We calculate trapping forces, trap stiffness and interference effects for spherical particles in optical tweezers using electromagnetic theory. We show the dependence of these on relative refractive index and particle size. We investigate resonance effects, especially in high refractive index particles where interference effects are expected to be strongest. We also show how these simulations can be used to assist in the optimal design of traps.
© 2008 Optical Society of America
While a large part of previous theoretical work on optical tweezers has made use of approximate methods such as Rayleigh scattering , suitable for small particles (radius<0.1λ), or geometric optics , suitable for large particles (radius > 5–10λ), neither of these are suitable for describing the trapping of particles of sizes on the order of the wavelength. However, an analytical solution to scattering of light by a sphere of arbitrary size—and hence the optical trapping of spherical particles—exists [3, 4], and the computational power needed to extract numerical results from the theory is now readily and cheaply available.
We use a computational implementation of Lorenz–Mie theory, in the form of an optical tweezers toolbox , to investigate the forces in optical tweezers as a function of the most important parameters for optical trapping: the size of the particle, the relative refractive index of the particle, and the numerical aperture of the microscope objective focussing the trapping beam. This reveals interesting physical effects, such as resonances in high refractive index particles, which determine whether or not the particle can be trapped.
While computational modelling has been used in the past to investigate specific cases of optical trapping, a broad overview of the performance of optical tweezers, such as we present here, does not appear to have been previously performed. Our results will assist in the selection of particles for particular applications in optical tweezers, and are useful for the optimization of optical traps.
2. Calculating optical forces
Optical trapping can be simply understood in terms of conservation of momentum. Forces on dielectric particles illuminated by a focussed beam occur when the momentum flux of the incident beam is changed by the particle. If the beam is deflected away from the original beam axis, a transverse force results; this force is responsible for trapping in the radial direction. Forces in the axial direction result, firstly from backscattering, which pushes the particle in the direction of propagation of the beam (the so-called “scattering force”), and secondly, if the divergence or convergence of the beam is changed. If a particle causes the beam to diverge more strongly, then the axial momentum flux is reduced relative to that of the incident beam. If the divergence or convergence is decreased, the momentum flux is increased, and reverse restoring forces can occur . This reverse restoring force is necessary in order to overcome the scattering force before trapping can occur.
If the sum of all forces on the particle at a particular location are zero and when displaced from this position a force acts upon the particle to restore it to equilibrium, then the particle can be considered trapped. As a typical Gaussian trapping beam will have a symmetric scattering profile for radial displacement away from the beam axis for spherical particles, it is the axial force that is most important for determining if a particle is trapped.
Finding the force acting upon a particle of a particular size as a function of position within the trap gives us all the information needed to determine if the particle is trapped, the maximum reverse axial force, which characterizes the strength of the trap, and the trap stiffness. The trap stiffness is of particular interest as it allows for the measurement of force through particle displacement within a trap . The relationship between these quantities and the axial and radial force acting on a particle are shown in Fig. 1.
Performing the calculations on a range of particles (0–3λ in radius, with 1–2 relative refractive index) in water we obtain a data set that can be presented in the form of a 2D intensity landscape or 3D surface. The calculations were performed over a spatial grid dependent on particle size, relative refractive index and numerical aperture of the modeled system. The dynamic grid size allows a coarse grid to be generated while still capturing significant features of a particle’s force profile. Significant features in a particle force profile include the maximum and minimum force and the equilibrium position of the particle in the beam.
To compensate for the discrete nature of the grid, spline interpolants were used at the maximum, minimum and equilibrium position. Splines have an advantage over the calculation of forces over fine grids in that the algorithm to produce a spline is much quicker than calculating the force over a finer spatial grid. The error in the dimensionless force efficiency Q(z) introduced by this is ΔQ = 10-5, well below the limits of practical experimental investigation. For example, the uncertainty in force would be 6.7×10−18N for a 200mW laser, as compared with a typical force of 4×10−11N.
3. Optical trapping landscapes
In Fig. 2(a), (b) and (c), we show the dependence of the trap strength (the maximum reverse axial force) on numerical aperture. Here, we calculate forces for water immersion objectives with numerical apertures of 1, 1.2 and 1.3. The trapping landscapes consist of two types of trapping regions, which are dependent on refractive index and particle radius. The first region is a range of refractive indices and particle sizes that always experience a positive restoring force within a trapping beam. Particles with relative refractive indices ranging from about 1.05 to 1.3–1.5 (the upper limit depends on the numerical aperture; here we have an upper limit of about 1.3 for N.A. = 1.0, 1.4 for N.A. = 1.2, and 1.5 for N.A. = 1.5) can be trapped for particle radii from the Rayleigh range well into the geometric optics regime. There is a limit to the smallest trappable particle as Brownian motion dominates the dynamics of small particles, while the trapping forces on Rayleigh particles are proportional to their volume .
For higher relative refractive indices (above 1.3–1.5), we see the second type of trapping regime where only particles of specific sizes can be trapped. The trapping landscapes show a quasi-periodic variation between trapping and non-trapping, with the alternation occurring over a very small change in particle size (approximately 0.1λ). The alternating trapping to non-trapping regions create the ability to select particles of particular sizes to trap.
Looking at the differences between the figures in more detail we find that trapping regions in high numerical aperture optical tweezers are far more accessible. High numerical aperture objectives are an optimal choice for optical tweezers as they have a larger standard trapping region for medium relative refractive indices (1.05–1.5 for N.A. of 1.3), and larger selected-size trapping regions for higher relative refractive indices.
To quantify the effect of numerical aperture on overall trapping strength we calculated the maximum trap strengths (i.e., the maximum values in trapping landscapes such as those shown in Fig. 2) restoring force over numerical apertures of 0.6–1.33 in water, which are shown in Fig. 3. Figure 3(a) shows the variation in the maximum trap strength for a given numerical aperture. The overall maximum trap strength acting upon a particle increases non-linearly with the numerical aperture. The data gives a clear illustration of why choosing high numerical apertures close to the limit imposed by the medium’s refractive index (which is the maximum value for illumination from a hemisphere) is more effective than picking even slightly smaller numerical apertures. Each data point in Fig. 3(a) corresponds to a relative refractive index and particle size of a maximally trapped particle shown in Fig. 3(b) (i.e., the location of the maximum value in the trapping landscape). Over the range of values that were calculated, the refractive index and particle radius scale quadratically.
Choosing two cases of relative refractive index corresponding to polystyrene (n rel = 1.19 in water) and diamond (nrel = 1.82) we can look at two different types of effects present in the low and high refractive index regimes. In Fig. 4(a) the trapping of a polystyrene microsphere, commonly used in optical trapping, can be seen over a range of numerical apertures and sizes. The alternating trapping/non-trapping regions in Fig. 2 can be seen more clearly for the two particles in Fig. 4. For polystyrene we see that the weakening and strengthening of trapping force is like an oscillation around a monotonic function. Trapped diamond has a much stronger dependence on the particle size due to resonances resulting from its high refractive index. In the Rayleigh regime, the optical force scales with the cube of radius, and in the geometric optics regime there is no dependence of trapping force on particle size. In both cases the force scales with the cube of radius in the small particle (<0.5λ) regime. However, as the large particle regime is approached (≥3λ) only the polystyrene has reduced dependence on particle size, consistent with a geometric optics interpretation.
In Fig. 4(a), the dependence of trap strength on particle size becomes more significant in the large particle regime as the numerical aperture of the optical tweezers is increased. With a high numerical aperture, such as N.A. = 1.3, the trap strength continues to increase with particle size at radii at which the trap strength of the lower numerical aperture trap (N.A. = 1) has ceased to increase.
Trapping in Fig. 4(b) of the diamond particle is very sensitive to particle size. On the graph it is seen as a rapid oscillation from trapping to non-trapping when the particle size is changed by less than a tenth of a wavelength. As the particle size is increased, the amplitude of the oscillation decreases due to a larger number of modes being being able to exist within the particle. Given that the refractive index of the media, numerical aperture or particle size can be chosen then it is conceivably possible to trap anywhere within regions of relative refractive index and particle size defined in Fig. 2.
4. Interference and resonances in high refractive index particles
In high contrast particles (nrel > 1.5), finger-like trapping regions in the particle trapping landscape are seen for high numerical aperture tweezers. Trapping of high refractive index particles is due to an interference-mediated enhancement (see Fig. 2). There are two factors to consider: the first is the high contrast nature of the trapped medium, which results in a high reflectivity.
The second is that the reflectivity depends on the size of the particle, and if it is low, the “scattering” force will not overcome the gradient force. In our simulations the force is the difference in momentum flux between the incoming and outgoing light. To show the differences in trapped and non-trapped particles we can graphically compare the radiation patterns of the incoming and outgoing light. The radiation patterns for a particular pair of similarly sized particles is shown in Fig. 5. For the non-trapped particle, there is a significant back-scattered component while for the trapped one there is not. Back-scattered momentum components in Fig. 5 occur due to constructive interference between reflections from the top and bottom surfaces of the particle. Thus, high refractive index particles are trapped due to a simultaneous reduction of back scattered light, resulting in a reduced scattering force, and increase of forward scattered light, giving rise to an enhanced gradient force. In our calculations we use the size of the particle as a parameter, and as we vary this parameter our calculation of unnormalized momentum in the far field shows a number of momentum maxima, increasing with the particle size.
Another way the effect of interference can be observed is through the extinction efficiency (i.e., the extinction cross-section, which is the sum of the scattering and absorption cross-sections, divided by the geometric cross-section of the sphere) . Intuitively one would expect that the Mie interference structure would in some sense correspond to the force on a trapped particle, although we can expect variation due to the non-plane-wave illumination. In Fig. 6(a) and (b) we can see that resonances in the Mie coefficients correspond to interference effects changing the trapping properties of an nrel = 1.82 particle. In Fig. 6(a), the extinction cross section for particles of 0 to 3λ can be seen. For small particles the extinction efficiency curve is devoid of sharp, fine structure, due to the limited number of modes able to exist within the particle. As we increase particle size, the fine features become more dependent on particle size. Following the trapping force profile along with the extinction efficiency we see that, qualitatively, they have coinciding features. The Mie coefficients are produced using Lorenz–Mie theory, so to a particular order, contain all possible transverse modes. As a result the extinction cross section does not take into account any focusing of the incident radiation as focusing selects the contribution from each mode to produce its structure. As a result the extinction cross-section and trapping force profile have some features in common, but they don’t necessarily exactly correspond. Mie resonances do not necessarily show whether constructive interference is in forward or backscattered light, so they do not indicate which resonance corresponds to trapping.
5. Absorbing particles
To make our predictions more accurately model some actual experimental observations, we need to account for the effect of absorption within trapped particles. For the high refractive index case (nrel = 1.82) we have added the effect of absorption to the refractive index. Figures 7(a)–(d) show the effect of the imaginary part of the refractive index, representing absorption in the medium, on the trapping force. It is clear from the figure that the addition of absorption does not significantly affect the occurrence of trapping enhancement. It does, however, affect whether or not the enhancement will cause a particle to be trapped. The trapping is not significantly affected until the absorption is on the order of n im = 0.01. The high absorption regime of nim = 0.1 completely overcomes restoring forces outside the Rayleigh regime.
6. Trap stiffness
Measurements of Brownian motion are commonly used to calibrate optical tweezers for for the measurement of piconewton forces in microscopic systems . Accurate measurement of force in optical tweezers requires sensitive position measurement and high trap stiffness to reduce Brownian motion.
Figure 8 shows the trap stiffnesses corresponding to trapping for particular numerical apertures in Fig. 2. In all graphs displayed in Fig. 8, the highest stiffness occurs in particles approximately half a wavelength in radius. At larger particle sizes, the trap stiffness gradually falls off. Regions shown without a trap stiffness in this instance denote regions where the particle cannot be trapped. Interference effects lead to large variations in trap stiffness over small differences in particle radius or relative refractive index. The trap stiffnesses shown here tells us what particle sizes are appropriate for particular situations and measurements.
Small particles (<0.5λ) in addition to strong stiffnesses also have trap stiffnesses which remain constant over the span of the particle radius as they act more like Rayleigh scatterers. For large particles (> 3λ) with low refractive indices, the particle experiences a weak trap that as a rule of thumb is linear only over half a radius from the equilibrium point. Figure 9 shows three different force profiles where we can see the departure from trap linearity.
7. Forces away from the beam axis
Due to the symmetry of beam profile and spherical particles many experiments use the axis transverse to beam propagation to make measurements as the response should be the same in both directions along an axis. However, we need to consider the effect on the trapping position in the direction of the beam when moving the particle away from the axis. If the particle’s equilibrium position changes, then so too must the trapping force the particle experiences along a transverse axis. Figure 10 shows the effect on the axial equilibrium position of a 1λ radius particle of varying relative refractive indices for a given radial displacement in a NA = 1.3 system. As we are using a highly focused beam in our calculations we have a small beam waist, approaching the 0.5λ limit. We can see from Fig. 10 that the axial trapping position remains fairly constant for all refractive indices over the beam waist. If a force measurement acts within this region a simple spring approximation is sufficient to obtain accurate results. If a high refractive index particle is moved radially away from the beam waist, the axial position where the axial forces are in equilibrium rapdily moves further from the focal plane—the axial position of such particles will change quickly if transverse (non-optical) forces act to move the particle off-axis. This effect is much smaller for lower index particles. In the case of n rel = 1.5 stable trapping can only be achieved when the particle falls into the trap from above the focus, as it is repelled if approaching from below. Our results indicate that if transverse forces are to be calculated computationally, a simple translation away from the axis is not always sufficient to describe the properties of trapping—the new axial equilibrium position needs to be re-calculated.
We have shown that even in a relatively simple case of trapping, such as the optical trapping of spheres in optical tweezers, there are many interesting physical effects that are not commonly considered. We have examined the variation of forces and trapping as a function of particle size and refractive index, and the numerical aperture of the focusing lens. We have also included the effects of absorption. Corresponding with other studies, we find that the strongest traps are created for particles of radius close to half a wavelength of the trapping beam within the suspending medium. We have also shown the possibility of trapping high-index particles of specific sizes, with relative refractive indices of over 1.5.
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