## Abstract

We characterize a single beam supercontinuum “white light” trap and determine the trap stiffness in the transverse trapping plane. We realize a holographic white light trapping system using a spatial light modulator, and explore the generation of a dual beam trap and characterize its performance. We also demonstrate optical trapping and rotation of particles using a supercontinuum vortex beam. It is shown that orbital angular momentum can be transferred to spheres trapped in a supercontinuum vortex. Quantified rotation rates are demonstrated.

© 2008 Optical Society of America

## 1. Introduction

Optical tweezers are an extremely versatile tools that have found many applications since their development in 1986 by Ashkin et al. [1, 2]. They rely upon the gradient (dipole) force in a tightly focused beam to allow particles ranging from the nanometer to the micron scale to be held in three dimensions. An emergent theme has been the ability to trap and manipulate a plurality of objects using holographic techniques [3], rapid laser scanning [4] or the interference of trapping beams [5]. Optical traps are excellent force transducers and have found major application in the study of molecular motors [2]. Furthermore, they have enabled major investigations into the fundamental physics of colloidal science and non- zero order light fields. In particular, optical angular momentum, namely orbital and spin angular momentum, has been transferred from light fields to trapped particles [6–8]. This has allowed the intrinsic and extrinsic nature of the field to be explored [9,10]. Virtually all optical micromanipulation studies have used monochromatic light sources. However, two recent studies in this field have used white light for trapping [11], and extended guiding of particles [12]. Here, we use a white light supercontinuum source offering a spatially coherent beam of light with a very broad bandwidth, which results in a very short temporal coherence length. The use of a supercontinuum (SC) source in optical micromanipulation adds an extra dimension to the possibilities of the field; for example one may obviate any issues related to interference, enable new studies of the coherence properties of light, and perform simultaneous spectroscopy and trapping [13]. Whilst a basic white light trap has been generated [11], no studies of the trap characterization or imaging of particles in such a trap have been reported. Furthermore, to fully exploit the properties of the SC trap it would be useful to realize multiple traps or generate novel light modes, such as optical vortices using a SC source. These are features that have proved powerful in monochromatic optical trapping. Finally, all SC traps suffer from spatial dispersion issues which need to be addressed.

In this paper, we address and demonstrate all three of these concepts. We commence by giving a theoretical framework within which we can understand our data. We then describe and characterize a single beam SC optical tweezers. To do this we determine the transverse trap stiffness for 1µm polymer spheres. We also realize a high quality imaging system that permits direct particle observation during optical tweezing. We then progress to implement a holographic SC optical trapping system using our source. In particular, we explore issues for multiple optical traps and holographically generate a dual trap system using a dispersion compensated SC beam. We conclude by generating a Laguerre-Gaussian (LG) supercontinuum trap, and demonstrate the transfer of orbital angular momentum from the temporally incoherent (broadband) supercontinuum vortex to groups of micron-sized spheres.

## 2. Theoretical considerations

Modeling the motion of microparticles interacting with a supercontinuum beam can be conceptually separated into three parts. The first part concerns the propagation of the incident beams, Gaussian and more generally Laguerre-Gaussian beams. We treat these beams using the first order paraxial approximation that we correct through the scattering fields within the numerical Finite Elements Method (FEM). In the second part, we deal with the electromagnetic linear and angular momentum transfer between the supercontinuum beam and the microparticles in its path. This momentum transfer is evaluated using the conservation relation involving Maxwell’s stress tensor and its associated angular momentum tensor.

#### 2.1 Laguerre-Gaussian beams

LG beams arise as the physical manifestation of an infinite family of solutions of the paraxial wave equation. These solutions form a complete orthogonal basis set of functions, which possess a cylindrical symmetry in intensity with a helical phase front around the main axis of the beam. The number of 2π azimuthal phase changes encountered when circulating around the main axis gives the charge or helicity of the beam, and is denoted by the azimuthal phase index *l*. The most general LG beams posses a number (*p*+1) of rings, where *p* is the radial mode index. Here, we only consider single ringed LG beams (*p*=0) with a vortex line on the *z*-axis defined by the complex solution of the paraxial equation [14]:

where *z _{r}*=

*n*

_{0}

*k*

_{0}

*w*

^{2}

_{0}/2 is the Rayleigh range,

*w*the waist of the Gaussian envelope,

_{0}*k*the vacuum wavevector,

_{0}*n*the index of refraction of the host media and

_{0}*u*the amplitude. The first coefficient gives the longitudinal shape and decay of the beam including the LG Guoy phase shift, while the second coefficient describes the vortex part of the LG beam.

_{0}The full description of the LG beam needs the definition of the electric and magnetic fields, which, within the first order approximation [15], can be defined through the use of the vector potential **A**=(*u ^{l}_{p=0}*(

*x,y,z*)exp(

*-in*

_{0}

*k*

_{0}

*z*),0,0), by:

$$\mathbf{H}=\frac{i}{{Z}_{0}{k}_{0}}\nabla \times \mathbf{A}$$

Here we consider a linearly polarized beam along the *x*-axis, and *Z _{0}* is the vacuum impedance. This fully vectorial first order description of the LG beam can further be improved by introducing an additional vector potential for the magnetic field, which renders the relationship between the electric field

**E**and magnetic field

**H**more symmetric. Integrating the optical period averaged Poynting vector <

**S**>=Re(

**E**×

**H***)/2 (the asterisk denotes the complex conjugate and the chevron denotes a time average) over the azimuthal (transversal) plane gives the total power

*P*of the beam, which can be used to calculate the amplitude coefficient [

_{0}*u*in Eq. (1)]

_{0}Here, *Z* is the wave impedance and *s*=1/(*n*
_{0}
*k*
_{0}
*w*
_{0}) is the Gaussian beam order parameter. This amplitude coefficient includes helicity changes and corrects for relatively tightly focussed LG beams as a function of the order parameter *s*.

Up to this point we have considered the presence of a single monochromatic beam, exp(*iωt*), where *ω* is the optical frequency and is given by the dispersion relation: *ω*=*ck*
_{0} where *c* is the speed of light. However, in the case of the supercontinuum we need to sum over all the spectral components when defining the vector potential,

where *A*(*ω*) is the spectral amplitude of the supercontinuum and where the parenthesis contain the three vector components of the vector field.

#### 2.2 Optical force and torque in the supercontinuum beam

To calculate the optical forces and torques acting on dielectric microparticles in LG supercontinuum beams, we need to integrate the electromagnetic force density *f _{i}*=

*-∂*-

_{t}g_{i}*∂*over the volume of the particle and average over the pulse duration. The subscripts i, j and k varying from 1 to 3, correspond to the three Cartesian coordinates. We further assume summation over repeating indices in products. The two parts in the force density correspond to the force arising from the variation of the electromagnetic momentum density,

_{j}T_{ij}*g*=

_{i}*ε*, and to the influx of momentum given by the divergence of Maxwell’s momentum-stress tensor [16,17]

_{ijk}D_{j}B^{*}_{k}Here, *E _{i}*,

*D*,

_{i}*H*and

_{i}*B*denote the electric field, the electric displacement, the magnetic field and magnetic flux respectively, and where we consider linear constitutive relations in SI units such as

_{i}*D*=

_{i}*ε*

_{r}ε_{0}

*E*and

_{i}*B*=

_{i}*µ*

_{r}µ_{0}

*H*. In these constitutive relations

_{i}*ε*,

_{r}*ε*

_{0}and

*µ*

_{0}refer to the relative dielectric constant, the vacuum permittivity and permeability.

The total optical force, time averaged over the pulse repetition period Δ*t*, is:

where the momentum density, *g _{i}*, cancels out in the averaging process. The surface integral is evaluated on a closed surface surrounding the particle with

*ds*, which denotes the normal surface vector pointing outwards. The pulse can be seen as the superposition of many monochromatic waves [Eq. (4)] and the total average force becomes:

_{j}where ${\stackrel{\u2322}{T}}_{\mathrm{ij}}={\stackrel{\u2322}{E}}_{i}{\stackrel{\u2322}{D}}_{j}^{*}+{\stackrel{\u2322}{H}}_{i}{\stackrel{\u2322}{B}}_{j}^{*}-\frac{1}{2}\left({\stackrel{\u2322}{E}}_{k}{\stackrel{\u2322}{D}}_{k}^{*}+{\stackrel{\u2322}{H}}_{k}{\stackrel{\u2322}{B}}_{k}^{*}\right)$ is Maxwell’s stress tensor for each individual spectral component. Eq. (6) includes the cross product terms between the different monochromatic waves having different optical frequencies. Note that, due to the beating between these terms (and neglecting dispersive effects) all these cross products cancel out. Similarly, the total torque acting on the particle or ensemble of particles is:

where *ε _{ijk}* is the Levi-Civita antisymmetric tensor.

#### 2.3 Numerical application

There are many methods to calculate the scattering electromagnetic fields when given the incident field and the scattering object. Due to the commensurable size between the object and the wavelength, we chose to use a fully vectorial Finite Elements Method (FEM) for directly calculating the scattered field. This robust approach is very general and compares very well with standard Mie and T-matrix techniques [18] for spherical particles. Differences can occur due to the truncation order and different beam description, such as angular spectral decomposition. Calculating the scattering field using FEM offers the added benefit of automatically correcting the paraxial incident beam to be a numerically exact solution of Maxwell’s equations.

To study the effects of the SC Gaussian beam we calculated the transversal trapping forces acting on a 1µm polymer sphere (refractive index *n*=1.59) as a function of the wavelength (Fig. 1), and then used Eq. (7) to sum the spectral contributions of the SC beam. Here, we consider a linearly polarized Gaussian beam having a waist of *w _{0}*=1.75µm. The 1µm polymer sphere is free to move in the focal plane of the beam and we assume the spheres to have a constant refractive index across the wavelength range of interest. We determined the global, transversal trap stiffness of 1.19pN/µm in the polarization direction and 1.14pN/µm in the perpendicular direction.

## 3. Experiments

#### 3.1 Experimental setup

A commercially available supercontinuum (SC) source was used in our experiments. The source (Fianium Ltd, 4ps, 10MHz) delivered 5.5 Watts of unpolarized radiation across a spectral range of 464–1750nm. Figure 2 shows the initial experimental setup used. The SC was employed in an inverted optical tweezers setup in order to investigate the characteristics of this supercontinuum trapping system. A wavelength range of up to 850nm was selected and focused to a Gaussian-like spot in the trapping plane. A Köhler illumination scheme was employed to illuminate our sample plane, which allowed independent control of the depth of field and the average intensity of the illuminated field [19]. This also avoided imaging the light source in the sample plane and allowed homogeneous illumination with adjustable contrast. A spectrum of the beam in the trapping plane was obtained using a spectrometer (AVASPEC 3648), also seen in Fig. 2, along with a detailed experimental setup.

Cylindrical sample chambers were made using spacers 80µm deep. Due to the short working distance (0.13mm) of the microscope objective, zero thickness coverslips were used.

## 4. Results

#### 4.1 Single beam trap stiffness

With the setup described above we were able to trap 1µm polymer spheres in 2D using a SC beam. The apochromatic objective lens largely compensated for chromatic aberrations. Imaging the microparticles at the same time has often been a problem with supercontinuum optical tweezers [11], this is due to the difficulty of blocking the broadband beam while allowing the trapped sphere to be viewed. We have solved this problem using our previously described illumination method. The SC light was focused to a Gaussian-like beam spot, shown in Fig. 3, which optically trapped the polymer spheres. These can clearly be seen throughout the process, thus facilitating the tracking of the particles in the supercontinuum trap. Optical trapping in 3D was possible using silica particles of 780nm, 970nm and 1.28µm diameter. Although, experiments were done with both polymer and silica particles, the data presented is taken with polymer spheres due to their well known refractive index.

Microparticles moving through the surrounding liquid are subject to Brownian and viscous drag forces. For spheres moving linearly the Stokes drag coefficient *γ*
_{0}=6*πηa* links the velocity to the force through the relation *F*=*-γ*
_{0}
*ν*. Here, *a* and *ν* are respectively the radius and velocity of the particle, and *η* is the viscosity of the liquid (8.9×10^{-4}sPa for water at 298K) [20]. The Brownian force acting on the particle can be treated using the Langevin equation
$m\frac{{d}^{2}x}{d{t}^{2}}+{\gamma}_{0}\frac{dx}{dt}+kx={f}_{b}\left(t\right)$
, where *f _{b}(t)* is a random particle force having a Gaussian distribution with the moments <

*f*(

_{b}*t*)>=0 and <

*f*(

_{b}*t*)

*f*(

_{b}*t*′)>=2

*γ*

_{0}

*k*. Here

_{B}T*k*and

_{B}*T*are respectively Boltzmann’s constant and the absolute temperature [21], and

*k*is the optical spring constant. The motion of the particle is overdamped leading to the inertial term, the first term in the Langevin equation, to cancel out. The stochastic nature of this equation makes it possible to treat the solution of this equation, thus providing the particle position probability [22]:

where *U(x)* is the optical trapping potential.

Data from the CCD (Basler A602f-2) camera was used in conjunction with a particle tracking algorithm to determine the trap stiffness of the supercontinuum trap. We used a polystyrene particle with a diameter of 1 µm and a trapping power of 11mW. Histograms of particle positions were fitted with the expected probability distribution given by Eq. (9) determining the values of the trap stiffness for both *x* and *y* directions, *k*
_{x}=1.1pN/µm and *k*
_{y}=1.3pN/µm respectively. The precise supercontinuum beam profile in the trapping plane is unknown due to uncertainty of the distance between the top coverslip and the focal plane of the Gaussian beam. However, a Gaussian beam with a spot radius of *w _{z}*=1.75µm gives the correct trap stiffness within 10%.

## 5. Holographic supercontinuum trapping

#### 5.1 Introduction

The range of possibilities for white light trapping broadens when considering the use of beam shaping technologies. Such developments would permit the simultaneous use of multiple traps, or the creation of novel beam shapes by imposing unusual wavefronts upon the trapping beam profile. A spatial light modulator (SLM) is a computer controlled liquid crystal display that imposes a phase profile on an incident beam, similar to a hologram, enabling us to generate any desired beam shape. Thus, SLM’s are able to convert a Gaussian beam into a wide variety of beam shapes [23]. It is therefore possible to use the SLM to generate multiple trap sites in an optical tweezers set up and simultaneously trap multiple particles [24]. Liesener et al. [25] demonstrate the ability to trap and manipulate arrays of spheres in three dimensions using a computer controlled liquid crystal display. The use of supercontinuum light coupled with such technology presents important opportunities to combine novel beam shaping with a broad bandwidth, along with the low temporal coherence properties of the white light source.

In addition to multiple beams, it is also possible to generate LG modes [26] using the SLM. As described in section 2 [Eq. (1)], such beams may possess an on-axis optical vortex and an annular intensity profile. These modes have generated much interest due to their spiral phase front. This means that the Poynting vector of these beams has an azimuthal dependence that gives rise to a well-defined orbital angular momentum of *l*ħ per photon regardless of wavelength, where *l* is the (integer) azimuthal index. A number of studies have seen the transfer of OAM from a monochromatic vortex to trapped particles by absorption [6] or scattering [8, 27]. However, it is interesting to explore the role of both temporal and spatial coherence in OAM experiments.

#### 5.2 Experiments

### 5.2.1 Holographic experimental setup

The previous experimental setup shown in Fig. 2 was adapted to include a Holoeye 2500 spatial light modulator (SLM, wavelength range 400–700nm), Fig. 5. The supercontinuum beam dispersion was compensated for by adjusting the SLM hologram grating spacing to match a 10° prism. Different lens relays were used after the SLM for the dual beam and LG beam traps due to each beam’s propagation requirements.

### 5.2.2 Dual beam supercontinuum trapping

A dual-beam trap was generated using the SLM. The back aperture of the objective was placed at the conjugate plane of the SLM, and a 10° prism was placed just before the objective’s back aperture to help compensate for dispersion. The grating spacing of the hologram was adjusted using a Labview program until the wavelengths overlapped in the two trapping beams and dispersion was corrected for, shown in Fig. 5(b). Using this method, it is difficult to compensate for dispersion for a higher number of multiple beam traps without a multi-prism configuration with accurate positioning.

The normalized profiles of the two beams can be seen in Figs. 6(a)–6(d) for x and y profiles, where the red, green, and blue spectral component profiles of each spot are shown. This demonstrates simultaneous dispersion compensation for both trap sites where we observe that the upper trap is slightly better compensated. The total power after spatial filtering of the unwanted diffraction orders was 24mW. Analyzing Fig. 5(b) we found that the upper beam contains about 1.1 times higher power than the lower beam. The Gaussian-like waist spot size of each trap is approximately *w _{0}*=1.2µm.

Optical trapping was achieved in two-dimensions, shown in Fig. 5 for two 2µm polymer spheres (Duke Scientific 4202A, refractive index *n*=1.59) immersed in water, where trapping occurred up against the top coverslip in the sample chamber. As for the single beam trap, we estimated the individual trap stiffness for each beam by using their respective particle position histograms shown in Fig. 8.

The stiffness of the lower trap is less than that of the upper trap, most likely due to the differing diffraction efficiency for individual beams on the SLM and imperfect dispersion compensation in that trap. As was previously seen in Figs. 5(b) and 6, the dispersion was well compensated for in the upper trap and only partially compensated for in the lower trap. Similar to the case of the single beam trap, we cannot obviate the uncertainty in the waist size, the wavelength dispersion, and the inaccuracy in the vertical positioning of the trapping plane. We compared the measured values with the theoretical prediction by adjusting the waist size of the considered Gaussian beam and used a waist of *w _{0}*=1.1µm. Using Fig. 5(b) to estimate the power ratio between the two traps, we find an average trap stiffness of 16.5pN/µm for the upper trap and 15pN/µm for the lower trap.

### 5.2.3 Supercontinuum vortex trapping

Here, we use the same approach as discussed above to generate a SC vortex beam as proposed by Stzul et al. [28]. A LG mode beam (*l*=3, *p*=0) was generated using the SLM as shown in Fig. 5(c). All SC vortices generated were analyzed [28] and the observed fringes confirmed that all spectral component of the broadband source possessed equivalent azimuthal phase commensurate with the azimuthal index encoded onto the SLM. A 10° prism was placed immediately before the objective in the SLM conjugate plane for dispersion compensation. The resulting LG beam [as defined in Eq. (1)] can be decomposed into a superposition of multiple higher order LG beams [29] (*l*=3, *p*=0,1,2,…) where more than 50% of the power is in the *p*=0 mode. We used the measured beam profile to decompose the observed beam into the higher *p* modes.

Trapping in the bright ring of the vortex was achieved with 1µm diameter polymer spheres (Duke Scientific 4010A, refractive index *n*=1.59). The spheres were trapped against the top coverslip of the sample and the optical confinement occurred in 2D. Rotation rates for the case of three trapped spheres, (inset Fig. 9), were determined using a fast camera (Basler pioneer plA640-210gm). The videos were then broken up into their individual frames and analyzed using a Mathematica tracking program to determine the rotation rates at different powers (Fig. 9). Each point on the graph is averaged over at least 50 rotations with a 10% error.

As expected, there is a linear relation between the power of the trapping beam and the rotation rate of the spheres (Fig. 9). The maximum power in the trapping plane was 30mW and the corresponding maximum rotation rate was 5.8 rotations per second (Hz). Even with the relatively low power of 10mW we achieve rotation rates of 2.4Hz. Taking into account only the power in the *p*=0 LG mode we find a rotation rate efficiency of 0.39Hz/mW. Some data point error bars lie outside the linear fit, which could be due to slight differences in the surface friction between measurements.

Laguerre-Gaussian beams of *l*=2,3,4 and 5 were generated using the SLM. The beam spot radius was determined by measuring the annular profile of the LG beam, see Fig. 10(a). We ensured that each spectral component of the temporally incoherent beam had an azimuthal phase step of appropriate order [30].

A stable configuration of three polymer spheres, 1µm in diameter, was used to determine the particle rotation rates, Ω, at different *l*-values. Figure 10(b) shows the rotation rates of three spheres trapped within the LG beam as a function of *l*. Again, at least 50 rotations were taken for each rotation rate at a power of 15mW measured in the trapping plane; the error was taken as 10% of the rotation rate.

To simulate the OAM transfer from the LG beam to the three spheres, we used the theoretical approach described in section 2, i.e. integration of the total torque acting on the three spheres [Eq. (8)]. As an object rotates in a viscous fluid it incurs a drag torque (*T*=-16*π*
^{2}
*ηa*
^{3}Ω for a sphere, where Ω is its rotation rate). There is no analytical formula when considering a group of spheres turning together as a fixed ensemble, but the problem can be tackled using multipole expansion of the flow velocity in a series of spherical harmonics [31]. For simplicity we chose to use FEM to solve numerically the incompressible Navier-Stokes equations. This was done for three 1µm touching spheres turning in the plane defined by their centres and around their centre of gravity. The constant angular velocity of the three spheres gives a drag coefficient of *-T*/Ω=9.4×10^{-20}
*NHz*
^{-1}
*m*. This drag coefficient is equivalent to the rotational drag coefficient of a sphere of diameter 1.77µm. Taking the wall effects into account increases this drag coefficient by 5% for a distance of 500nm and by 20% for 100nm distance. Additionally to the optical torque, each particle is subject to a force pulling it towards the axis of the LG beam. This centripetal force *F _{c}* is represented in Fig. 11(b) and shows that the group of three rotating particles forms an optically stable configuration.

Using this drag coefficient in conjunction with the optical torque we can determine the theoretical rotation rate of the three spheres as a function of wavelength, Fig. 11(a). Further, using the spectrum of the SC beam in the focal plane (Fig. 2) we can sum all the spectral rotation contributions giving the red line in the graph, and which corresponds to a rotation rate power efficiency of 0.44Hz/mW. This is approximately 10% higher than the measured efficiency after correcting for the power in the *p*=0.

## 6. Conclusions

We have demonstrated an appropriate optical system for viewing trapped spheres in a supercontinuum single beam trap. We have characterized the trap and recorded the transverse trap stiffness values for a 1µm polymer sphere.

We then progressed to implement a holographic supercontinuum trapping system. We set groups of particles into rotation using the OAM of a supercontinuum LG trapping beam. This study conclusively shows the transfer of OAM from a temporally incoherent (broadband) light source to the trapped particles. In addition, we demonstrate and characterize the first spectrally compensated holographic supercontinuum dual beam trap system. Dual supercontinuum beam traps might be used to avoid optical cross-talk when studying hydrodynamic cross-correlations [32].

## Acknowledgments

We thank W. M. Lee, S. E. Skelton, M. Mougenot for assistance with early aspects of the experiment. We would also like to thank the UK Engineering and Physical Sciences Research Council for the funding of this work. JEM, AEC and MM contributed equally to the work presented.

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