Cooperative microlenses

We demonstrate that microspheres above a substrate act as microscopic lenses. Their Brownian motion causes the focal point to change abruptly, thereby creating characteristic intensity fluctuations which depend on the interaction between the spheres. To this end, superparamagnetic spheres in a magnetic field assemble into long pearl chains, where the intensity fluctuations depend on the stiffness of the chain. Upon assembling the superparamagnetic beads into a two-dimensional colloidal crystal, the fluctuations are restricted in two dimensions, and temporal network structures develop. © 2004 Optical Society of America OCIS codes:(080.3630) Lenses; (120.6810) Thermal effects; (350.3950) Micro-optics References and links 1. M. Born and E. Wolf,Principles of Optics(Cambridge University Press, UK, 1980). 2. D.W. Pohl, W. Denk and M. Lanz, “Optical stethoscopy: Image recording with resolution λ /20,” Appl. Phys. Lett. 44,651–653 (1984). 3. S.M. Mansfield and G.S. Kino, “Solid immersion microscope,” Appl. Phys. Lett. 57,2615–2616 (1990). 4. M. Oikawa, K. Iga and T. Sanada “Distribute-index planar microlens array prepared from deep electromigration,” Electronics Letters. 17,452–454 (1981). 5. M.H. Wu and G.M. Whitesides, “Fabrication of arrays of two-dimensional micropatterns using microspheres as lenses for projection photolithography,” Appl. Phys. Lett. 78,2273–2275 (2001). 6. N. Chronis, G.L. Liu, K.H. Jeong and L.P. Lee, “Tunable liquid-filled microlens array integrated with microfluidic network,” Optics Express 11,2370–2378 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-192370. 7. J. Rosen and D. Abookasis, “Seeing through biological tissue using the fly eye principle,” Optics Express 11, 3605–3611 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-26-3605. 8. M. Sasaki, T. Kurosawa and K. Hane, “Micro-objective manipulated with optical tweezers,” Appl. Phys. Lett. 70,785–787 (1996). 9. Z. Chen, A. Taflove, V. Backman, “Photonic nanojet enhancement of backscattering of light by nanoparticles: a potential novel visible-light ultramicroscopy technique,” Optics Express 12, 1214–1220 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-7-1214. 10. J.P. Brody and S.R. Quake, “A self-assembled microlensing rotational probe,” Appl. Phys. Lett. 74, 144–146 (1999). 11. H. Takei and N. Shimizu, “Gradient sensitive microscopic probes prepared by gold evaporation and chemisorption on latex spheres,” Langmuir 13,1865–1868 (1997). 12. J.N. Anker and R. Kopelman, “Magnetically modulated optical nanoprobes,” Appl. Phys. Lett. 82, 1102–1104 (2003). 13. J. Choi, Y. Zhao, D. Zhang, S. Chien and Y.H. Lo, “Patterned fluorescent particles as nanoprobes for the investigation of molecular interactions,” Nano. Lett. 3, 995–1000 (2003). 14. C.J. Behrend, J.N. Anker and R. Kopelman, “Brownian modulated optical nanoprobes,” Appl. Phys. Lett. 84, 154 (2004). 15. D.C. Prieve, “Measurments of colloidal forces with TIRM,” Adv. Colloid. Interfac. 82,93–125 (1999). 16. L.E. Helseth, T.M. Fischer and T.H. Johansen, “Paramagnetic beads surfing on domain walls,” Phys. Rev. E 67, 042401 (2003). (C) 2004 OSA 26 July 2004 / Vol. 12, No. 15 / OPTICS EXPRESS 3428 #4732 $15.00 US Received 1 July 2004; revised 13 July 2004; accepted 13 July 2004


Introduction
Optical microscopy is a versatile technique for investigating phenomena on the micrometer scale [1].The resolution of a conventional microscope is usually comparable to the wavelength of light, but modern approaches have been able to improve this by e.g.taking advantage of the near-field radiation [2,3].
Microlenses may expand the applicability of both optical microscopy and lithography, and have therefore attracted considerable attention in recent years [4,5].Tunable microlens arrays have been fabricated using for example soft lithography on top of microfluidic networks, and may therefore be useful in adaptive optics [6].Moreover, such arrays can also be used to image through diffuse media, which may advance the quality of image recovery systems [7].Sasaki et al. found that single microscopic spheres manipulated by optical tweezers can be used as objectives for imaging details a conventional microscope cannot observe [8].A recent theoretical study by Chen et al. suggested that the waist and focal depth of light focused by a microscopic dielectric cylinder is controlled by the incident wavelength and size of the cylinder [9].Interestingly, it was also suggested that the backscattered light from nanoparticles near the cylinder is enhanced by many orders of magnitude due to the strongly focused light.Thus, using microscopic particles may improve both imaging and light scattering systems.
Microspheres have also been useful for probing the local properties of materials.To this end, Brody and Quake showed that by attaching a small microsphere to a larger microsphere using the strong biotin-streptavidin coupling, one obtains an asymmetric system that is sensitive to alignment with the incident light [10].In this way they were able to probe the rotational diffusion constant in various viscous liquids.In 1996 Takei and Shimizu invented an interesting method for monitoring localized chemical bond formation using electric forces [11].First they evaporated a thin gold film on one side of a fluorescent latex sphere, and then coated the gold film with a self-assembled monolayer.The microsphere could be aligned in an electric field, which allowed them to probe specific binding to the underlying susbstrate using asymmetric fluorescent emission.More recently, Anker and Kopelman used this method to demonstrate that halfly coated fluorescent magnetic microspheres controlled by a modulated magnetic field may probe the local chemistry more efficiently [12].Later it was shown by Choi et al. and independently by Behrend et al. that such asymmetric microspheres exhibit characteristic intensity fluctuations due to rotation Brownian motion, which may be used to probe molecular interactions and local rheology [13,14].
The methods of Refs.[13,14] depend on the asymmetric scattering properties of halfly reflecting spheres.Here we take advantage of the lensing action of spheres performing Brownian motion close to an underlying substrate.Since the spheres have symmetric absorbtion properties, only linear Brownian motion perpendicular to the substrate is of importance.It is demonstrated that thermally excited microspheres show collective intensity fluctuations upon interacting with each other.In this way we are able to probe their collective dynamics as well as their interaction with the underlying substrate.We also believe that the approach presented here could help us understanding how to self-assemble one and two-dimensional structures of microlenses.

Fluctuating microlens
A glass ring with diameter about 1 cm was put on top of a transparent substrate (glass of refractive index 1.5 or garnet of refractive index ∼ 2), and spherical beads immersed in deionized, ultrapure water were confined within the walls of this ring.The paramagnetic beads used here had an iron content of about 15%, an effective susceptibility of about χ ≈ 0.17, a radius of a=1.4 µm (±5%) and were manufactured by Dynal (Dynabeads M270 coated with a negatively charged carboxylic acid group).The beads eventually approach the solid-water interface due to gravity, but do not stick.The glass-water interface is negatively charged due to formation of silanol groups, and therefore prevent the negatively charged beads from being irreversibly adsorbed.Due to electrostatic double layer repulsion the microlens is not in close contact with the glass slide, but instead levitates a small distance h above it, as can be seen in Fig. 1.The Fig. 1.A single bead focusing and collecting the light.microlens is on average located a distance h m above the glass slide as determined by competing electrostatic and gravitational forces [15] where l D is the Debye length (l D ≈0.9 µm in deionized water), B may be about 50 kT (k is Boltzmann's constant and T=300 K is the temperature) and G = 69 fN is the gravitional force on the bead.This gives h m ≈ 1.6 µm, i.e. comparable to the bead radius.The beads are illuminated from above using linearly polarized halogen light passing through a microscope objective, and the whole event was detected by a Hamamatsu CCD camera of temporal resolution of 1/30 s.We clearly observe that the beads exhibit transversal Brownian motion in the xy-plane (parallel to the substrate).The beads also undergo Brownian motion perpendicular to the glass slide (z-direction), but this is often undetectable due to the limited (C) 2004 OSA 26 July 2004 / Vol. 12, No. 15 / OPTICS EXPRESS 3430 focal depth of the microscope objective.However, we found that the brightness of the spheres fluctuates when they are observed in reflection mode with crossed polarizers, as can be seen in Fig. 2.Here the normalized intensity is shown over a period of about 3 s.The normalized intensity cannot be lower than about 0.15 due to the background scattering from the bead, which may be related to polarized backscattered light, spatially inhomogenities in the refractive index, or a rough bead surface (the roughness may typically be a few nanometers).Note also that the fluctuations are rather fast, on the order of the temporal resolution of the camera.
The intensity fluctuations cannot be due to backscattering from the upper surface of the sphere (i.e., the surface facing the camera), since the amount of backscattered light is largely independent of the height h.We believe that the reason for the fluctuations is that in reflection mode the light passing through the transparent bead is focused down onto the glass slide, reflected, and then propagated back through the lens to the CCD camera.The fluctuations were only visible under crossed polarizers, which suggests that the effect is weak (compared to the background) and is dependent on the state of polarization of the reflected light.It is well-known that high-aperture focusing systems change the state of polarization of the incident light, and here we detect this new polarization component by using crossed polarizers.In transmission no such intensity fluctuations were observed.To this end, it should be pointed out that the relative position of the excitation light and the camera plays a role, and one could therefore imagine other geometries were the transmission mode may be used to detect Brownian motion.
For beads of radius smaller than 2 µm the light absorption is negligible, and they therefore act as nearly transparent microlenses focusing the light onto the glass slide.To this end, consider a single microscopic ball lens of radius a and refractive index n b is located in water of refractive index n w = 1.3.The paraxial focal length of the lens (for small n b − n w ) can be estimated to be For a lens with the paraxial focal point in water we must require n b ≤ 2n w .If this criterion is not fulfilled, the focal point will be located within the lens.Here we use polystyrene beads with a refractive index of n b ≈ 1.6 (at a wavelength λ = 550 nm), thus resulting in f ≈ 2.5a.
Accounting for the finite size of the microlens our rough estimate predicts the focal point to be located a distance ∼1 µm away from its surface, which is comparable to the average height h m estimated above.
In classical optics the transversal resolution ∆r is usually given as [1] ∆r and the expected resolution obtainable in the current system should therefore be of the order λ .On the other hand, the focal depth of a microspherical lens is approximately given by [1] ∆z ∼ λ 2 i.e. ∆z ∼ 2λ .The light reflected by the substrate is reflected back through the bead, and finally emerge at the camera.Using a sensitive intensity detector or camera one should be able to resolve a fraction of the focal depth of the microlenses, i.e. detect microlens motion of the order 0.1λ or smaller.It should be clear that these are only hand-waving arguments neglecting aberrations, polarization of the light as well as the detailed wavepropagation through the lens.
The polarization of the light changes upon propagating through the microlens, which allows a fraction of the reflected ligh to be detected even in presence of crossed polarizers.It should also be mentioned that the intensity fluctuations were most pronounced when using objectives with sufficiently small numerical aperture (NA=0.5 or smaller), since otherwise the focal point may appear inside the microlens.Since we use a pair of nearly crossed polarizers, the beads appear as bright spots with a thin dark cross due to geometric depolarization.Numerical simulations like those of Ref. [9] must be performed in order to understand the system better, but this is outside the scope of the current study.Nonetheless the simple arguments above give us a feeling of the focusing properties of such microscopic lenses, and suggest that our method may be useful for probing small thermal fluctuations.It should be pointed out that Total Internal Reflection Microscopy (TIRM) is a very sensitive method which is able to detect Brownian motion with a resolution of better than 1 nm [15].However, a significant advantage of our method is its simplicity and the ability to use the whole field of view of the microscope objective, which allow us to follow the intensity fluctuations of a large number of microlenses simultaneously.

One-dimensional microlens assembly
Since the microlenses are paramagnetic, they tend to align their magnetic moment with an external magnetic field.In order to minimize the magnetic energy, the beads also try to form  long chains in the direction of the external field.We here took advantage of this fact by using a small electromagnet to generate a field parallel to the underlying substrate.Figure 3 shows a several chains of beads in the presence of an external magnetic field, H = 500 A/m.It is seen that each bead in the chain has a different brightness due to its individual height h above the glass slide.
Figure 4 shows the intensity fluctuations of three different beads in a chain when H = 500 A/m.We note two interesting features here.The fluctuations of the two nearest neighbours (dashed and dash-dotted lines) are correlated, whereas the intensity fluctuations of a bead located about 6a away (solid line) from the nearest of the two beads are not correlated.Thus, the chain is sufficiently loose so that only nearest neighbours show correlated motion.We also observe that the intensity fluctuations are smaller for beads in a chain than for a single isolated bead, since each bead is now inhibited by its neighbours.We would like to point out that the beads do not assemble into chains if the magnetic field is below 80 A/m (1 Oe), and the chains disassemble when we remove the magnetic field.
Upon increasing the magnetic field most of the beads are found to have the same brightness and the chain becomes stiffer, as can be seen in Fig. 5.Here the magnetic field is approxi- mately H = 3000 A/m, and the normalized intensity varies only within 10%.Thus, simply by increasing the magnetic field we may increase the chain stiffness and decrease the intensity fluctuations.

Two-dimensional microlens assembly
A two-dimensional assembly of microlenses will change the correlated motion between the beads, since each bead now is allowed to interact with more than two neighbours (i.e., typically six nearest neighbours).However, for this to happen, we must bring the beads together in a twodimensional crystal in order for them to interact with each other.In a recent paper we demonstrated that Bloch walls in magnetic films (garnet films of composition Lu 3−x Bi x Fe 5−x Ga x O 12 ) can be used to trap or repel paramagnetic colloidal particles, sensitively depending on the external magnetic field [16].A Bloch wall is in principle a microscopic bar magnet inside a homogenous magnetic film which gives rise to a localized magnetic field.Figure 6 shows our setup, were a long one-dimensional domain wall of width w ≈ 50 nm and magnetization vector in the z-direction is located along the y-axis (x=z=0).We deposited beads immersed in deionized water on the top garnet surface.The beads experience an attractive force drawing them toward the nanomagnet (the direction of the force is denoted by arrows attached to the beads).Interestingly, we found that the intensity fluctuations decrease when the beads are approaching the domain wall, as can be seen in Fig. 7.It is seen that the normalized intensity is fluctuating randomly when the bead far away from the domain wall, whereas at x = 0 the fluctuations are typically 20% or less.This is explained by the magnetic force in the z-direction trying to drag the bead towards the garnet surface.As the height h decreases, the focal point moves away from the reflecting garnet surface, and the amount of light passing back into the detection system decreases.This observation therefore provides additional evidence for our assumption about the lensing effect of the microbeads.
In a high density solution of beads (10 8 beads/ml) many of them are attracted toward the domain wall.In fact, a few minutes after deposition a two-dimensional crystal of size 0.1 mm 2 was formed, see Fig. 8.The size of the crystal formed is limited by the finite range of the attractive force generated by the domain wall, but we found that the crystal can be made larger if the attraction process is assisted by liquid flow (of the surrounding water).However, this will be reported elsewhere.Figure 8 and the associated video shows the crystal as observed in reflection mode.It is seen that the beads interact along the accessible crystallographic directions, thus creating network structures with dark and bright areas.These correlated structures are created and anihilated rather quickly, and usually involve clusters ranging from two to six beads.In Fig. 9 (a) we show a part of the crystal as observed in transmission mode (no intensity fluctuations), whereas (b) is the corresponding Fourier Transform.We note that the crystal is highly regular  and that the Fourier Transform has distinct peaks as one could expect for a hexagonal lattice.Figure 9 (c) shows an image of the same area as in (a), but now taken in reflection mode.Note that we have inverted the gray levels so that black becomes white and vice versa.The crystal may therefore appear to have several vacancies.However, this is not case, and the bright areas only represent darker beads.Taking the Fourier Transform of Fig. 9 (c) results in Fig. 9 (d), where it is seen that the peaks are smeared out in basically all crystallographic directions.This suggests that there is no preferred direction for the correlated fluctuations.

Conclusion
We have demonstrated self-assembly of microscopic paramagnetic beads which exhibit lensing action when exposed to light.Due to Brownian motion, this lensing action leads to characteristic intensity fluctuations, which depends on the detailed interaction between the bead and its neighbours.We believe that the intensity fluctuations can help us not only understanding the magnetic and nonmagnetic interaction between beads, but could also assist in the study of other interfacial phenomena.Finally, it gives us an unique opportunity to study how microscopic lenses assemble on a substrate.

Fig. 2 .
Fig. 2. Intensity fluctuations of a single bead diffusing on top of a garnet surface.

Fig. 3 .
Fig. 3. (AVI 199 KB) Blinking chain above a glass slide in presence of a magnetic field of about 500 A/m (6 Oe).

Fig. 4 .
Fig. 4. Intensity fluctuations of three beads in a chain consisting of about 30 beads.

Fig. 5 .
Fig. 5. Chain above a glass slide in presence of a magnetic field of about 3000 A/m (38 Oe).

Fig. 6 .
Fig. 6.Schematic drawing of the beads moving toward the domain wall.

Fig. 7 .
Fig. 7.The intensity increases as the bead is getting closer to the domain wall.