Optical trapping of director structures and defects in liquid crystals using laser tweezers

We demonstrate optical manipulation of structures and defects in liquid crystals (LCs). The effective refractive index depends on the LC molecular orientations and the laser beam’s polarization. We use the orientation-mediated refractive index contrast for the laser trapping in LCs with a homogeneous composition, but with spatially-varying patterns of molecular orientations. Tightly-focused polarized beams allow for optical trapping of disclinations and their clusters, dislocations and oily streaks, cholesteric fingers and focal conic domains, etc. We calculate the optical gradient forces for typical structures and explain the trapping properties at low laser powers. We also show that when a high-power beam causes local molecular realignment, the laser trapping properties change for two reasons: (1) the refractive index pattern and optical gradient forces are modified; (2) additional elastic structural forces arise to minimize the elastic free energy. ©2007 Optical Society of America OCIS codes: (160.1190) Anisotropic optical materials; (140.7010) Laser Trapping; (160.3710) Liquid crystals; (180.6900) Three-dimensional microscopy; (260-1440) Birefringence. References and links 1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288 (1986). 2. P. N. Prasad, Introduction to Biophotonics (Wiley, New York, 2003). 3. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787 (2004). 4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810 (2003). 5. Y. Roichman and D. G. 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Introduction
Optical trapping with tightly focused laser beams has been broadly applied to objects as diverse as biological molecules, colloidal particles, and living cells [1][2][3].The manipulation of multiple objects has been demonstrated by the use of holographic tweezers and time-shared trapping [2][3][4][5].These optical trapping approaches allow for the measurement of weak pico-Newton forces and open new applications [2,3].Optical trapping is often described in terms of the gradient force, which pulls the object in the direction of increasing intensity of the focused beam, and the scattering force derived from photon backscattering, that pushes the object along the beam axis.A focused Gaussian beam has light intensity gradients both along the beam axis and perpendicular to it.The resulting optical gradient forces pul the high-index objects towards the focus with the highest intensity, and a stable trapping is achieved when they overcome the scattering and gravity forces.Usually, these objects (such as particles or biological cells) have a material composition and, thus, refractive indices different from that of the surrounding medium, as needed for noncontact manipulation [3].
In this article, we demonstrate laser manipulation of director structures in LCs when no foreign inclusions are present.The studied samples have homogeneous composition, but spatially-varying structures of molecular orientations.Since the LC's effective refractive index depends on the LC director orientation and on the light polarization state, the spatiallylocalized director structures usually have an effective refractive index different from that of the surrounding LC.This refractive index contrast allows the structures to be manipulated using polarization-controlled optical gradient forces that can be varied from attractive to repulsive.In most of the experiments, the used laser powers are below the threshold for optical Freedericksz transition, when the laser-induced realignment can be neglected.For this regime, extending the approach of Tlusty et al. [31], we calculate the optical gradient forces for typical director structures and find a fair agreement with the experiments.Optical trapping properties at high laser powers are modified by the laser-induced changes of the refractive index pattern and, thus, the modified optical gradient forces.In addition, elastic interactions between the laser-induced director distortion and the structure give rise to elastic forces comparable with the optical forces.We also show that polarized high-power beams allow one to exert torsion on LC disclinations.Optical manipulation of the director structures is important for fundamental LC research and for applications in displays, telecommunications, spatial light modulators, tunable photonic crystals, and artificial negative-index materials.
LC cells were assembled from plates coated with either polyimide PI2555 (HD MicroSystems) alignment layers buffed to set the uniform in-plane far-field director 0 N , or with the polyimide JALS-204 (JSR, Japan) for homeotropic anchoring (with 0 N perpendicular to the substrates).The sample thickness m μ ) 50 5 ( − was set using strips of mylar films placed along cell edges.After the cell was sealed using a UV-curable glue, the cell gap thickness was measured using the interference method [33,34].The samples were prepared using either room-temperature smectic LC 8CB (octylcyanobiphenyl, obtained from Aldrich) or a nematic LC AMLC-0010 (obtained from AlphaMicron Inc.).Refractive indices of AMLC-0010 are the following: ordinary index .To obtain the cholesteric LC, AMLC-0010 was doped with a chiral additive ZLI-811 (obtained from EM Industries) with the helical twisting power of 10.47 1 − m μ [34].By adding different amounts of the chiral agent (up to 10 wt.%), the cholesteric pitch was varied within m μ ) 20 1 ( − .For the FCPM studies, the LCs were doped with ~ 0.01 % of a fluorescent dye N,N'-Bis(2,5-di-tert-butylphenyl)-3,4,9,10perylenedicarboximide (BTBP).Using a 60× objective (NA=1.4) and the confocal microscope working in the FCPM mode [35], BTBP was excited by an Ar-laser at 488nm and fluorescence was detected within the spectral range 510-550nm.All used materials are transparent at the trapping wavelength nm 1064 , so that no significant laser-induced sample heating (>1 0 C) was observed.The LC cells were filled by the capillary forces in the isotropic phase (to avoid any flow influence on the LC alignment) and then cooled down to the room temperature.After the material is brought into the nematic, cholesteric, or smectic phase, the defects and structures spontaneously nucleate; in this work, these director structures are studied from the standpoint of their optical manipulation.

Optical microscopy of director structures
Molecular orientations in LCs form a wealth of complex three-dimensional (3-D) patterns of the director ) ( ˆr N , that depend on the treatment of confining surfaces, presence of inclusions and chiral agents, flow and temperature changes, etc. Figures 1-4 show the typical director structures, which we selected to explore optical trapping properties.The structure shown in Figs.1(a), 1(d) contains the domain with the uniform N ˆand the domain with 0 180 -twisted N ˆacross the cell, separated by a disclination [32].Cholesteric fingers in a homeotropic cell, Fig. 1(b), contain the local in-plane twist of N ˆ, which is matched to the homeotropic boundary conditions at the substrates by the two twist disclinations, Fig. 1(e) [34].In the axially-symmetric structure of the toric focal conic domains (TFCDs) in a smectic LC, Figs.1(c), 1(f), the smectic layers fold around two defect lines, a circle and a straight line passing through the circle's center [35,36].
The cholesteric circular domains (CDs) shown in Fig. 2 are observed in LC cells with homeotropic treatment of the confining substrates and the cell gap approximately equal to the cholesteric pitch p.The director structure of the CD is rather complex, but, for simplicity, can be represented as in Fig. 2(f), where N ˆ is orthogonal to the cell substrates in the center of the domain, and twists by 0 180 as one moves to the domain's perimeter in all radial directions.Thisπ -twist of the director matches overall untwisted homeotropic ( N ˆ perpendicular to the cell substrates) director field in the rest of the LC cell.The dislocation core is split into two λ -disclinations of half-integer strength and opposite signs [32,33].The director N ˆ in the center of the escaped λ -disclination is along the defect line, Fig. 3, making it non-singular [32].Another director structure in the lamellar cholesteric system shown in Fig. 4 contains a quadrupole of the non-singular half-integer λ -disclinations.

Laser trapping and manipulation with low-intensity beams
Linearly-polarized optical traps can manipulate all structures described above.As an example, we demonstrate laser-controlled transportation of cholesteric CDs along the computerprogrammed trajectories using time-shared dynamic laser traps [37], Fig. 2.These structures start to be attracted to the laser trap at distances less than m μ 3 from their perimeter.At the used power mW 8 per one trap, no laser-induced director distortions are observed, Fig. 2. Optical trapping depends on the structure's in-plane size, refractive index distribution, and the beam's polarization state.For example, the CDs much larger than the beam waist are trapped at the two equally-probable high-index regions, where the beam's linear polarization matches the in-plane director, Fig. 2(f); the CDs of radius m μ 2 and less are trapped at their center.The boundary between the untwisted and twisted domains in Figs.1(a), 1(d) can be displaced along the thickness gradient using a focused laser beam of power mW 10 ~.Noticeable interaction of the defect line with the trap starts at a separation distance less than m μ 3 ; the structure remains intact if the focused beam is farther away.When the laser beam is polarized along the rubbing direction, Figs.1(a), 1(d), the untwisted domain of the structure can be slowly shifted (at ) by the beam in the direction of increasing cell thickness.When the beam's linear polarization is orthogonal to the defect line (along the thickness gradient), one can shift the π -twisted domain into the direction of decreasing thickness of the wedge cell, Fig. 1(a), 1(d).In other words, the untwisted domain is attracted to the laser trap with polarization along the rubbing direction and repelled from the trap polarized along the thickness gradient; the vice versa polarization dependence of the trapping properties is observed for the twisted domain.The linear light polarization does not follow the director twist while the beam is traversing through the sample (cell thickness .They can be stretched or continuously shifted when the beam is displaced perpendicular to the finger.Moving the laser trap along the finger leaves the structure intact, unless the trap approaches one of the finger's ends.In the later case, the optical trapping forces cause the finger drift along its axis in the direction of the displacing laser trap.When the beam polarization is orthogonal to a finger, no trapping is observed at powers mW 15 < , as expected considering that there is no refractive index contrast in this case, Fig. 1(b), 1(e).Manipulation of the axially-symmetric TFCDs [Fig.1(c), 1(f)] in the matrix of planar smectic layers is direction insensitive.Regardless of the trap's polarization state, the TFCDs always have the effective refractive index higher than that of the surrounding smectic LC with N ˆorthogonal to the substrates.Optical trapping is efficient when the beam's focus is in the plane of the TFCD's circular line defect, because of the abrupt change of the director from horizontal (within the domain) to vertical (around it), corresponding to a large refractive index contrast, Fig. 1(f).The index contrast is weaker near the confining plates, Fig. 1(f); therefore, optical trapping is less effective if the laser is focused close to one of the substrates.
Optical manipulation of the director structures shown in Figs. 1 and 2 is two-dimensional, because these structures span across the entire cell gap.The situation is different for dislocations of Burgers vector p b = in the planar system of cholesteric layers, Fig. 3.For a beam with linear polarization along the disclination, the effective refractive index within the structure is close to e n and larger than that of the surrounding LC.This allows the dislocations to be optically trapped, Fig. 3.The structure is repelled from the beam if its polarization is orthogonal to the disclination, because the refractive index difference between the dislocation and the surrounding LC is negative in this case.Axial optical trapping forces are, in general, weaker than the trapping forces in the lateral plane, but sufficient for manipulation.For example, at powers mW 30 and larger, they allow one to produce dislocation kinks and anti-kinks [32] that mediate dislocation shift across the layers (along the microscope's optical axis), Fig. 3(h).Manipulation of the quadrupole of λ -disclinations, Fig. 4, also employs polarization dependence of the structure's effective refractive index, which can be varied from values smaller to higher than that of the LC around it.Both attractive and repulsive forces can be effectively used for optical manipulation.For example, the director structure in Fig. 4 is repelled from the linearly polarized laser trap with polarization perpendicular to it, but can be stretched and efficiently manipulated using two time-shared laser traps located at the opposite sides of the cluster.

Optical manipulation using high-intensity beams
Optical trapping with high-intensity focused laser beams is different from that described above.At high powers (usually mW ) 45 30 ( − ≥ , depending on the LC and cell thickness), a tightly-focused polarized laser beam [38,39] locally reorients the director and produces elastic distortions, causing the optical Freedericksz transition [11,12,27,29,[40][41][42][43].This alters the optical trapping properties.For example, at powers mW 40 < , the cluster shown in Figs. 4 and 5 is repelled from a focused beam with the linear polarization orthogonal to it.At higher powers, this beam locally reorients the director and exerts torsion on the structure, Fig. 5(a), 5(c), so that the disclinations in the cluster are locally parallel to the beam's polarization.The cluster with locally realigned director is then trapped by the beam and can be translated within the sample's plane.In a similar way, a cholesteric finger [Fig.1(b)] can not be manipulated by a low-intensity beam with polarization perpendicular to the finger, as described above.However, the LC director is locally reoriented and trapping becomes possible at high powers; the focused laser beam causes local reorientation of the finger at powers mW 100 > , similar to the case of the cluster shown in Fig. 5. Clearly, high-power laser beams allow the director structures to be not only spatially translated but even locally reoriented by changing the focused beam's polarization.

Refractive index contrast in liquid crystals with spatially-varying director
Polarized laser manipulation of the LC structures, Figs.1-4 interface), the effective refractive index in LCs usually changes on the scale of microns.This is because the abrupt changes of LC director would be costly in terms of the elastic free energy [17,18].The exceptions are the structures containing defects with a singular core (such as point defects, line defects, and defect walls), at which the LC domains with strongly misaligned N ˆ are joined, Fig. 1.These defects usually have the size of a core with undefined N ˆ in the nm ) 100 10 ( − range [17,18]; from an optical standpoint, this corresponds to instant spatial changes of N ˆ and of the effective refractive index.For example, the in-plane orientation of N ˆjumps by 0 90 at the disclination separating the uniform and twisted domains in the wedge cell [Fig.1(d

Optical gradient forces
Modeling of optical trapping of the director structures requires the knowledge of the 3-D refractive index patterns, which can be calculated only for known ) ( ˆr N .However, analytical expressions for of the localized structures are often not available.To get qualitative insights into the optical trapping properties of the director structures, we model them as dielectric objects with an effective refractive index DS n and shape mimicking that of the structure, Fig. 6, surrounded by the LC with an effective index SLC n .The simple models in Figs.6(a)-6(c) correspond to director structures such as those shown in Figs.1(d)-1(f).We calculate the optical gradient force exerted by a tightly focused beam on a director structure using the approach of Tlusty et al. [31].This approach was already successfully applied for complex anisotropic systems such as LC droplets and colloidal particles in LCs [14,26].The interaction energy between the laser trap and the structure reads: is the electric field of the tightly focused laser beam, 0 ε is vacuum permittivity, and integration is performed over an effective volume of the director structure DS V .In general, one would need to consider the actual electric field distribution in a tightly-focused laser beam [38,39] and account for aberrations due to focusing through dielectric interfaces, defocusing in the birefringent medium, as well as the change of the beam's polarization state while traversing the LC.However, the resulting beam intensity distribution is complex.Instead, to obtain a qualitative insight, we follow Ref. [31] and assume that the optical trap is formed by an axially-symmetric three-dimensional Gaussian beam.We then perform integration in Eq.( 1) and calculate the trapping force in the lateral plane perpendicular to the beam axis z ˆfor typical shapes of director structures, Fig. 6.
For the structure modeled as in Fig. 6(c) with (a cubic domain), the optical gradient force at distances r from the trap center reads: where c is the speed of light, ω is the size of the beam waist, W is the laser power, g is a coefficient of the order of unity that depends on the LC refractive indices, and ξ is the ratio between the focused beam size along z ˆ and perpendicular to it [31].Equations ( 2 6); (e) for the SD fragment using Eq.( 4) and sizes marked on the figure; (f) for the structure shown in (c) using Eq.( 2) and the marked sizes   , which is sufficient to alter the optical trapping properties.The laser-modified pattern of an effective refractive index can be controlled by changing the polarization and power of the trapping beam, allowing one to further tune the optical trapping properties.In addition, laser trapping is modified by elastic structural forces between the structure and the laser-induced distortions, tending to minimize the elastic free energy ( ˆr N comprises the structure and the laser-induced director distortions, 1 K , 2 K , and 3

K
are Frank elastic constants for splay, twist, and bend deformations of N ˆ [17], respectively, and integration is performed over the entire sample's volume.Since the LC elastic constants are typically pN 10 ~ [17,18], one can expect the elastic forces within pN ) 100 1 ( − , comparable to the optical gradient forces at high powers.Manipulation by high-power focused laser beams is, thus, mediated by a combination of both optical gradient forces arising due to the laser-modified refractive index patterns and the elastic structural forces. The opto-elastic trapping mediated by the director realignment is especially useful for manipulation of the structures with weak refractive index contrast spanning over large distances, which can not be effectively manipulated solely by optical forces at low laser powers.Moreover, the laser-induced realignment allows one to exert torsion and locally reorient disclinations and structures, Fig. 5 decreasing optical anisotropy and/or increasing the bend elastic constant 3 K [40].In planar cells (Figs.1(a), 3, 4, 5), th W tends to be larger because of the LC's effect on the beam polarization [34,41].th W depends on the cell thickness h and the beam waist ω , is larger for 1 / << h ω than for 1 / >> h ω [41,42], and can be within (10-300)mW, depending on the used trapping beams and LC cells.Finally, at very high powers (typically >500mW, depending on the LC's absorption at the trapping wavelength), the laser-induced heating may play an important role, even if the LC's absorption is low.In addition to the known effects in isotropic fluids [44], laser-induced heating in LCs may also change the order parameter, or even cause phase transitions, and, thus, can further complicate the interactions between the focused laser beam and the LC structures.

Conclusion
We have demonstrated optical trapping of director structures in liquid crystals.At low laser powers, this trapping is mediated by the optical gradient forces exerted by a focused beam on the structures with spatially-varying molecular orientations.The refractive index contrast in the polarized-state laser trapping originates solely from the patterns of molecular orientation.The calculated optical gradient forces reproduce experimental observations for typical structures.When a high-power beam causes director distortions, the trapping properties become qualitatively different for two reasons: (1) the laser-induced director reorientation modifies the refractive index pattern and, thus, the optical gradient forces; (2) the elastic structural forces arise to minimize the distortions in the elastic LC medium.Unlike in the case of particles, gravity forces are completely irrelevant even for trapping of large structures, because there are no density gradients; therefore, large structures (tens or hundreds of micrometers) can be manipulated at low powers of the order of 10mW.Optical gradient forces should be accounted for when imaging LCs using laser scanning probe techniques such as FCPM [35]: the laser power should be kept below 1mW in order to avoid pulling the highindex director structures into the focused beam of a confocal microscope.The described optical and opto-elastic trapping approaches are not restricted to LCs, and are applicable to all anisotropic materials.Potential applications include composite systems in which at least one of the components is anisotropic, such as polymer dispersed and polymer stabilized LCs.

Figs. 1 .
Figs. 1. (a).-1(c).Polarizing microscopy textures: (a) domains with a uniform (left) and the 0 180 -twisted across a cell director (right); (b) cholesteric fingers in a homeotropic cell; (c) toric focal conic domains in a smectic LC. (d-f) Director fields in vertical cross-sections of the structures: (d) for a wedge cell shown in (a) with a disclination line separating the two domains; (e) for a cholesteric finger of the CF3 type; (f) for the toric focal conic domain.The defect lines in (d-f) are marked by the red lines or filled circles.Note that four different types of cholesteric fingers are known [34], and (e) illustrates the simplest one.Crossed polarizers in (a-c) are along the vertical and horizontal image edges.

Fig. 2 .
Fig. 2. Optical manipulation of cholesteric CDs: (a, 2 MB) video sequence and (b-e) extracted images.Using a single beam that is time-shared between four traps, four CDs (marked by white arrows in (a)) are first positioned in vertices of a square and then are set to follow computerprogrammed concentric motion trajectories as shown in video frames (a-e) taken at about 1s time intervals.(f) The director structure of the CD in the plane of the LC cell.Textures (a-e) were obtained with the crossed polarizer and analyzer along the image edges.The black ellipses in (f) mark the two equally-possible trapping cites for large CDs when the trap's linear polarization is along the horizontal direction (marked by the black bar and "P").

Figure 3
Figure 3 shows a dislocation of Burgers vector p b = in the cholesteric layered structure, obtained in a cell with planar boundary conditions and thickness much larger than the pitch.

Fig. 3 .
Fig. 3. Stretching of an optically trapped dislocation in a lamellar LC by moving the sample using a stage: (a, 1.1 MB) video sequence and (b-g) extracted representative image frames.The directions of the sample displacement are shown by the black arrows in (a-g).(h) Layers profile in the vertical FCPM cross-section that was obtained (before the optical trapping experiments) in the plane orthogonal to the dislocation, as shown by the h-h line in (d).Schematics of the director field around the defect core shown by the red circle in (h).Linear polarization direction of the trapping laser beam is along the y-axis in (a-g).FCPM polarization direction in (h) is along the defect line.Polarizing microscopy textures (a-g) were obtained with the crossed polarizer and analyzer along the vertical/horizontal image edges.

Fig. 4 .
Fig. 4. Laser manipulation of a disclination quadrupole: (a, 1.9 MB) video sequence and (b-f) extracted frames.A single beam is time-shared between the two IR traps visualized in (a) by a HeNe laser beam co-localized with the trapping beam.The traps are slowly shifted as shown by white arrows in (a) and the disclinations are stretched (a-f) until they escape from the traps (f), because of their line tension.(g) Layers structure visualized by the FCPM for the vertical crosssection perpendicular to the cluster, as marked by the g-g line in (b).(h) The director field around the disclination cluster shown by the red circle in (g).Linear polarization of the laser beam used for manipulation is along the y-axis in (a-f).FCPM polarization direction is normal to the image and along the cluster in (g).Textures (a-f) were obtained with the crossed polarizer and analyzer along the image edges.
]. Therefore, the above observations are natural and correspond to attraction of the high-refractive-index LC domains to the laser trap and repulsion of the lowindex ones.Cholesteric fingers, Figs.1(b), 1(e), are attracted to the laser trap with a linear polarization parallel to their axes starting from distances

Fig. 5 .
Fig. 5. (a).An image showing that a linearly-polarized beam of power mW W 100 ≈ exerts torsion and locally reorients an initially straight disclination cluster shown in Figs. 4. (b), 4(c) Schematics of the molecular alignment in the central part of (b) the undistorted cluster [similar to that shown in Fig. 4(h)] and (c) with the local laser-induced director reorientation that results in the structure trapping at high powers, as shown in (a).Crossed polarizers are along the image edges in (a).
, utilizes the spatiallyinhomogeneous refractive index patterns.Depending on the local director N ˆ (optic axis), the effective refractive index eff n can vary from o n to e n , which results in the polarizationcontrolled refractive index contrast between the LC domains.Unlike in the case of colloidal particles suspended in liquids (where the refractive index changes abruptly at the particle-fluid #78218 -$15.00USD Received 21 December 2006; revised 25 February 2007; accepted 26 February 2007 (C) 2007 OSA )] and at the circular defect line of the TFCD [Fig.1(f)]separating the regions with N ˆ orthogonal (outside the domain) and parallel (within the TFCD) to the bounding plates.Respectively, the local effective refractive index at the defect lines can jump between o n and e n or can be varied within some fraction of optical anisotropy

.
) and(3) explain the attraction of the director structure to the trap for With increasing the center-to-center distances r between the beam and the structure, the force linearly increases at small r , is maximum when the beam is close to the structure'decreases at larger r .When the structure is much larger than the beam's waist, ω >> l , the gradient force is strong only at the structures' edges ( f), in agreement with the experimental observations.

Fig. 6 .
Fig. 6.Models of the LC structures and the calculated optical gradient forces.(a) A semiinfinite domain (SID) of thickness z l , extending from x=0 (thick lines in (a)) to infinity ∞

15 =
and indices marked in (d) in calculations for the SID.Indices 55 the calculations for the SD and the cubic domain in (e,f).Optical gradient forces acting on anisotropic structures [Figs.6(a), 6(b)] are directionsensitive.For the stripe-like domains (SDs) with dimensions l the beam's lateral plane are different when measured in the directions along (y) and perpendicular (x) to the elongated structures (Fig. $15.00 USD Received 21 December 2006; revised 25 February 2007; accepted 26 February 2007 (C) 2007 OSA 2 April 2007 / Vol. 15, No. 7 / OPTICS EXPRESS 4368regime).Therefore, the laser trapping of director structures can be very complex and hard to describe analytically, especially in the LCs of strong optical anisotropy and at large depths of optical trapping.

Fig. 7 .
Fig. 7. Optical gradient forces calculated for the cholesteric CD and for power mW W 15 = .The values of pitch p are marked for each curve.The trapping positions (zero force) are shown by the arrows.Note that large CDs can be trapped at a finite distance from their center.

4. 3
Optical manipulation assisted with the laser-induced director distortionsA laser beam of power exceeding the threshold th W locally distorts the director N ˆ [40-43] and changes the refractive index pattern, which then alters the optical gradient forces, Fig.5.The maximum laser-induced change of local effective refractive index associated with the director realignment is , which is not possible at small laser powers