern, where the edge of the image bent outward. The latter assembly whose lattice constant was smaller, shown in Fig 1(b), had a larger diffraction angle and hence experienced more distortion, as shown in Fig 2(b). The third order of the diffraction intensity maxima had a diffraction angle of approximately 29 degrees. It was natural to expect a large uncertainty in mapping such a big angle onto the flat screen, as shown in Fig 2(b), where the outlying spots were obviously distorted. A significant improvement could be obtained by adding a spherical projection screen concentric with the sample cell.

The significance of the diffractive approach in analyzing the optically-bound assemblies lies in its independence of tracking any individual particle. Rather than collecting images in real space, the diffraction patterns are the images in reciprocal space, where increasing the number of particles generally results in sharper diffraction patterns. Although the absolute diffraction intensity is dependant on the size of the particles, the relative intensity distribution between various orders of diffraction maxima in a multiple-particle system is not size-dependent, to the first order. Therefore analysis the diffraction patterns is an effective way to study small-particle arrays. It is the interferometric nature of the diffraction pattern that offers the possibility to obtain statistical characteristics of the whole assembly from very few orders of the diffraction pattern.

3. Diffractive Image of Dynamic Systems

To initiate the study of the dynamic particle assemblies, we applied this imaging technique on a lattice changing over a time scale where it is nearly impossible to track the location of all particles in situ with traditional microscope images. Two coherent laser beams were initially adjusted so as that the lattice constant was approximately equal to the diameter of the spheres. A square lattices was observed being optically-bounded in the interferometric optical field. One of the beams was then alternatively blocked and opened. The diffraction pattern of this time-evolving particle assembly was recorded, as shown in Fig. 4 (a)–(h) and Fig. 5. All diffraction images below were taken with short exposure at a negligible time scale relative to the particles’ movement.

In Fig. 4, a diffraction pattern of the square lattice was initially observed, as shown in the frame (a). One of our two beams was blocked at T=4.15 s, as shown in frame (b), and then unblocked, as shown in frame (g). The optical trenches exist only when both beams are present, and therefore no significant gradient force existed in the horizontal dimension from frame (b) to frame (g). On the other hand, a centripetal gradient force, due to the Gaussian intensity profile of the remaining beam, did exist. Our observation demonstrated that the organized lines of aligned polystyrene spheres relaxed under the Brownian motion. They were also trapped toward the center of the light intensity maximum by the radial gradient force of the remaining light beam. The lines were relaxed in less than 0.5 seconds from the ordered square lattice to an approximately random structure, as shown at T=4.67 seconds in (c). The concentric ring-like diffraction pattern, shown in frame (c), clearly demonstrated a random distribution of the spheres. It is noticeable that the diffraction intensity in each ring was not perfectly angularly symmetrical. It was because the number of the particles in our assembly is relative small. The dynamic assembly reported in this work consist approximately 70 micro-spheres. In contrast, we expect that the ideal diffraction ring from a random system to be observed with an assembly with larger number of particles. Frame (b), frame (d) and frame (g) demonstrate the transient states from the periodic structure to the random structure, from the random structure to the close-packed hexagonal structure, and from the close-packed structure back to the square structure, respectively. It is clearly seen that the diffraction pattern at the instants had both character of a random structure and an ordered structure. The lines were promptly reorganized after the second laser beam was reapplied at 15.87 s. Although the relaxation and re-organization times depend upon both laser power and the definition of the optical structures, we note from the frames shown below that the spheres moved from random positions into the light traps with a speed greater than 10 µms^-1.

 

Fig. 4. Diffraction patterns from a dynamic optically-bound assembly.

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We note again that the diffraction approach is not limited by the field of view of an objective lens. Instead, the only limit on the region being sampled is set by size of the laser spot, which in turn is set by the (independent) requirement that the intensity be sufficient to trap the array as a whole. The technique is therefore ideally suited to characterize large areas, rather than a local region. Although the assembly studied in this work is still within the size of field of view of a conventional microscope objective lens (for the convenience to compare both diffractive measurement and the direct measurement), our approach demonstrated the potential applications toward much larger assembly. More interesting is that the transverse size of the intensity maxima in the square pattern was larger than that of the hexagonal pattern, as shown in Fig 4 and Fig. 5. This resulted from the weaker optical bonding in the assembly trapped onto square lattices. The Brownian oscillation of the individual particles in respect to the otherwise square trapping lattices leaded to a “distortion” of the crystalline colloidal structure, and hence a broader diffraction peak. In contrast, the oscillation of individual lattices is prohibited in a close-packed assembly. Therefore, the Brownian motion of the close-packed assembly trapped, over the entire assembly, is much weaker than that acting on the individual particles in the square lattice. As shown in the Fig. 5, the Brownian motion of the whole assembly was observed as a random rotation of the orientation of the hexagonal assembly from T=5.87 s to 13.00 s. Further studies of such Brownian rotation and the analysis of the diffractive images for the statistic information of the dislocation of the particles (defects) are under investigation.

 

Fig. 5. (2.3MB) Diffractive approach to image a dynamic optically-bound assembly.

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4. Conclusion and Discussion

We have reported on an alternative approach to optically characterizing the optically-bound assemblies of colloidal polystyrene spheres and its utilization in the analysis of a dynamic colloidal assembly. This is a method to analyze optically bound assembly from its diffraction patterns at far zone without the need to scan through the assembly with high power microscope objective lenses. The technique represents a significant departure from the conventional microscope-based imaging techniques. Therefore, the diffraction approach is not limited by the small field of view from the objective lenses. More importantly, the diffraction patterns allow one to readily extract statistical properties of the whole assembly. Therefore, the time needed to track the individual particles is significantly reduced. It is the interferometric nature of the diffraction patterns that offers the fast analysis of a dynamic assembly. And hence it offers the possibility to observe and analyze dynamic optical assemblies of large numbers of particles in situ. The pincushion distortion was found in this work; an improvement of adding a spherical projection screen concentric to the sample cell is suggested.