Influence of experimental parameters on the laser heating of an optical trap

In optical tweezers, heating of the sample due to absorption of the laser light is a major concern as temperature plays an important role at microscopic scale. A popular rule of thumb is to consider that, at the typical wavelength of 1064 nm, the focused laser induces a heating rate of B = 1 °C/100 mW. We analysed this effect under different routine experimental conditions and found a remarkable variability in the temperature increase. Importantly, we determined that temperature can easily rise by as much as 4 °C at a relatively low power of 100 mW, for dielectric, non-absorbing particles with certain sets of specific, but common, parameters. Heating was determined from measurements of light momentum changes under drag forces at different powers, which proved to provide precise and robust results in watery buffers. We contrasted the experiments with computer simulations and obtained good agreement. These results suggest that this remarkable heating could be responsible for changes in the sample under study and could lead to serious damage of live specimens. It is therefore advisable to determine the temperature increase in each specific experiment and avoid the use of a universal rule that could inadvertently lead to critical changes in the sample.


Relation between power spectrum and light momentum trap calibration with temperature
In previous work 16 , we showed the validity of interpreting, under certain strict conditions, back focal plane (BFP) interferometry signals as measurements of light momentum changes; that is, as direct readings of the trapping force. The most significant requirements for such an interpretation are: 1) the use of a high-NA, aplanatic collecting lens that captures all the light from the optical traps; and 2) to track the light intensity distribution at the BFP of the collecting lens with a position sensitive detector (PSD). The usual approach to measuring forces in optical tweezers consists of calibrating the trap stiffness κ (pN/µm), in accordance with: F = -κ·x, and the position sensitivity β (µm/V), such that: x = β·S x (where S x is the sensor positional voltage signal). In particular, we proved that if the aforementioned conditions hold, the product α trap = κ ·β (pN/V) is invariant and equal to the constant and permanent momentum calibration of the sensor, α detector = R D /ψf'c, where R D is the detector radius, f' and ψ (V/W) are the focal length and the responsivity of the instrument, respectively, and c the speed of light: In contrast to α trap , which must be calibrated, for example using the power spectrum method (Supplementary Ref. 1), α detector is obtained from first principles and is determined by the optical parameters of the beam detection system alone. This suggests that, if the required instrument design conditions are fulfilled, no new in situ trap calibration is necessary when the experiment changes 15,16 and force can be directly obtained as: F x,y = -α detector ·S x,y . The momentum calibration of the force sensor, α detector , is independent of the geometry of the trapped object and of the structure of the trapping beam 17 ; moreover, and importantly for the purpose of this paper, it is not dependent on laser power or chamber temperature. Figure S1 | Discrepancies between α detector and α trap =κ·β, obtained from the power spectrum analysis, provide evidence of change in the sample when the laser power is increased. When the temperature increase is considered in the power spectrum (PS) fitting (Supplementary ref. 1), the calibration is compensated and α trap is constant.
In Fig. S1, we used the discrepancy between the two schemes to determine changes in sample temperature. A close look at the results for α trap = κ ·β reveals that the equivalence α trap = α detector starts to fail when the laser power is increased, with α trap deviating from the constant value R D /ψf'c. This is indicative of a local temperature increase due to laser absorption (of 4 ºC/100 mW in Fig. S1), which leads to incorrect κ and β calibration if overlooked. As discussed by Peterman et al. 9 , temperature affects the power spectra of optically trapped microspheres, both as a thermal variable governing Brownian motion and through the viscosity of the solvent, which importantly is dependent on it. We studied the transfer function of the control electronics and found that, even at low frequencies (1-10 Hz), the amplitude of the output voltage sent to the stage differed by ~10%-20% from the input signal (see Fig. S3a). The ratio of the two values was quite independent of the amplitude, but decreased with frequency following a singlepole-like function with a roll-off frequency of 20 Hz. The stage velocity therefore had to be corrected to take into account the deformation introduced in the triangular signals by the low-pass filtering of the electronics (see  To do this, we correlated the monitored output voltage with the triangular input signal and we found that only in a region of 40%-80% of the semi-period (shadowed area) was the velocity of the stage constant. Moreover, the actual velocity was larger than the theoretical value and it depended on the amplitude, A, and frequency, f, of the signal (see Fig. S3b (inset) and Table S1). All measurements were kept below 6 -7 Hz to ensure a constant velocity time frame. In Eq. 1 (see Main Text), we used this calibrated velocity value for the calculation of the viscosity change.
Furthermore, we analysed the variability of the measured viscosity with the stage velocity and found no significant change, which indicated that the dissipation of heat was faster than the motion of the fluid, so the temperature "experienced" by the particle was constant (Fig. S4).  (Table S1). The solid red line and the grey shadow are the mean and standard deviation of the quotient η/η 0 .
The oscillation parameters were chosen so that they produced similar drag forces on the microspheres used, which had different radii and were given by their corresponding manufacturers (Table S1). The diameter of the smallest microspheres (0.61 µm) was also confirmed using dynamic light scattering (DLS). Finally, we checked the long-term stability of our fibre laser output power. We observed large oscillations (10%-20%) for long periods of time (see Fig. S5), whose origin was the fluctuation in polarization of the beam. These oscillations vanished when no polarizing elements were introduced along the optical path, or by orientating the beam polarization parallel to the transmission axis of the polarizing beam splitter.

Laser heating measurements in assorted optical tweezers laboratories
A large number of studies have been carried out to assess the phenomenon of sample warming due to infrared laser absorption in optical tweezers. In Fig. S6 and Table S2, we indicate some of the reported results that represent the state of the art in this matter. It is of special significance that the strategies undertaken are based on a wide variety of principles. First, several heating studies have been based on the temperature dependence of the buffer viscosity (either water or glycerol), which can be determined with a microsphere trapped in the optical trap through direct drag force measurements based on the trapping light momentum 12 , power spectrum calibration 9 or active-passive calibration 11 . Second, one can find studies consisting of sample thermometry through thermally-dependent fluorescence performed on different optically-trapped samples 2,3,4 . Thirdly, experiments on empty traps based on different approaches have also been undertaken 5,6,7,8 .

Heat transport simulations
We made use of the MathWorks Partial Differential Equation (PDE) Toolbox to simulate the heating of the sample due to a laser trap. Given that the geometry we used exhibits cylindrical symmetry, we adapted the heat equation to include the Jacobian along the radial component and solved the problem on a 2D surface. As mentioned in the Main Text, the modelling adopted by Peterman et al. of B(z) 9 , which was conceived in spherical geometry in which the trap is created at r = 0 and the ΔT = 0 condition is fixed at r = z, exhibits a nonstopping increase (Fig. S7a). In contrast, the choice of cylindrical symmetry and the Dirichlet boundary conditions at two parallel surfaces corresponding to the coverslips yields a constant B value after an abrupt rise over the first 10 µm (Fig. S7b). This is especially evident in the real sink case with thermal conductivity K glass (Fig. S7c), due to the coverslip only being capable of cooling the sample sufficiently when the trap is placed very close to the interface. In Fig. S7d and Table S3, we show two models in the literature that describe the radial temperature profile with similar accuracy. The model of Celliers et al. 8 was simulated in the same cylindrical geometry and coincided closely with our simulations, with a slightly greater temperature increase, ΔT(ρ), for the NA = 1.3 objective than for the NA = 1.2 objective. The analytical expression provided by Mao et al. 12 also coincides with our measurements and exhibits an even greater difference between the two objectives.
; > tan !!  Finally, we simulated the effect of reducing the effective NA of the trap (Fig. 2d and Fig. S7e). Although expressed in terms of the trap power, the heating rate, B (ºC/100 mW), is more related to the local irradiance, which eventually explains the observed variation. Irradiance is contained in the shape of the heating source, q(r), in models of Peterman et al. 9 and Celliers et al. 8 , as well in the R factor in that of Mao et al 12 .
Models in refs. 8 and 12 seem to capture the main behaviour of the heating dependence on NA eff for the 3.00-µm beads. As compared with the experimental measurements, they reveal an ascending pattern that can be directly connected to the wider light cone illuminated, i.e. wider heat source. For the smaller, 0.61-µm beads, one could think of the smaller beam waist created with higher NA eff to conclude that the therefore higher irradiance leads to greater heating as well. However, as mentioned in Results (see Main Text), the optical field at the bead-medium interface is here bound to a number of aberrations and deviates from an ideal Gaussian shape, which leads to the measurements notably deviating from the simulations. Besides, our reducing NA eff by means of a diaphragm at the back of the trapping objective leads to a different overfilling, thereby producing higher variations in the local optical field.