1.

Introduction

“Optical tweezers” is the common term used during the last two decades to describe the optical force generation and confinement of microscopic particles by a highly focused laser beam, as was first demonstrated by Ashkin ^{1, 2} An optical trap, apart from its use as a particle manipulator, when calibrated can be used to measure the forces exerted on trapped objects as a result of various molecular motors within the cell.^{3, 4, 5} Other important biological applications are the trapping of cells, bacteria and viruses,⁶ manipulation of DNA molecules^{7, 8, 9} and erythrocytes,^{9, 10, 11, 12} cell sorting,¹³ cell fusion,^{14, 15} intracellular surgery,^{16, 17} and neuronal growth in studying how neural circuits are formed.¹⁸ Finally, in biotechnology, the “laser-guided direct writing” technique is used to deposit particles of $100\mathrm{nm}\text{to}10\mu \mathrm{m}$ onto solid surfaces.^{19, 20}

For most of the preceding applications, an exact knowledge of the optical forces applied to the trapped particle is useful and different methods exist for force calibration. Two of them that are widely used are the Brownian motion calibration and the viscous drag force calibration. In the first method, the power spectrum of the position of an isolated particle in an optical trap that experiences random forces due to thermal fluctuations is studied,^{21, 22, 23} while the second method examines the relative motion of a trapped particle with respect to the surrounding liquid.^{24, 25} Although there has been more than a decade of intense activity, the agreement between theory and experimental force determination is not usually fully satisfactory,¹⁵ and only in a few cases a good agreement has been found.²⁶ Theoretical estimations of the trapping forces exist only for a spherical geometry of the trapped particle in two limited size regimes. The geometric optics model is in good agreement with measured forces if the diameter of the trapped object is well above the wavelength of the laser light,²⁷ whereas the electromagnetic theory approach must be used for particles that are small compared with the wavelength (Rayleigh approximation). In the intermediate regime, where the particle sizes are of the order of the wavelength of the trapping laser, the electromagnetic theory seems to yield better results than the geometric optics,²⁸ but there remain discrepancies between theory and experiment for the trapping forces in this regime.

In this paper, we describe a different calibration method for an optical trapping system. The method uses the dielectrophoresis, combined with optical trapping, to determine the relation between the output laser power and the corresponding exerted force. Dielectrophoresis is the term used to describe the polarization and associated motion induced in dielectric particles by a nonuniform electric field. The nonuniformity of the electric field induces a dipole moment, due to the convergence of the field lines. Analytical expressions for the dielectrophoretic force, which depends on the geometry of the used system, can be derived and thus the optical force (which is counterbalanced by the dielectrophoretic one) can also be calculated, in a fast and easy way. Additionally, the study of the dimensionless trapping efficiency is performed by changing the cell diameter and the numerical aperture of the objective lenses.

2.

Dielectrophoretic Forces

The term “dielectrophoresis” was first introduced by Pohl in 1958 to describe the forces induced by a nonuniform electric field on a small polarizable but uncharged particle.^{29, 30} Time-periodic inhomogeneous electric fields induce polarization and subsequent movement of dielectric particles. For a spherical particle of radius $r$ , suspended in a medium of electrical permittivity ${\epsilon }_m$ , the dipole moment $\mathbf{m}$ arising from the boundary charges is given from the equation:

3.

Materials and Methods

The optical trapping setup is schematically shown in Fig. 1 . We used an ${\mathrm{Ar}}^{+}$ laser emitting at $\lambda =488\mathrm{nm}$ , with a nearly Gaussian $2\text{-}\mathrm{mm}$ -diam beam. The laser beam was guided through a lens, with a focal length $f=100\mathrm{mm}$ . The lens was placed at the required distance allowing the beam to overfill the objective of a modified commercial microscope (Motic, B1 series), as overfilling of the objective is a very crucial parameter to achieve high trapping efficiency. The laser beam was tightly focused to a spot using either a $40\times$ long-distance working or a $100\times$ immersed oil objective lens, with numerical apertures of 0.65 or 1.30, respectively. The particles were observed through the same optical port that the beam enters the microscope objective, and thus it was necessary to insert a dichroic mirror between the eyepiece and the objective to inject the laser beam. The laser power that was entering the objective varied from $5\text{to}35\mathrm{mW}$ , with the objective loss being 10% loss in power. The sample image was observed through the same focusing objective with a CCD video camera on a personal computer and recorded for further analysis. A filter was placed between the eyepiece and the CCD camera to prevent the CCD array from damage. The particles we used for trapping were either yeast cells of diameters from $3.5\text{to}7.0\mu \mathrm{m}$ , washed and diluted in distilled water, or $8\text{-}\mu \mathrm{m}$ -diam polystyrene beads (Fluka Chemie), also diluted in distilled water. The diameter of the cells was measured with an error 5%, while that of the beads were known with an accuracy of about 1%. The particle diameters used in every case were in the range from $3.5\text{to}8\mu \mathrm{m}$ , which is larger than the laser wavelength used $(\sim 0.5\mu \mathrm{m})$ , so we are closer to the geometrical optics regime.

Fig. 1

Experimental setup for optical tweezers based on an ${\mathrm{Ar}}^{+}$ laser, emitting at $\lambda =488\mathrm{nm}$ .

The dielectrophoretic chamber consisted of two thin-film electrodes, of thickness $\sim 6000\mathrm{A}$ , placed at a distance of $100\mu \mathrm{m}$ , fabricated using conventional microfabrication processes (photolithography) on the top of a microscope slide (Fig. 2 ). This configuration was chosen as it is easy to construct and also facilitates the theoretical extraction and numerical calculation of the produced potential and electric field. Note that special care was taken to ensure the maximum possible electrodes uniformity and photolithographic quality, as established through optical examination microscopy of $100\times$ magnification. To this aim, the best set of electrodes between the several constructed was chosen to be employed in the experimental setup. A quality inspection picture of the electrodes is shown as an insert in Fig. 2. In most cases, the nonuniformity did not exceed $1∕10$ of the particle size. Despite the preceding, care was taken to trap the particles and repeat the measurements in the most uniform areas of the electrode spacing. Sine wave excitation of $1\mathrm{MHz}$ and $12\mathrm{V}$ was applied from a signal generator. Time-periodic inhomogeneous electric fields induce polarization and subsequent movement of the dielectric particles. Frequencies above $50\mathrm{kHz}$ ensure that distortions of the electrical double layer, induced at the boundary between the particle and the surrounding medium by the electric field, become negligible. The excitation at $1\mathrm{MHz}$ was selected as it is quite far from $50\mathrm{kHz}$ and additionally, as shown in Fig. 3 , in this area any possible frequency fluctuations do not affect the real part of the Clausius-Mossotti factor $\mathrm{Re}\left(f_{\mathrm{CM}}\right)$ , which is the frequency-dependent parameter of the dielectrophoretic force. In addition, the $1\text{-}\mathrm{MHz}$ frequency enables force measurements to be obtained at moderate voltages.

Fig. 2

Schematic presentation of the dielectrophoretic chamber setup and quality inspection picture of the electrodes through optical microscopy examination of $40\times$ magnification.

Fig. 3

Frequency dependence of the real part of the Clausius-Mossotti factor for a $6.0\text{-}\mu \mathrm{m}$ -diam yeast cell and for a polystyrene bead. In the inset, we can see the dependence for the case of the polystyrene bead in detail.

The resulting force on a spherical particle suspended in the medium, due to the dipole moment arising from the boundary charges, enables the particle to move toward regions of high- or low-field intensity, depending on the kind of dielectrophoresis we have. The dielectrophoretic force depends on the intensity of the electric field and the volume of the particle. In our setup, with the easy-to-fabricate $100\text{-}\mu \mathrm{m}$ gap between the two electrodes a voltage of maximum $10\mathrm{V}$ was required to move the particle.

To measure the optical trapping force, we used optical tweezers to trap the particle in the space between the two thin electrodes, when no voltage was applied. Then, by increasing the voltage, the particle experienced a dielectrophoretic force, which could be calculated from the determined electric field. The determination of the field between the two planar electrodes was made by using analytical methods based on Green’s theorem.³² Considering the boundary conditions of the electrodynamic problem, which describes our electrode configuration, we derived an expression for the electric potential and from that for the root mean square (rms) of the electric field:³³

Fig. 4

Field gradient in the $z$ plane for a voltage value of $5\mathrm{V}$ . For $z\sim 35\mu \mathrm{m}$ we can see the abnormality of the gradient, which creates the change in direction in the movement of the bead.

The experimental procedure was repeated over 500 times for different cells or beads and for each different numerical aperture of the microscope objective that was used, by increasing the laser power. For the electrical properties of the polystyrene the following values were used: ${\epsilon }_p=2.5{\epsilon }_0$ , for the dielectric permittivity, where ${\epsilon }_0$ is the dielectric permittivity of the vacuum space, and ${\sigma }_p=0.24\mathrm{mS}∕\mathrm{m}$ , for the electric conductivity. The case of the yeast cells is more complicated, as we have to deal with a biological particle that has a complex structure. Thus, we must consider different values for the electrical properties of the cell membrane and the cytoplasm and then to combine them, taking into account also the thickness of the membrane, to extract the proper value for the complex permittivity.³¹ The expression for a particle with shell becomes

A membrane thickness of $12\mathrm{nm}$ was considered, while for the values of the dielectric permittivity and the conductivity for the membrane and the cytoplasm, respectively, we have ${\epsilon }_{\mathrm{mb}}=11.3{\epsilon }_0$ , ${\sigma }_{\mathrm{mb}}={10}^{-3}\mathrm{mS}∕\mathrm{m}$ , and ${\epsilon }_c=50{\epsilon }_0$ , ${\sigma }_c=500\mathrm{mS}∕\mathrm{m}$ . Finally, the respective values for the distilled water are ${\epsilon }_{\mathrm{m}}=80{\epsilon }_0$ and ${\sigma }_{\mathrm{m}}=6.15\mathrm{mS}∕\mathrm{m}$ . The real part of the Clausius-Mossoti factor, which determines the kind of dielectrophoresis we have, for the case of the polystyrene beads is $\mathrm{Re}\left(f_{\mathrm{CM}}\right)=-0.47$ , which means that we have n-DEP, while for the yeast cells the mean value for different cell radii is about $\mathrm{Re}\left(f_{\mathrm{CM}}\right)=0.87$ , which means that in that case we have p-DEP.

4.

Results and Discussion

Trapping in three dimensions was achieved with both $40\times$ and $100\times$ objectives. Trapping was assured for a quite long distance on the $z$ axis, up to $35\mu \mathrm{m}$ . The system, of the dielectrophoresis calibration experiment, detects a possible misalignment of optical tweezers. When the laser beam was not very well focused, the particle was trapped at a relatively high distance on the $z$ axis. The field gradient shows a discontinuity for distances above $\sim 35\mu \mathrm{m}$ , as we can see in Fig. 4. In this case, the particle is trapped in such a high $z$ value that when the electric field is applied it starts moving upward at higher $z$ levels, as the axial trapping force is not sufficient to keep the particle well confined in the trap. In Fig. 5 we can see the movement of an $8\text{-}\mu \mathrm{m}$ -diam polystyrene bead in a misaligned optical tweezers, when the voltage is increased. At $t=0\mathrm{s}$ the bead experiences only the optical force. By increasing the voltage the bead is guided upward but remains trapped until $t=18\mathrm{s}$ . Then the dielectrophoretic force slightly exceeds the optical one and the bead starts moving to the middle area of the electrodes. After a few seconds, we can see that the bead changes direction and moves toward the electrode’s edge, while at the same time it keeps moving also upward (elevation being farther away from the focal point of the microscope objective). At this point, further elevation is due to the form of the field gradient in the middle between the two electrodes, where it has also a component in the $z$ direction and thus the force pushes the particle in a different direction. Note that in the area of the abnormality of the field gradient, the dielectrophoretic force shows also a discontinuity. We can thus conclude that it is better to work in relatively small $z$ distance $\left(5\text{to}25\mu \mathrm{m}\right)$ , where we can find an interval for the distance from the middle of the electrodes, where the dielectrophoretic force has a proper value.

Fig. 5

Influence of the misalignment on the optical tweezers performance. The frames show a polystyrene bead changing its direction of travel under the action of the dielectrophoretic effect, as explained in detail in the text.

For practical considerations, a dimensionless quality factor $Q$ was introduced by Ashkin to give the measure for the efficiency of the laser trapping: $Q=Fc∕P{n}_1$ , where $F$ is the trapping force, $P$ is the laser power, $c$ is the light velocity, and $n_1$ is the refractive index of the medium.³⁴

Figure 6 shows the transverse force against the laser power measured for polystyrene beads with radii of $8\mu \mathrm{m}$ , for the two microscope objectives used. The measured forces varied between $\sim 6$ and $40\mathrm{pN}$ when the laser power was increased from $5\text{to}32\mathrm{mW}$ . At this point, note that at low laser powers, the uncertainty in the estimation of the escape voltage is greater than at higher powers, so measurements at this region should be made very carefully. From the slopes of the linear approximation of the graphs the trapping efficiency was estimated. The efficiency measured for the $100\times$ objective is $Q=0.22\pm 0.03$ , while that for the $40\times$ objective is $Q=0.28\pm 0.03$ .

Fig. 6

Maximum transverse force determined through dielectrophoresis for $8\text{-}\mu \mathrm{m}$ -diam polystyrene beads, using the $40\times$ and the $100\times$ objectives.

Figure 7 presents the respective graph for the case where the setup was tested by using yeast cells, and Table 1 shows the values for the trapping efficiency $Q$ . Forces in the range from $0.5\text{to}38\mathrm{pN}$ were measured. In the case of the yeast cells, where a large number of measurements were performed, we had a larger variety of particle diameters in the range from $3.5\text{to}7.0\mu \mathrm{m}$ in each sample. However, as we had an uncertainty of 5% in measuring the diameter, the measured particles were studied in groups of those who had close diameters according to the estimated error and in the graphs we show only few representative values. The values that are mentioned in the table and the graphs are mean values for each group.

Fig. 7

Maximum transverse force determined through dielectrophoresis for yeast cells of several diameters, using the (a) $100\times$ and (b) $40\times$ objectives.

The efficiency values presented in this work are in the range of the $Q$ values reported by other researchers.^{34, 35, 36} Wright ³⁵ measured efficiency values, for different experimental setups, below 0.1, whereas Ashkin³⁴ reports trapping efficiencies up to 0.30. Fuhr ³⁶ report efficiencies of 0.077 for $4.9\text{-}\mu \mathrm{m}$ -diam beads, 0.096 for $6.4\mu \mathrm{m}$ , 0.122 for $7.8\mu \mathrm{m}$ , and 0.174 for $12\mu \mathrm{m}$ , using a setup with two quadrupole-electrode systems and also taking advantage of the dielectrophoretic effect. Generally, we measured greater efficiency values than Fuhr, but the trapping setup and the dielectrophoretic field cage are totally different, so the comparison cannot be direct. However, a common notice could be that in both works the $Q$ factor increases by increasing the particle diameter. From our results, it seems also that further increment of the particle diameter leads to a plateau. Fuhr also mentions that when a different objective was used, a possible misalignment led to nonlinear plots of the force versus power. By changing the objective in our setup, we did not noticed any important change in our measurements. Small deviations from linearity could be present in any case and that was obviously due to systematic errors.

In our results, we can also notice that trapping efficiencies obtained with the $40\times$ objective tend to be generally higher than those obtained with the $100\times$ . In the case of polystyrene beads this is obvious, while in the measurements performed with the yeast cells we can see (Fig. 8 ) that, apart from two or three cases, the general view that we get is the same. This result is in contradiction to what we would expect from the theory of optical tweezers. The use of a $100\times$ immersion oil microscope objective is a crucial factor to obtain a very well focused and powerful trap, due to its high numerical aperture. However, it is not the first time that a similar observation is made. In the literature someone can find both views. Wright, ³⁵ reports higher efficiencies for higher numerical aperture objectives, but of the same magnification. The work of Malagnino, ²⁴ measuring the trapping efficiency of a tweezers system using a diode laser, and the viscous drag Stoke’s forces as a calibration method, is closer to the work presented here, as they performed measurements with both $100\times$ and $40\times$ objectives of the same numerical aperture (NA) as that of the objectives that we used. This paper shows also that for particles with diameters greater than $\sim 4\mu \mathrm{m}$ , the $40\times$ objective was more efficient. The qualitative explanation that is given is based on the ray optics model. The trapping efficiency for particles larger than the beam waist is higher than for smaller particles because they take advantage of the whole beam. But, for the tightly focused beam, the smaller particles also experience the whole cross section of the more convergent rays. It is also mentioned that comparison with numerical calculations of the axial and transverse forces performed by Harada and Asakura³⁷ for a wide range of particle size between ${10}^{-3}$ and $10\mu \mathrm{m}$ , considering a wavelength of $514\mathrm{nm}$ , showed a qualitative agreement in terms of orders of magnitude with this result. In addition, Malagnino also noted that the $Q$ factors for both NAs approached a plateau at large particle sizes (larger than $6\mu \mathrm{m}$ diameter). In any case, the $Q$ values of that work, also, did not exceed the 0.1.

Fig. 8

Trapping efficiency versus cell diameter obtained with the $100\times$ and $40\times$ objectives.

Table 1

Trapping efficiency Q measured for the yeast cells.

Thus, it is obvious that there is a wide spectrum of experimental data concerning force calibration of optical tweezers. ^{26, 27, 34, 35, 36, 38, 39} This fact can be attributed to several technical characteristics, which diversifies every experimental setup used. The most important characteristic is the optical quality of the microscope objective lens, as immersed objectives are sensitive to spherical aberrations.^{35, 40} Since the optical forces cannot be calculated exactly, it is very important to find easy ways to measure them directly and calculate the efficiency of laser trapping. The method that we describe here is an electrically controlled repetitive noncontact method. The whole concept is very simple and the setup is easy to construct. The two thin film parallel electrodes are easy to manufacture and handle, because they are in the form of a microscope slide. Their simple geometry makes the whole procedure a lot easier by simplifying the analytical and numerical calculations. More sophisticated geometries can lead to more complicated analyses, and more mathematical concessions.

Moreover, it was mentioned³⁶ that thermal gradients, induced by the laser focus inside the cell, change its permittivity and conductivity. The calculation of these gradients is also possible in similar setups and is very important, as they contribute in increasing the electric field gradient and they introduce anisotropies, enabling the creation of space charges.

Finally, in our setup, low voltages and a frequency far above the charge relaxation time were used, as already mentioned. Thus, hydrodynamic effects near the electrodes of high field strength are restricted and thermal streaming or electrophoretic effects become negligible.

5.

Conclusions

An optical tweezers system based on an ${\mathrm{Ar}}^{+}$ laser emitting at $\lambda =488\mathrm{nm}$ was tested for two microscope objectives of different numerical aperture, namely, 0.65 ( $40\times$ long-distance working) and 1.30 ( $100\times$ immersion oil), by using $8.0\text{-}\mu \mathrm{m}$ -diam polystyrene beads and $3.5\text{to}7.0\text{-}\mu \mathrm{m}$ -diam yeast cells. Transverse optical forces were measured by taking advantage of the dielectrophoretic effect that appears when a dielectric particle is inserted in a nonuniform electric field. The dielectrophoretic force equals the optical one, which in this way can be easily calculated. The dielectrophoresis setup is simple in construction and easy to use, consisting of two thin film electrodes in the form of a microscope slide that simplifies analytical and numerical calculations of the electric field. To our knowledge this is the first time that so simple, but accurate, a dielectrophoretic chamber was used for optical tweezers calibration.

Forces in the range between $\sim 6$ and $40\mathrm{pN}$ were obtained for the $8.0\text{-}\mu \mathrm{m}$ -diam polystyrene beads and in the range between 0.5 and $38\mathrm{pN}$ for the yeast cells, by increasing the laser power from $5\text{to}32\mathrm{mW}$ . The measured trapping efficiencies are generally in the same range reported by other researchers.^{27, 34, 35, 36} We have measured $Q$ factors from 0.05 to 0.28, with the higher efficiencies achieved for the larger diameter particles. Comparing the trapping by two different NA lenses, we have noticed that, in the used diameter range, the $40\times$ objective generally was found more efficient. Finally, by increasing the particle diameter the $Q$ values showed a tendency to reach a plateau for both microscope objectives used.