Photovoltaic versus optical tweezers

Javier Villarroel; Héctor Burgos; Ángel García-Cabañes; Mercedes Carrascosa; Alfonso Blázquez-Castro; Fernando Agulló-López

doi:10.1364/OE.19.024320

1. Introduction

Optical tweezers are a well-established tool [1,2] to trap particles and/or molecules and organize them into ordered arrangements (patterning). It uses the coupling energy between an electomagnetic wave field (light) and the induced polarization in an electrically neutral particle (dielectrophoresis [3,4]). However, parallel manipulation of many particles is not feasible due to the high intensities required. Static electric fields provided by electrodes can be also used [5] but the method lacks flexibility. A similar criticism may apply to the use of dielectrophoretic forces generated by ferroelectric polydomain patterns [6]. Recently, the possibility of using electric fields generated by illumination such as those associated to the bulk photovoltaic (PV) effect or more generally to the photorefractive (PR) effect, has shown great promise. These fields are produced in certain, suitably doped, ferroelectric crystals, such as LiNbO₃ (LN), when illuminated with moderate levels of light of an appropriate wavelength [7]. LN presents a high PV drift and so constitutes a good choice to apply those fields to a variety of scientific and technological purposes. For example, several test experiments have been successfully performed on the trapping and manipulation of particles on crystal surfaces [8–11]. Due to the higher fields reached in photovoltaic materials like LiNbO₃, we shall call such trapping devices photovoltaic tweezers more properly than photorefractive tweezers. Moreover, the biomedical implications of such PV fields to kill cancer cells and provide a possible cancer therapy have been demonstrated [12]. The purpose of this work is to explore the potential of the method and discuss the advantages and shortcomings in relation to optical tweezers. In relation to previous works, a more detailed and meaningful analysis of the PV electric field patterns and the resulting dielectrophoretic forces has been carried out. In particular, the role of particle (e.g. geometrical) anisotropy and high modulation of the light pattern is addressed. Our discussion focuses on LiNbO₃ because of its high PV response [13] and relevance for electro-optics, nonlinear optics and optical waveguide fabrication [14].

2. Illustrative experiments on PV tweezers

First, let us present two representative experiments of photovoltaic microparticle trapping. The experiments have been carried out in 1 mm thick x-cut congruent LiNbO₃ crystal highly doped with iron (0.1% wt) in order to have a strong photovoltaic effect. Single beam and holographic configurations have been tested. After illumination with 532 nm laser light, chalk (CaCO₃) micrometric particles, with a size range of 1–10 μm, were blown out from below on the bottom sample surface. Then, the particle distribution was visualized either by an optical photograph or a micro-photograph. The results are shown in Fig. 1 . The microparticle distribution when a light interference pattern (spatial periodicity Λ = 30 μm) generated by two laser beams illuminated the sample is presented in Fig. 1a. Particles are trapped periodically with the same Λ as the light pattern. In turn, Fig. 1b shows the particle distribution when the sample is illuminated with a single Gaussian beam with a diameter at half height 2σ = 0.6 mm. It can be seen that the particles are trapped at both sides of the beam intensity maximum whereas the center of the previously illuminated region is free of particles. This repulsion has been also very briefly mentioned in [2] although no explanation was provided.

Fig. 1 Chalk micro-particles trapped on the surface of LiNbO₃:Fe after illumination with the interference of two beams with periodicity 30 μm and modulation m ≈1 (a) and with a single laser beam (b). In (b) the illuminated region (diameter 4σ) has been indicated with a circular dashed line.

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3. The bulk photovoltaic effect

The bulk PV effect is a singular drift mechanism [7,13] that appears in some non-centrosymmetric crystals when suitably doped, e.g. LiNbO₃, KNbO₃, BaTiO₃ and LiTaO₃. It involves a non-symmetric light-induced excitation of carriers that gives rise to a photo-current density even without any applied voltage. The PV effect is, generally, associated to certain impurities (e.g. Fe and Cu) and charge states. For LiNbO₃ it has been associated to optical transitions between a localized donor center such as Fe²⁺ (Li sites) to the conduction band, involving a net directional electronic transport along the polar axis. It has a completely different origin to the PV effect found in semiconductor junctions. It allows for the generation of very high electric fields under moderate levels of illumination. The active wavelength is dependent on the material and doping. Although the responsible microscopic processes are not yet sufficiently well understood the effect may find very interesting practical applications.

After light excitation at the absorption band of the active center a photovoltaic electrical current is immediately generated along the c trigonal crystal axis. Assuming a scalar formulation, the current density is written

j_{P V} = q (I / h ν) σ N_{D} l_{P V}

with q being the electron charge, I the light intensity, ν the light frequency, N_D the donor concentration, σ the absorption cross-section and l_PV the photovoltaic length. This is the current measured in closed circuit. Under open circuit conditions the current proceeds until an opposite electric field is generated (photovoltaic field) that compensates the effect. The field is given by

E_{P V} = \frac{J_{P V}}{q μ n} = \frac{σ I N_{D} l_{P V}}{h ν μ n} = \frac{l_{P V} γ N_{A}}{μ}

with μ being the electron mobility and

n = \frac{σ I N_{D}}{h ν γ N_{A}}

the photogenerated free electron density. In this later expression N_A is the electron acceptor density and γ an electron trapping coefficient. Using reasonable values for the various parameters of highly doped LiNbO₃ (σ = 2×10⁻¹⁸ cm², N_D = 10¹⁸ cm⁻³, N_A = 10¹⁹ cm⁻³, γ = 10⁻⁹ cm³/s, μ = 5×10⁻³ cm²/Vs, l_PV = 5 Å) and assuming a light intensity of 10 W/cm², one obtains

j_{P V} = 500 {nA/cm}^{2}, n = 5 \times 10^{9} {cm}^{- 3} = 10^{9} {cm}^{- 3}, E_{P V} = 100000 V/cm

Rough estimates for the times required to establish such values are 0.1 ns for the steady electron density and 5×10³ s for the photovoltaic field (dielectric relaxation time).

4. Field profiles under periodic illumination

The most interesting situation appears when the light distribution is inhomogeneous and follows a given spatial pattern. The simplest case corresponds to a periodic light distribution with period Λ, as illustrated in Fig. 2 ,

I = I_{0} (1 + m \cos K x)

with K=2π/Λ, m being the modulation of the light pattern. It is generally obtained by interference of two coherent plane waves impinging on the active material. Then, an electric field pattern develops that matches (except for a certain phase mismatch) the imposed light pattern [7]. For a PV crystal like LiNbO₃ with the c axis along X, a modulated (sinusoidal) photovoltaic current immediately starts growing with time that is in phase with the light intensity pattern. Under short-circuit conditions (to avoid blocking fields at the ends of the illuminated region), the associated space-charge field at short times is

E_{P V} = \frac{j_{P V}}{ε ε_{0}} t = \frac{σ N_{D} I_{0} e l_{P V}}{h ν ε ε_{0}} (1 + m \cos K x) t

which is also modulated in phase with the light. After this initial stage, other transport processes, diffusion and electric-field drift, set in to determine the final steady state solution of the problem. By solving the transport equations [7] for the one-dimensional case (i.e. excluding the role of limiting surfaces), including all transport processes, one arrives at a periodic field distribution inside the crystal plate with period Λ. Assuming a low modulation factor m, the steady-state first-order Fourier component of such field has a complex amplitude [7]:

E_{0} = i m \frac{E_{D} + i E_{P V}}{1 + \frac{E_{D} + i r E_{P V}}{E_{q}}} ≅ \frac{m E_{q} E_{P V}}{E_{q} + i r E_{P V}} ≅ m E_{P V}

where

Fig. 2 Evanescent photovoltaic electric fields E (represented by arrows) as a function of the position (x,z) generated by a sinusoidal light pattern.

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\begin{array}{l} E_{D} = \frac{k_{B} K T}{q} (diffusion field) \\ E_{q} = \frac{q N_{D} r}{ε ε_{0} K} (limiting field) \\ r = \frac{N_{A}}{N_{D} + N_{A}} (oxidation/reduction ratio) \end{array}

Note that the presence of the imaginary unit in expression (6) indicates that a certain phase mismatch generally develops between the light and the field patterns. Moreover, the field amplitude is the photovoltaic field except for the modulation factor m.

The above linear approach has not taken into account the higher harmonics appearing for high modulation m (non-linear terms in the rate equations [15,16]) and it has also ignored the edge effects at sample surfaces [17–20] that become relevant when one is interested in the fields outside the crystal (see section 5). For high modulations [15,21] the field profiles are strongly distorted from the sinusoidal shape. Finally, any bias voltage due to a defective short-circuiting of the crystal introduces a bias in the field profile and leads to non-equivalent maxima and minima fields.

5. Fringe (evanescent) fields outside the crystal

For patterning and manipulation purposes of particles on a LN surface one requires the evanescent field pattern at and above the surface in order to act on the adsorbed or deposited particles or molecules. The edge effects in the PR space-charge field appearing near the limiting faces of the sample under sinusoidal illumination have been previously addressed [17–20]. They constitute a two-dimensional coupled problem between the Maxwell and material transport equations, so that it results a bit cumbersome. However, for our purposes, one can simply consider a sinusoidal electrostatic potential distribution along the X-axis in the z = 0 plane that attenuates exponentially with the distance z to the sample surface, $V (x, z) = \exp (- β z) V_{0} \sin K x$ . From Laplace equation, β = K and, then, we easily arrive at the fields present in the vacuum half-space (z > 0):

E_{X} (x, z) = - K e^{- K z} V_{0} \cos K x

E_{Z} (x, z) = K e^{- K z} V_{0} \sin K x

containing both parallel E_X and perpendicular E_Z field components. The two field components show sinusoidal profiles with a π/2 phase-mismatch and evanescent amplitudes when one gets away from the surface. Moreover, it results from (8) that the modulus of the total field E has only a z dependence and it is constant along the X-axis.

The corresponding field distributions inside the crystal (z < 0), including edge effects, obey the Poisson equation and can be written as

E_{X}^{i n} (x, z) = (E_{0} - E_{B} e^{K z}) \cos K x

E_{Z}^{i n} (x, z) = E_{B} e^{K z} \sin K x

Both the edge and the bulk PV contributions are apparent with amplitudes, E_B and E₀ = mE_PV, at the surface, respectively. From the boundary conditions one may relate the boundary and vacuum field amplitudes to the effective PV field (for an infinite medium) and arrive at

E_{B} = E_{0} \frac{ε}{ε - 1}, V_{0} = \frac{E_{B}}{K}

Now, the field distributions are fully characterized. The field lines are illustrated in Fig. 2. Note that the electric field strength decays with z in a scale governed by the grating period (typically a few micrometers). This means that the PV trapping effect could be selectively cancelled by using a thin non-PV covering layer.

6. Photovoltaic tweezers

Let us now examine the effect of the photovoltaic fields on particles, molecules or cells placed on the surface of an illuminated crystal.

6.1 Electrical (Dielectrophoretic) forces on particles

The dielectrophoretic PV forces arise because of the energy coupling between the PV field at the surface and the induced dielectric polarization of the particle [4–6]. Assuming that the wavelength of the field inhomogeneity is much larger than the size of the particle (Rayleigh approximation), so that the field can be considered constant inside the particle, the gradient force associated to the coupling energy is

\vec{f} = - \nabla W = - \nabla (- \vec{p} \cdot \vec{E})

being

\vec{p}

the field induced particle dipolar moment that in turn can be written in terms of the particle polarizability. In general, the polarizability α_ij is a tensor due to either crystalline or geometrical anisotropy (p_i = ε₀α_ijE_j). However, in simple cases such as for spherical particles of an isotropic material (see below), it is a scalar and

\vec{p} = ε_{0} α_{} \vec{E}

leading to

\vec{f} = ε_{0} α \nabla ({\vec{E}}^{2})

. For this scalar case one expects maximum forces at the inflexion points of the field pattern and equilibrium positions for the particles at the maxima (or minima) of the electric field modulus maxima, depending on the sign of the effective polarizability.

For isotropic spherical particles, taking into account the local field effects associated to the polarization of a homogeneous surrounding medium of dielectric permittivity ε_M, one has for the scalar particle polarizability [6,9]:

α = 2 π_{} r^{3} ε_{M} (0) \frac{ε_{P} (0) - ε {}_{M}{(0)}}{ε_{P} (0) + 2 ε_{M} (0)}

where r is the particle radius and the values for the dielectric permittivities correspond to static fields. Depending on the relative sign of the two dielectric permittivities, for the particle ε_P and the surrounding medium ε_M, one obtains either positive or negative values for α. Anyhow, the dielectrophoretic force is given by Eq. (12).

In general, one may use particles with arbitrary shapes leading to a geometrical anisotropy in the effective polarizability. Many smooth shapes can be approximately represented by an ellipsoid having three principal axes (X, Y, Z) and different polarizabilities along them. One, then, has a polarizability described by a second-order tensor. For example, for the limiting case of disk-shaped particles of volume V [22] (parallel to the XY plane):

α_{X X} = α_{Y Y} = V (ε_{P} - ε_{M})

α_{Z Z} = V ε_{M} \frac{ε_{P} - ε_{M}}{ε_{P}} = V ε_{M} (1 - \frac{ε_{M}}{ε_{P}})

showing that for ε_P >> 1, the polarizability along X is remarkably larger than along Z. In this case, the dielectrophoretic force would be

\vec{f} = \nabla (\vec{p} \cdot \vec{E}) = ε_{0} \nabla (α_{X X} E_{X}^{2} + α_{Z Z} E_{Z}^{2})

A similar situation arises when the particles have crystalline (dielectric) anisotropy. Then, the particle has a tensor effective polarizability with three principal values along principal crystal axes, although the anisotropy is generally expected to have a smaller effect.

6.2 Particle trapping under periodic light intensity distributions

Let us consider, first, the situation found in a PR experiment (see section 2), with two coherent beams impinging on the XZ cut of a LiNbO₃ crystal and producing an interference sinusoidal pattern, Fig. 1a. This is a convenient case readily obtained in the laboratory. Let us also consider, as in previous works [9,10], that spherical particle of volume V with dielectric permittivity ε_P is placed on the surface of the crystal plate and is embedded in an outside medium of permittivity ε_M (e.g. vacuum, ε_M = 1). According to the Eqs. (8) for the evanescent PV fields, it is clear that

{\vec{E}}^{2} = E_{X}^{2} + E_{Y}^{2} = E_{0}^{2} {(\frac{ε}{ε - 1})}^{2} e^{- K z}

In other words, the square modulus of the field has only a z dependence and the dragging force is directed along the vertical direction Z as illustrated in Fig. 3a . Therefore, it can only be used to trap the particles above the sample surface, but not to periodically structure the particles along the X-axis.

Fig. 3 Dielectrophoretic forces for isotropic spherical particles under periodic photovoltaic fields: (a) sinusoidal electric field pattern inside the crystal and (b) non-sinusoidal field pattern, that are illustrated in the bottom part of the figures.

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However, that negative prediction has been obtained for purely sinusoidal fields that are produced in the low-m (linear) regime. The situation is different for high modulation m where the field profiles reach higher values and the profiles may depart greatly from sinusoidal [15,16,22] (due to the higher-order Fourier components) and lead to X- dependent forces. A very simple example is provided by considering two harmonic components for the PV field with wave-vectors K and 2K, and amplitudes A_K/A_2K = 2 (typical ratio for m ~0.9 [20]). Now ${\vec{E}}^{2}$ has a sinusoidal X-profile with grating vector K, equal to that of the exciting light, due to the crossing of the K and 2K components. The situation is illustrated in Fig. 3b where the dielectrophoretic force is plotted as a function of the two spatial coordinates X and Z. One clearly sees, now, an induced periodicity in the magnitude of the dielectrophoretic forces along the X-axis with wavevector K in accordance with the experiment of Fig. 1a and the data reported in ref [8]. Therefore, one has the two options: trapping at the sample surface (due to the Z component of the gradient $\nabla {\vec{E}}^{2}$ ) and displacement along the surface (due to the X component of the gradient). The situation may be comparable to that achieved for ferroelectric polydomain fields [6]. In this case, the field profiles may present large gradients in the X direction and so one might achieve significant dragging forces for spherical particles.

In the more general case of non-spherical particles, one should consider also the geometrical anisotropy to determine the dragging force along the sample surface (see Appendix). For the example of disk-shaped particles, it is obtained that the PV forces along X show a dominant component with wavevector 2K, i.e. half the periodicity Λ of the light (even for sinusoidal PV fields. This may account for the results found in some experiments reported in references [9–11] that present a trapping period Λ/2. Some simple numerical estimates can be now made for disk-shaped particles of 10 μm = 10⁻⁵ m diameter placed on top of the LN crystal during a realistic photorefractive experiment. Assuming the interference of two coherent light beams of 100 mW/cm², producing a grating pattern of Λ = 100 μm and a field amplitude of 10⁵ V/cm, one obtains a rough estimate for the dragging force f ≥ 100 nN. One should note that the gravity force (weight) on such particle will be f ~10⁻¹¹ N i.e. around 10,000 times smaller. Moreover, the dragging force can be made relatively much larger by using smaller particles in tighter patterns.

The crystalline anisotropy may be also relevant to determine the PV dragging forces but, in principle, should be less critical than the geometrical anisotropy. Anyhow, in practical situations, one should have a broad distribution of particle shapes and orientations leading to diffuse particle patterns.

In summary, it can be concluded that the pattern of trapped particles on the surface of the PV crystal is very sensitive to the shape of the particles, the orientation of the dielectric tensor and the modulation index m of the light profile.

6.3 Single beam experiments

This configuration, illustrated in Fig. 1b for a single laser beam with a roughly Gaussian intensity profile, is particularly interesting to clarify some aspects of the physics involved. One clearly sees two regions of trapping on both sides of the X direction but outside the bright spot. In order to interpret the observed trapping pattern we have calculated, using a simple approach, the spatial pattern of PV electrical forces assumed to be generated by a distribution of elementary PV dipoles whose distribution follows the light intensity profile. The calculated light intensity and (photovoltaic) E² profiles at a z = constant plane close to the sample surfaces are shown in Fig. 4a and Fig. 4b, respectively, by means of color maps. For particles of higher dielectric permittivity than the surrounding medium (air) one expects, from either formulae (9) or (10), positive effective polarizabilities and so particles should concentrate on the regions of highest E² that roughly correspond to the borders of the light beam profile along the PV X-axis. In other words the particles escape from the center of the light beam and accumulate on the spots where E² reaches a maximum value. These two regions (Fig. 4b) clearly correlate with the concentration spots in Fig. 1b. Note that in this case there is no restriction on the shape of the trapped particles.

Fig. 4 Two-dimensional plot of the Gaussian light intensity distribution (a) and the square of the calculated photovoltaic electric field generated by that beam illumination (b). Note that x stands for the PV axis.

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7. Comparative analysis of optical and photovoltaic tweezers

Now, one can try a comparison of our proposed PV tweezers with the best known case of optical tweezers [1,2]. This trapping tool uses the gradient forces acting on a particle placed in an inhomogeneous light field whose amplitude is E₀(x,z). In the Rayleigh approximation the local energy of the field at the particle site is given by the same expression (6) as applied to a wave field of frequency ω and amplitude E₀,

W = - ε_{0} α (ω) < \vec{E} {(ω)}^{2} > = \frac{- 1}{2} ε_{0} α (ω) ({\vec{E}}_{o}^{2})

where α(ω) is the polarizability at the light frequency ω and one has to perform an average over a wave cycle. The corresponding optical gradient force on a polarizable particle is

\vec{f} = \frac{1}{2} ε_{0} α (ω) \nabla ({\vec{E}}_{0}^{2})

As a first example, it is interesting to evaluate the gradient force associated to an interference pattern such as that considered in the photorefractive experiment of the previous section.

Let us, first, consider the modulated light intensity pattern ${\vec{E}}^{2} = {\vec{E}}_{0}^{2} (1 + m \cos K x)$ , m being the modulation index. The gradient force on a polarizable particle is:

\vec{f} = \frac{1}{2} ε_{0} α (ω) \nabla {\vec{E}}^{2} = - \frac{1}{2} ε_{0} α (ω) m {\vec{E}}_{0}^{2} \sin K x

i.e. maximum forces appear at the inflexion points of the pattern, having maximum intensity slope. Therefore, the situation is similar to that described for PV tweezers. The difference is the lower value of the purely electronic polarizability α(ω) versus the static polarizability α and the lower fields in comparison to PV fields for moderate light powers. For example in the previous PR experiment, the amplitude of the wave field produced by the light interference of two beams with I = 100 mW/cm² is around 10 V/cm (instead of 10⁴-10⁵ V/cm induced by PV effect), leading to an optical gradient force that is at least six orders of magnitude lower than that one associated to the PV fields (i.e. below 0.1 pN). The difference is even larger for more compact gratings. Stronger optical forces can be achieved at higher light intensities such as those obtained with focused laser beams i.e. above 100 kW/cm².

Anyhow, the light contributes with another force, scattering force, corresponding to the net momentum transfer to the illuminated particle. One has f = I/c. For I = 1 kW/cm² = 10⁷ W/cm² and a particle offering a surface of 10⁴ nm² = 10⁻¹⁴ m², the scattering force is f = (1/3)×10⁻¹⁵ N.

8. Comparative performance parameters

To conclude our comparative analysis we are now summarizing the main performance parameters for optical and PV tweezers.

a) The PV tweezers only operate for light wavelengths that lie within the spectral region that triggers the photovoltaic effect. For the case considered in this paper of Fe:LiNbO₃, it extends through the wavelength range 400 - 650 nm.
b) For moderate light intensities (< 1 W/cm²) the PV forces are larger than the optical ones. The relative importance depends on the light intensity, modulation index, grating period and crystal parameters. Optical forces can be comparable or even higher at rather high light powers (≥ 100 kW/cm²).
c) The response time for optical gradient forces respond as the electronic polarizability (≤ 1 ps). PV forces require a certain energy deposition by optical absorption and eventually electronic transport. Then, response times are light intensity dependent and much longer, in the scale of seconds for light intensities < 1 W/cm².
d) The space-charge fields induced by the PV effect decay with the dielectric dark relaxation time and so they can remain for times of the order of days-months depending on a number of crystal parameters such as doping or reduction state of Fe impurity. Moreover thermal fixing techniques allow generating nearly permanent electric fields [23,24] (with lifetimes of about 10 years).
e) PV tweezers can only apply for particles very close to the surface of a PV crystal (some micrometers) whereas optical forces have not this limitation.

9. Conclusions and new perspectives

The physical basis and performance of PV tweezers have been now discussed in some details and compared to the best known case of optical tweezers. So far, very few preliminary experiments have been reported. The field is still quite open and new developments are expected. On the theoretical side the detailed analysis of the PV dragging forces under high m conditions is still needed. On the other hand, new experiments with a more accurate control of the shape, size and orientation of the particles are necessary. Finally, a theoretical and experimental study of the effects of moving light interference patterns on the trapping and manipulation capabilities of PV tweezers would be very promising.

Appendix A: Dielectrophoretic PV force for anisotropic disk-shaped particles

Let us consider as a relevant example the case of disk-shaped particles deposited with their principal dielectric axes along the X and Z axes on the PV crystal surface as discussed in section 6.2. For simplicity, we also assume a periodic non sinusoidal electric field having only two harmonic components with wavevectors K and 2K consistent with a sinusoidal light illumination with wavevector K and high modulation (m~1). Then, the PV electric field is written as

\begin{array}{l} E_{X} (x, z) = - V_{0} K e^{- K z} \cos K x - 2 V_{1} K e^{- 2 K z} \cos 2 K x \\ E_{Z} (x, z) = V_{0} K e^{- K z} \sin K x + 2 V_{1} K e^{- 2 K z} \sin 2 K x \end{array}

where V₀ >> V_1. According to expression (14), the dielectrophoretic force for disk-shaped particles is

\vec{f} = \nabla (\vec{p} \cdot \vec{E}) = ε_{0} \nabla (α_{X X} E_{X}^{2} + α_{Z Z} E_{Z}^{2})

with α_XX and α_ZZ given by expressions (13).

In order to analyze the periodicities of the trapped particle pattern, let us focus our attention on the dielectrophoretic force component f_X, along the X-axis, i.e. parallel to the surface:

f_{X} = ε_{0} (α_{X X} \frac{\partial E_{X}^{2}}{\partial x} + α_{Z Z} \frac{\partial E_{Z}^{2}}{\partial x})

After some algebra we obtain

\begin{array}{l} \frac{\partial E_{X}^{2}}{\partial x} = - V_{0}^{2} K^{3} e^{- 2 K z} \sin 2 K x - 4 V_{1}^{2} K^{3} e^{- 4 K z} \sin 4 K x - 4 V_{0} V_{1} K^{3} e^{- 3 K z} \cos 2 K x \\ - 8 V_{0} V_{1} K^{3} e^{- 3 K z} \cos K x \sin 2 K x \\ \frac{\partial E_{Z}^{2}}{\partial x} = V_{0}^{2} K^{3} e^{- 2 K z} \sin 2 K x + 4 V_{1}^{2} K^{3} e^{- 4 K z} \sin 2 K x + 4 V_{0} V_{1} K^{3} e^{- 3 K z} \cos K x \sin 2 K x \\ - 8 V_{0} V_{1} K^{3} e^{- 3 K z} \sin K x \cos 2 K x \end{array}

And so, the component of the force is

\begin{array}{l} f_{X} = ε_{0} (α_{Z Z} - α_{X X}) K^{3} [V_{0}^{2} e^{- 2 K z} \sin 2 K x + 6 V_{0} V_{1} e^{- 3 K z} \sin 3 K x + 4 V_{1}^{2} e^{- 4 K z} \sin 4 K x] \\ - 2 V_{0} V_{1} ε_{0} (α_{Z Z} + α_{X X}) K^{3} e^{- 3 K z} \sin K x \end{array}

That presents spatial modulations along the X-axis with wavevectors K, 2K, 3K, 4K, the most significant one being the zero order contribution 2K, found in a number of experiments [9–11]. In fact, for low modulation of the exciting light pattern, the PV field becomes sinusoidal (V₁~0) and only this 2K component is nonzero.

Finally, note that if α_XX=α_ZZ = α₀ (isotropic case), the X component of the force simplifies to

f_{X} = - 4 ε_{0} α_{Z Z} V_{0} V_{1} K^{3} e^{- 3 K z} \sin K x

recovering the result for spherical particles, i.e. trapping wavevector K as that of the light pattern that only exists if the PV field includes a second harmonic contribution (nonsinusoidal PV field).

Acknowledgments

We acknowledge financial support from the Ministerio de Ciencia e Innovación (MICINN) under grants MAT2008-06794-C03-01 and MAT2011-28379-C03-01.

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Photovoltaic versus optical tweezers

Abstract

1. Introduction

2. Illustrative experiments on PV tweezers

3. The bulk photovoltaic effect

4. Field profiles under periodic illumination

5. Fringe (evanescent) fields outside the crystal

6. Photovoltaic tweezers

6.1 Electrical (Dielectrophoretic) forces on particles

6.2 Particle trapping under periodic light intensity distributions

6.3 Single beam experiments

7. Comparative analysis of optical and photovoltaic tweezers

8. Comparative performance parameters

9. Conclusions and new perspectives

Appendix A: Dielectrophoretic PV force for anisotropic disk-shaped particles

Acknowledgments

References and links

Cited By

Figures (4)

Equations (26)

Optics Express