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* Division of Physics and Astronomy, Vrije Universiteit, Amsterdam, The Netherlands, and Department of Physics, Washington State University, Pullman, Washington 99164-2814, USA
Correspondence: Address reprint requests to Erwin J. G. Peterman, Division of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081 HV, Amsterdam, The Netherlands. Tel.: +31-20-4447987; Fax: +31-20-4447899; E-mail: erwinp{at}nat.vu.nl.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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; Mehta et al., 1999; Molloy and Padgett, 2002; Svoboda and Block, 1994) and other fields (Ashkin, 1997; MacKintosh and Schmidt, 1999). For example, the force generation by molecular motors (Finer et al., 1994; Svoboda et al., 1993) has been studied as well as the mechanical properties of biopolymers (Kellermayer et al., 1997; Smith et al., 1996; Tskhovrebova et al., 1997). In a typical optical trapping experiment a near-infrared laser beam is focused close to the diffraction limit using a high-power, high-numerical-aperture microscope objective lens (Ashkin, 1997). The laser beam exerts a force on micrometer-sized particles with a refractive index larger than that of the surrounding medium, such that there is an effective potential minimum near the focus. The focus therefore serves as a three-dimensional trap for the particle, and forces of tens of piconewtons can be readily exerted on micrometer-sized polystyrene or silica particles in an aqueous medium. In order to obtain these forces, laser powers of typically tens to hundreds of milliwatts are used, leading to focal intensities exceeding MW/cm2 (for comparison the intensity of bright sunlight on the surface of the earth is on the order of 100 mW/cm2). The potential of thermal and nonthermal damage caused by these high intensities to (biological) samples has been a matter of concern and investigation (Ashkin et al., 1987; Liu et al., 1995, 1996; Neuman et al., 1999). One way of reducing nonthermal photodamage to many biological materials is to use near-infrared lasers (such as Nd:YAG, Nd:YLF, diode- or Ti:Sapphire lasers) rather than visible lasers. Using a vitality assay measuring the rotation rate of trapped bacteria, it was recently shown that single-photon-induced processes involving the presence of oxygen are responsible for the remaining nonthermal photo damage by near-infrared laser light (Neuman et al., 1999). The temperature increase due to trapping a particle in water has been roughly estimated to be rather low, i.e., on the order of 1 K/W (Block, 1990). At an air-water interface, Wurlitzer et al. measured 
5 K/W heating in the focus (Wurlitzer et al., 2001). In trapped human sperm cells, hamster ovary cells, and liposomes the temperature increase was measured to be on the order of 10°C/W, 11.5°C/W, and 14.5°C/W respectively (Liu et al., 1995, 1996). These temperature changes were measured by observing phase transitions in lipid monolayers and bilayers respectively. The temperature increase was described by both groups with a simple model involving heat generation by absorption of light in the focus and subsequent heat dissipation to the bulk solution. A more elaborate model was presented by Schönle and Hell (1998) without accompanying experimental data. In their model heat generation by absorption and conduction in the whole light cone is taken into account. However, they do not obtain steady-state solutions, but rather focus on short-time effect with an eye on short-pulse applications. Here we report a more direct method to observe temperature changes in a focused laser beam which is based on the analysis of the thermal motion of a trapped bead. Trapping a polystyrene or silica bead is commonly used in optical tweezers experiments. In addition, we will present a model that takes into account more accurately the entire spatial profile of the focused beam in a low numerical aperture (NA) approximation. By comparing data taken in water and glycerol, we show that light absorption by and dissipation in the solvent is the primary determinant of the temperature change, rather than heat absorbed by the trapped particle. The (cooling) effect of the sample cell wall is part of the model and is also demonstrated in the experiments. Presenting data for two cases of solvents with rather different heat conductivities demonstrates the applicability of the model. The model can be used for all other cases (with not too high absorption) as long as heat absorption and conductivity are known, and the results are largely independent of trapped particle properties, provided again that the absorption is not too high. Goals of the paper are to present a correct physical picture of heating in optical tweezers and to provide a practical model for experimenters to use to calculate potential heating effects in their particular situations.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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). In short, it consists of a custom-built inverted microscope, capable of trans-illumination differential interference contrast microscopy and optical trapping with near-infrared light from a diode-pumped Nd:YVO4 laser (1064 nm, Topaz 106C, Spectra Physics, Mountain View, CA).
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To visualize the beads, the sample was illuminated with the 546-nm line of an Hg arc lamp (100 W, Zeiss). The transmitted light was collected with an objective (Zeiss, NeoFluar 100x, oil immersion, NA 1.3) and imaged onto a camera (VT1000, Dage-MTI).
The laser-beam path consisted of two 3x beam expanders, a combination of a 
/2 plate and a Glan-laser polarizer (to adjust laser intensity), a 1:1 telescope (to position the trap focus in x-, y-, and z-directions), and beam steering mirrors. The laser beam was focused in the sample by the objective. The near-infrared and visible beam paths were separated using dichroic mirrors. Transmitted laser light was collected by the condenser (Zeiss, oil immersion, NA 1.4). Displacement of a bead trapped in the laser focus was measured in two dimensions, normal to the optical axis of the microscope (x, y) by imaging the back-focal plane of the condenser onto a quadrant photodiode (SPOT-9DMI, UDT Sensors) (Gittes and Schmidt, 1998a). The signals of this photodiode were amplified and anti-alias filtered using custom-built electronics, digitized with a PC board (AT-MIO16X, National Instruments or AD16/ChicoPlus, Innovative Integration), and further processed with custom software (LabView, National Instruments).
To exactly determine the laser intensity in the objective focus, the transmission of the objective was measured by replacing the condenser with an identical objective and measuring the transmitted light. We measured the transmission of this particular objective at 1064 nm as 62 ± 2%, in agreement with values published by others for the same brand and type of objective (Liu et al., 1995; Neuman et al., 1999; Svoboda and Block, 1994).
Optical absorption of water and glycerol
The optical absorption of glycerol and water at 1064 nm was determined by measuring the transmitted intensity of a laser beam passing through a cuvette filled with a variable path length of the respective liquid. The cuvette was made from an acrylic glass cylinder, oriented vertically, open on the top and closed with a cover glass at the bottom. Liquid was pumped in with a syringe connected to the side of the cylinder, close to the bottom. Another coverslip was placed inside the cylinder floating on top of the liquid layer, in order to obtain a flat meniscus. The water and glycerol where filtered through a 0.2-µm syringe filter and care was taken to avoid air bubbles. The height of the liquid column was measured with a precision ruler. Laser light (from a diode-pumped Nd:YVO4; Topaz 106C, Spectra Physics, Mountain View, CA) was sent through the cuvette and the transmitted light intensity was measured with a laser power meter (Newport 1815-C with detector head 818T-10) as a function of the variable liquid path length in the cuvette as shown in Fig. 2. The extinction coefficients could be extracted from these data by fitting a single exponential function. The extinction coefficients were found to be 14.2 m-1 for water (similar to the 12 m-1 obtained from the graph in Hale and Querry (1973) and 21.4 m-1 for glycerol.
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, Clarkson University, NY) were used. Unless stated differently, single beads were trapped 10 µm above the coverslip-liquid interface to avoid thermal and hydrodynamic surface effects. The effect of heating in the focus of the trapping beam was measured in two ways. First, power spectra of the Brownian motion of a bead in the trap were measured as function of the laser power, and second, displacements of trapped beads were measured upon exertion of viscous drag (by moving the sample chamber using the piezo-actuated stage), also as a function of laser power.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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),
 | (1) |
= 3
x 
x d is the Stokes' drag of the bead with diameter d in a solvent with viscosity , kB is the Boltzmann constant, T is the absolute temperature, 
is the trap stiffness, and f0 is the corner frequency of the spectrum. Some typical power spectra of a 502-nm-diameter polystyrene bead in glycerol are shown in Fig. 3. If the laser power only affected the trap stiffness (see Eq. 1), the corner frequency f0 would increase with increasing power. For high frequencies, the f02 term in the denominator of Eq. 1 becomes negligible, and the spectral density would become independent of trap stiffness and thus independent of laser power. If, on the other hand, the temperature, T, in the vicinity of the bead were to change with power, the spectral density would change also at high frequencies. Such an effect is evident in the data (Fig. 3) from the fact that the slanting parts of the spectra (f > f0) do not overlap.
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; Weast, 1973). In order to quantify these effects, power spectra, like the ones in Fig. 3, were fitted by Eq. 1. Examples of such fits are shown in Fig. 3.
From Eq. 1, the laser–power dependence of the two fitted parameters (f0 and S0 x f02) can be expressed as
 | (2) |
 | (3) |
) and glycerol (Daubert and Danner, 1989) are:
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The power dependence of P/f0 and 1/(S0 x f02) for a polystyrene bead with diameter 502 nm in glycerol and a silica bead with diameter 444 nm in water are shown in Fig. 4. Also shown are fits of the data by Eqs. 2 and 3. The fit results for 502-nm (the experiments shown in Figs. 2–4), 840-nm, and 2200-nm polystyrene beads in glycerol and 444-nm (the experiments shown in Fig. 2) and 500-nm silica beads in water are listed in Table 1. These experiments indicate that in the case of glycerol the temperature increase is 41.1 ± 0.7 K/W laser light for the 502-nm beads, 43.8 ± 2.2 K/W for the 840-nm beads, and 34.2 ± 0.1 K/W for the 2200-nm beads. There is only a small effect coming from the size of the trapped bead. In water the temperature increase is substantially smaller than in glycerol, namely 7.7 ± 1.2 K/W for 500-nm beads and 8.1 ± 2.1 K/W for 444-nm beads. This is due to the lower absorption of 1064-nm light by water and the higher thermal conductivity of water (see below).
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A value for the temperature coefficient B in glycerol was obtained in an independent way (for a distance of the bead from the coverslip of 10 µm). The displacement of a trapped bead with respect to the center of the trap as a function of the laser power was measured while moving the sample stage (thus the surrounding glycerol) back and forth with constant speed. This displacement (
x) is proportional to the force exerted on the bead and is equal to
 | (4) |
 | (5) |
Displacements (multiplied with the laser power) as a function of laser power are shown in Fig. 5 C. Also shown is a fit of the data by Eq. 5. From this fit a temperature factor of 36.6 ± 2.0 K/W was obtained (see Table 1). For the 444-nm beads a temperature factor of 39.2 ± 2.2 K/W was measured. The values obtained for the laser-induced temperature increase obtained from the fluid-drag experiments are in agreement with those obtained from the power spectra (Table 1).
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THEORETICAL MODEL |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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 | (6) |
is the extinction coefficient of the solvent, defined by I(x) = I0 x e-x.
The heat generated by the absorption will dissipate. The dissipating heat flow 
obeys the local differential equation
 | (7) |
1.4 W/(m x K); Weast, 1973) and 
is the temperature deviation from ambient temperature due to heating at position 
. In steady state the heat dissipated is equal to the heat generated: 
J = 
Q/t. Using Eqs. (6) and (7), this becomes:
 | (8) |
, a model for the intensity profile of the laser is needed. Here a more accurate approximation of the intensity profile is considered than in a model published before (Liu et al., 1995). Suppose a laser of total power W0 is brought to a focus at r = 0 (using spherical coordinates r, 
, and 
, with = 0 along the incoming optical axis). To describe the beam intensity I, suppose that
 | (9) |
< /2, outgoing for > /2). At high NA, f() could be quite a broad function of angle. At large distances, this incoming or outgoing intensity falls off as an inverse-square law,
 | (10) |
) is normalized over incoming solid angle,
 | (11) |
=sindd. (The attenuation in Eq. 6 is assumed to be small, so that the outgoing power has been set equal to the incoming power.) To clarify Eq. 9, we derive the function f() and the constant a for the case of a Gaussian beam, which is a low NA approximation for a focused laser. Manipulating well-known expressions (Siegman, 1986), one can write the far-field intensity distribution for a Gaussian beam in the form,
 | (12) |
0 is the equivalent angular radius of a uniformly illuminated aperture with the same total power. In terms of 0 one finds that the intensity of the Gaussian beam through the focus (along the = 0 axis) has the form
 | (13) |
/202. The NA in this approximation is just 0 
sin 0, so a plausible extrapolation to NA 1 gives a = /2 in Eq. 9.
We are interested in solving Eq. 8 for the temperature in the focus, T(0). The Green's function for Eq. 8 satisfies
 | (14) |
Here, 
is the Dirac delta function. The Green's function with its argument ' set to zero and obeying the boundary condition G = 0 at some large radial distance R, is
 | (15) |
The G = 0 boundary condition might represent, for example, distant glass surfaces of higher heat conductivity (and lower attenuation). In terms of G(r), the solution T(r) of Eq. 8 at the origin is
 | (16) |
V = r2 dr d,
 | (17) |
/2 to get
 | (18) |
= 1064 nm), close to the experimentally determined values. It is important to note that the experimentally found difference between glycerol and water is reproduced by the model, indicating that indeed the determining parameters for the heat generation are absorption of light, as governed by the extinction coefficients, , and dissipation of the heat, as governed by the thermal conductivity, C. It should be noted that the dependence of the temperature difference on R is not very strong: if R was 100 µm, the temperature increase would be 21 K/W for water and 66 K/W for glycerol. Our experiments show that, indeed, the temperature increase is larger if the distance to the cover glass is larger. In general, in diffusion problems the shortest length scale dominates the effect, which is reflected in the logarithmic distance dependence. In reality the cutoff (i.e., the glass coverslip) is close only on one side. To obtain readily solvable equations, a symmetric cutoff R was assumed (as if the bead were in the center of a chamber, with varying thickness 2R). In Fig. 4 E the data is compared to the distance dependence of Eq. 8. Within the error margins of the experiment this simplified model describes the data well without any fitting parameters.
In the preceding discussion we ignored the influence of the trapped particle. To obtain a more accurate description of the heating effect, a correction term can be added to Eq. 18 that incorporates the difference in absorption and thermal conductivity of the bead compared to the solvent. We represent these effects as an effective change in the coefficient b over the volume of the bead:
 | (19) |
). For polystyrene we estimate the absorption factor to be 6 m-1 (Lytle et al., 1979), and the thermal conductivity is 0.12 W/m K (C.J.M. Lassence on http://www. electronics-cooling.com/html/2001_may_techdata.html). This means that the correction term to the heating coefficient (calculated with Eq. 19) due to a 500-nm silica bead in water is -2.2 K/W (i.e., 11–19% of that calculated with Eq. 18, assuming an R cutoff of 10 µm). For the polystyrene beads in glycerol the correction term is from -7.9 K/W (i.e., 21% of the uncorrected value for 1094-nm beads and -2.6 K/W (i.e., 7% of the uncorrected value) for the 502-nm beads. This shows that the effect of the trapped particle is relatively small. In our calculations it produces less of an effect than the choice of the R cutoff. Note that the calculation predicts that the heating in the vicinity of small (502-nm-diameter) polystyrene particles is larger than that around the larger (2200-nm) ones, and likewise for silica, inasmuch as absorption in the particles is weaker than in water or glycerol. Qualitatively, this is observed in the experiments.
Our experiments on fixed, trapped beads gave the same results as the drag experiments (in which the bead was held in place and the solvent was moved) suggesting that the heat equilibration is faster than the movement in these experiments. To estimate time scales for the equilibration time for any given temperature distribution, we need not only C (with units J/sxmxK) but also the heat capacity per volume cV (for water 4.2x106 J/m3xK, for glycerol 3.0x106 J/m3xK). On dimensional grounds, the equilibration time to a given distance r must be on the order of r2xcV/C. Thus, for both water and glycerol, temperature equilibration out to 10 µm takes roughly 1 ms. The time-dependent heating model by Schönle and Hell (1998) does not apply to our situation, because they assume an infinite sample size, which causes the infinite equilibration times they report. Temperature kinetics should thus not be relevant even if the solvent is moved fairly quickly. In our drag experiments the sample is moved with a speed of 910 nm/s, slow enough to allow temperature equilibration.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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This value for the laser-induced heating is comparable to the values measured before (5 K/W, Wurlitzer et al., 2001; 10–15 K/W; Liu et al., 1995, 1996). We think that our measurements are likely to be more accurate since well-defined solid particles (polystyrene, silica) were trapped, in contrast to some of the previous experiments where vesicles and cells were used. We have shown both experimentally and in our model that the effect of the trapped (polystyrene, silica) particles is only minor; the main contribution to the heating is light absorption by the surrounding solvent. The distance from the glass-solvent interface, however, has a substantial effect on the heating. We have shown that the heating increases substantially when the bead is trapped further away from the interface.
It remains to be discussed for which kind of experiments these results are relevant. Most optical trapping experiments use water (or watery solutions) as solvent and laser powers on the order of 100 mW (at 1064 nm). This leads to a temperature increase of only 0.8 K in the focus, which, depending on the solvent, may change the viscosity more dramatically. In many cases the Lorentzian fit to a power spectrum of a trapped bead is used for the calibration of the trap and detector response (Gittes and Schmidt, 1998b). If the heating effect is not taken into consideration using this calibration method, the trap stiffness—which is proportional to the estimated viscosity () times the measured corner frequency (f0)—is overestimated 2% (10%) when a laser power of 100 mW (500 mW) is used. Here we assume a temperature increase of 8 K/W and a base temperature of 294.55 K. The detector response (in m/V) is proportional to the temperature (T) divided by the viscosity () and the zero-frequency intercept of the power spectrum (S0 f02) and is in the same circumstances underestimated 2% and (11%). Consequently, heating effects due to laser-light absorption by the solvent in optical trapping experiments even in watery solution have a small but measurable effect, and should be taken into consideration, especially when laser powers higher than 100 mW are used.
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ACKNOWLEDGEMENTS |
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TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSTHEORETICAL MODELDISCUSSIONACKNOWLEDGEMENTSREFERENCES |
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, Clarkson University, New York, for the supply of silica particles. This research was supported by the Foundation for Fundamental Research on Matter (FOM). E.P. is supported by a Postdocs Universitaire Loopbaan Stimulerings Programma (PULS)-fellowship from the Research Council for Earth and Life Sciences (ALW) with financial aid from the Dutch Organization for Scientific Research (NWO).
Submitted on November 7, 2001; accepted for publication October 16, 2002.
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REFERENCES |
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Ashkin, A. 1997. Optical trapping and manipulation of neutral particles using lasers. Proc. Natl. Acad. Sci. USA. 94:4853–4860.
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Block, S. M. 1990. Optical tweezers: a new tool for biophysics. In Noninvasive Techniques in Cell Biology. S. Grinstein, K. Foskett, editors. Wiley-Liss, New York, New York. 375–401.
Daubert, T. E., and R. P. Danner. 1989. Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation. Hemisphere Pub. Corp., New York, New York.
Finer, J. T., R. M. Simmons, and J. A. Spudich. 1994. Single Myosin molecule mechanics—piconewton forces and nanometer steps. Nature. 368:113–119.[Medline]
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Gittes, F., and C. F. Schmidt. 1998a. Signals and noise in micromechanical measurements. Methods Cell Biol. 55:129–156.[Medline]
Gittes, F., and C. F. Schmidt. 1998b. Thermal noise limitations on micromechanical experiments. Eur. Biophys. J. Biophys. Lett. 27:75–81.
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