We demonstrate an optical tweezers using a laser beam on which is imprinted a focusing phase profile generated by a Devil’s staircase fractal structure (Cantor set). We show that a beam shaped in this way is capable of stably trapping a variety of micron- and submicron-sized particles and calibrate the optical trap as a function of the control parameters of the fractal structure, and explain the observed variation as arising from radiation pressure exerted by unfocused parts of the beam in the region of the optical trap. Experimental results are complemented by calculation of the structure of the focus in the regime of high numerical aperture.
© 2015 Optical Society of America
Optical tweezers  are well-established as a powerful and versatile tool for the study of matter on the micron and sub-micron scale [2, 3, 4]. In the most common implementation an optical tweezers uses a strongly focused gaussian beam as a three-dimensional trap which can hold, move, guide and exert a precisely calibrated force on a wide range of micro- and nanoscopic objects ranging from single biomolecules , through nanomaterials such as carbon nanotubes  and graphene flakes  to cells . More advanced trapping schemes can be realised by controlling the phase  or polarization  structure of the trapping beam to enable, for example, trapping of low-index particles in the core of a phase vortex beam , enhanced trapping of anisotropic particles with a predominantly longitudinally polarised beam , or control of the trap geometry for spherical particles by shaping the focal volume .
An exact fractal  is defined as “an object which appears self-similar under varying degrees of magnification, in effect, possessing symmetry across scale, with each small part replicating the structure of the whole” . Several structures exhibiting such self-similarity over length scales have been considered for use as optical elements. Saavedra et al  analysed the focusing properties of a fractal zone plate (FZP), finding that the axial irradiance also exhibited self-similar properties. Such a FZP was realised experimentally by Davis et al  using a liquid crystal display, and further modified by Tao et al  with the inclusion of a spiral phase which thereby produced a sequence of focused optical vortices. High efficiency fractal zone plates have also been fabricated by femtosecond laser two-photon photopolymerization . Fractal structures have also been used to design the phase profile of conical  and spherical lenses  and spherical spiral phase lenses , which again can be realised experimentally using a liquid crystal spatial light modulator . The application of fractal structures to the manipulation of light has even been extended to plasmonics, where a two-dimensional “Sierpinski nanocarpet” structure was shown to produce sub-diffraction limit confinement of the optical field .
In this paper we report on the experimental demonstration of a fractal structure lens, the so-called “Devil’s Lens”, and its use in optical tweezers. While the use of a Devil’s Lens for controlling the structure of optical traps has been suggested previously  it is not straightforward that a stable trap will be formed in this situation, since radiation pressure from rays originating in different zones of the lens may act to push particles away from the intended trap location. We first calculate the field distribution of the beam formed by such a lens in the focal region of a high numerical aperture microscope objective, showing how the structure of the focus is determined by the parameters of the fractal generating function. We then implement a Devil’s Lens using a spatial light modulator in an optical tweezers set-up, and show that the Devil’s Lens focus can indeed be used to stably trap a variety of microscopic particles. We characterise the strength of the Devil’s Lens trap by determining the trap spring constants, and measure these for several different values of the fractal generating function parameters.
A Devil’s Lens is one whose phase profile is derived from a devil’s staircase function. The exemplar devil’s staircase is the Cantor function FS(x) which can be generated from a Cantor set  as illustrated in Fig. 1. At the first stage (S = 0) the initiator, namely a straight line of unit length is defined. The generator of the set is then applied, namely the line is divided into m equal parts and some of them removed (stage S = 1), illustrated in Fig. 1(a) for the triadic Cantor set where m = 3. The process is repeated to leave 2S line segments of length 3−S remaining, shown in Fig. 1(a) up to the stage S = 3. The gaps between segments are specified by the start and end-points (pS,l, qS,l), with l = 1 ... 2S − 1. The resulting devil’s staircase Cantor function FS(x) increases linearly in the regions where line segments remain, and is constant in the gaps as plotted in Fig. 1(b) for S = 3, so is specified by:Fig. 1(c). With the addition of a phase which increases with azimuthal angle, ϕ such that the net phase retardation is Φ = ΦS + mϕ, where m is an integer known as the topological charge, the resulting element is known as a Devil’s vortex lens. Examples of the calculated two-dimensional phase retardation patterns used in the subsequent experiments are shown in Fig. 1(d) Fig. 1(e) for Devil’s lenses with S = 2 and S = 3 respectively, and in Fig. 1(f) for a Devil’s vortex lens with S = 3 and m = 3. We note here that the measure of the Cantor set, i.e. the length of the remaining line segments at each stage of generation, tends to zero as S → ∞, and so in this limit the beam will not be modulated by a Devil’s lens function.
The field distribution around the focus of a high numerical aperture objective lens is calculated following the method of Richards and Wolf . Calculations are performed using a wavelength of λ = 1.064 μm (Nd:YAG laser wavelength) and a numerical aperture of NA = 1.48 for a beam initially polarised in the x–direction. Example results are shown in Fig. 2 for focusing by a Devil’s Lens for increasing stage S = 0...3, and Fig. 3 for a Devil’s Vortex Lens with m = 1 and increasing stage S = 0...3. In both cases for S > 0 multiple foci can be seen to along the optic axis. The existence of multiple foci can be understood as arising from rays originating from different radius intervals in the Devil’s lens where the phase profile has different curvatures (focal lengths) are brought to a focus at different distances from the Devil’s Lens.
At the stage of generation S = 0, where the beam profile is unmodified by the Devil’s lens function the principal focus is close to z = 0, as shown in Figs. 2(a) and 3(a). At the stage S = 1 a large fraction of the beam area is modulated by the Devil’s lens which produces a focus at an axially shifted location, seen in Figs. 2(b) and 3(b). Due to the decreasing measure of the Cantor set at increasing stages of generation for S > 1 an increasing area of the beam is not modulated by the Devil’s lens, producing a shift in the focus back towards z = 0, as seen in Figs. 2(b)–(c) and 3(b)–(c). We note here, however, that the location of the principal focus is slightly shifted from z = 0 by aberration introduced by focusing through a planar interface which is included in the calculation to reproduce the effect of the microscope cover slip.
While the extent of the principal focus appears largely unchanged by increasing Devil’s Lens stage S, it is not necessarily obvious that a stable optical tweezers will still be formed. Stable three-dimensional trapping requires that the backward-directed gradient force be sufficient to overcome the forward directed scattering force (radiation pressure) to confine a particle at (or near) the beam waist. This condition is usually met by strong (high numerical aperture) focusing of the beam. In the case of Devil’s lens focusing however, near the principal focus, which is formed by rays emerging from a particular annular region of the Devil’s Lens and where a trap may be expected to occur, a particle is also subject to additional scattering forces from rays that emerge from other annuli and focused to a different axial position. These additional forward-directed forces may be high enough to destabilise the optical trapping potential of the principal focus leading to an inability to form an optical tweezers with these beams. Our experiments, therefore, set out to determine whether a stable optical tweezers could be formed using a Devil’s Lens or Devil’s Vortex Lens.
Our experimental apparatus is described in detail elsewhere , but briefly consists of an inverted fluorescence microscope equipped with a ×100, NA = 1.3 oil immersion objective lens. The trapping laser beam is derived from a single mode Nd:YAG laser with maximum output power of 3 W. The Devil’s lens is implemented by displaying the appropriate phase retardation pattern, such as those shown in Fig. 1(d) – Fig. 1(f) on a spatial light modulator, which is then expanded and relayed into the microscope to form the optical tweezers. A 5 mW helium-neon laser beam is used as a probe for particle tracking in the optical tweezers and trap calibration. This is combined with the trapping beam at a dichroic mirror just below the objective which also separates the probe laser light that is backscattered from the trapped particle, and directs it toward a quadrant photodiode (QPD). Scattered light from an additional Gaussian probe laser beam is used for particle tracking rather than the trapping beam to avoid difficulties in interpreting the relationship between the scattered light distribution and particle position when using a phase and amplitude structured beam. A diagram of the complete optical set-up is shown in Fig. 4.
We have tested the ability of the Devils lens optical tweezers to confine micron- and submicron-sized particles with a number of different samples. Figure 5 shows images from a number of experiment that clear demonstrate stable trapping for a variety of sample materials which are suspended in a water-filled sample cell. These include silica microspheres with diameters 1.5 μm and 1.0 μm, and an aggregate of gold nanorods (individual nanorods are 10 nm ×45 nm)  shown in Figs. 5(a) – 5(c) respectively in an optical tweezers made using a Devil’s lens with S = 2. Furthermore, stable trapping in a Devil’s vortex lens optical tweezers with S = 2, m = 1 is shown to be possible as illustrated in Fig. 5(d) for the aggregate of gold nanorods.
Calibration experiments on the Devil’s lens optical tweezers are performed by particle tracking using the backscattered light from the probe beam as detected on the QPD. For these experiments only 1.5 μm diameter silica microspheres have been used. Single particles are trapped in the optical tweezers and steered to a location distant from the fixed boundary of the microscope cover slip and any other particles to minimise the effects of hydrodynamic coupling. The signals from the segments of the QPD are combined via analog electronics to produce tracking signals that are proportional to the trapped particle’s displacements from equilibrium in x, y and z, and these signals are sampled at 50 kHz by a data acquisition board. The autocorrelations of the position tracking signals are calculated, and are found to be well-fitted by an exponential decay, as is characteristic of a particle in an overdamped harmonic potential performing Brownian motion. We extract the spring constant of the optical potential in each direction, κi, (i = x, y, z), from the time constant of the corresponding autocorrelation signal.
Results of the measurement of spring constant are shown in Fig. 6(a) and Fig. 6(b) for the transverse (the average of κx and κy) and longitudinal (κz) directions respectively. In both cases the spring constant for a Devil’s lens (m = 0) and Devil’s vortex lens (with m = 1) were measured as a function of the Cantor set stage S from S = 0...6, and normalised by the optical power. In both cases a similar behavior with the order of the Cantor set, S, is observed, namely that the stiffness of the optical trap (in both directions) slightly increases with increasing S in the case of the Devil’s lens (m = 0), and decreases with increasing S in the case of the Devil’s vortex lens (m = 1). One possible interpretation of these results is the destabilising effects of radiation pressure from the parts of the beam that arise from different focusing annuli in the Devil’s lens and do not contribute to the principal focus, but to the secondary foci. For lower orders of the Cantor set, as seen from Fig. 2 and Fig. 3 these foci are closer to the principal focus, thus the radiation pressure from the unfocused rays is greater, pushing the particle from the region where the optical trap is stiffest. For the Devil’s vortex lens the additional amplitude variation of the Laguerre-Gaussian beam profile plays a role, as the lack of on-axis rays in the dark core of the beam mitigates the destabilising radiation pressure. We also note that the decreasing measure of the Cantor set with increasing S decreases the area of the annuli that contribute to the Devil’s lens focusing and also changes the radii at which the annuli are located, thus contributing an additional apodization of the beam in the radial direction.
We have calculated the focusing of a beam on which is imprinted a lensing function governed by a devil’s staircase fractal structure (Cantor set) in the limit of high numerical aperture. In the calculated focal volume we observe a series of subsidiary foci along the optical axis, although we note that this in itself is not sufficient to ensure stable optical trapping. We have further demonstrated that an optical tweezers that is formed by a beam focused by a fractal structure (Devil’s lens) can be used to stably trap micron-sized particles. We have investigated the dependence of the optical trap strength on the parameters of the fractal lens (order of Cantor set and topological charge), and suggest that the observed variation in trap stiffness may be due to the action of unfocused rays emanating from different annular regions of the fractal lens. Exact calculation of optical forces using a full electromagnetic scattering (T-matrix) theory  would in future permit verification of this suggestion. Having successfully demonstrated that the Devil’s Lens Optical Tweezers can stably trap micron and sub-micron sized particles this suggests that it may find application the simultaneous manipulation of particles at different aial positions in the subsidiary foci, where the self-similarity of the foci  may prove advantageous.
This research is supported by the National Natural Science Foundations of China (NSFC) under grant numbers 61178015 and 11304104.
References and links
2. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). URL http://scitation.aip.org/content/aip/journal/rsi/75/9/10.1063/1.1785844. [CrossRef]
4. O. M. Maragó, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013). URL http://www.nature.com/nnano/journal/v8/n11/full/nnano.2013.208.html. [CrossRef] [PubMed]
5. F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011). URL http://www.nature.com/nphoton/journal/v5/n6/full/nphoton.2011.100.html. [CrossRef] [PubMed]
6. O. M. Maragó, P. G. Gucciardi, P. H. Jones, R. Saija, F. Borghese, P. Denti, and M. A. Iatì, “Optical trapping of carbon nanotubes,” Physica E: Low-dimensional Systems and Nanostructures 40(7), 2347–2351 (2008). URL http://www.sciencedirect.com/science/article/pii/S1386947707007308. [CrossRef]
7. O. M. Maragó, F. Bonaccorso, R. Saija, G. Privitera, P. G. Gucciardi, M. A. Iatì, G. Calogero, P. H. Jones, F. Borghese, P. Denti, V. Nicolosi, and A. C. Ferrari, “Brownian motion of graphene,” ACS Nano 4(12), 7515–7523 (2010). [CrossRef] [PubMed]
8. H. Zhang and K. Liu, “Optical tweezers for single cells,” J. R. Soc. Interface 5(24), 671–690 (2008). URL http://rsif.royalsocietypublishing.org/content/5/24/671.full. [CrossRef] [PubMed]
12. M. G. Donato, S. Vasi, R. Sayed, P. H. Jones, F. Bonaccorso, A. C. Ferrari, P. G. Gucciardi, and O. M. Maragó, “Optical trapping of nanotubes with cylindrical vector beams,” Opt. Lett. 37(16), 3381–3383 (2012). [CrossRef]
13. S. E. Skelton, M. Sergides, R. Saija, M. A. Iatì, O. M. Maragó, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38(1), 28–30 (2013). [CrossRef] [PubMed]
14. B. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman, 1982).
15. M. Segev, M. Soljaic, and J. M. Dudley, “Fractal optics and beyond,” Nat. Photonics 6, 201–209 (2012). URL http://www.nature.com/nphoton/journal/v6/n4/full/nphoton.2012.71.html. [CrossRef]
17. J. A. Davis, L. Ramirez, J. A. R. Martín-Romo, T. Alieva, and M. Calvo, “Focusing properties of fractal zone plates: experimental implementation with a liquid-crystal display,” Opt. Lett. 29(12), 1321–1323 (2004). [CrossRef] [PubMed]
18. S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Sequence of focused optical vortices generated by a spiral fractal zone plate,” Appl. Phys. Lett. 89(3), 031105 (2006). URL http://link.aip.org/link/?APL/89/031105/1. [CrossRef]
20. J. A. Monsoriu, C. J. Zapata-Rodrguez, and W. D. Furlan, “Fractal axicons,” Opt. Commun. 263(1), 1–5 (2006). URL http://www.sciencedirect.com/science/article/pii/S0030401806000484. [CrossRef]
23. M. Mitry, D. C. Doughty, J. L. Chaloupka, and M. E. Anderson, “Experimental realization of the devil’s vortex Fresnel lens with a programmable spatial light modulator,” Appl. Opt. 51(18), 4103–4108 (2012). [CrossRef] [PubMed]
24. G. Volpe, G. Volpe, and R. Quidant, “Fractal plasmonics: subdiffraction focusing and broadband spectral response by a Sierpinski nanocarpet,” Opt. Express 19(4), 3612–3618 (2011). [CrossRef] [PubMed]
25. W. D. Furlan, F. Gimenez, A. Calatayud, L. Remon, and J. A. Monsoriu, “Volumetric multiple optical traps produced by Devil’s lenses,” J. Eur. Opt. Soc. Rap. Pub. 5, 10037s (2010). URL https://www.jeos.org/index.php/jeos_rp/article/view/10037s. [CrossRef]
26. G. Cantor, “De la puissance des ansembles parfaits de points,” Acta Math. 4, 381–392 (1884). [CrossRef]
27. B. Richards and E. Wolf, “Electromagnetic Diffraction in Optical Systems. II. Structure of the Image Field in an Aplanatic System,” Proc. Royal Soc. London A. 253(1274), 358–379 (1959). URL http://rspa.royalsocietypublishing.org/content/253/1274/358. [CrossRef]
28. P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragó, “Rotation Detection in Light-Driven Nanorotors,” ACS Nano 3(10), 3077–3084 (2009). [CrossRef] [PubMed]