Forces due to pulsed beams in optical tweezers: linear effects

Nathaniel du Preez-Wilkinson; Alexander B. Stilgoe; Thuraya Alzaidi; Halina Rubinsztein-Dunlop; Timo A. Nieminen

doi:10.1364/OE.23.007190

1. Introduction

Optical tweezers [1] has developed into a widely used and powerful quantitative tool in biophysics [2] and other fields of research. The combination of optical tweezers and pulsed laser beams offers the possiblity of extending the capabilities of optical trapping systems to include the use of non-linear effects. For example, a pulsed beam can be used to excite two-photon fluorescence [3], hyper-Raman, or hyper-Rayleigh effects [4]. Such systems can either combine a continuous wave (CW) beam for trapping with a pulsed beam for production of non-linear effects, or use the same pulsed beam for both trapping and non-linear effects.

It is known that trapping by pulsed beams can result in different forces compared with CW beams of the same average power [5–8]. In at least some of these cases, the difference is likely to be due to non-linear effects. However, we can also expect differences in the forces due to linear effects. It has been shown that in the Rayleigh [8, 9] and ray optics [10] regimes, the forces due to pulsed beams and CW of the same average power should be identical, if nonlinear effects can be neglected. In the Rayleigh limit, this follows from the identical scattering patterns of dipoles, and in the ray optics limit, from the wavelength-independence of scattering (neglecting dispersion). However, at intermediate sizes, the forces depend on the ratio of the size of the particle to the wavelength [11–13], and we expect difference forces due to pulsed and CW beams, even without non-linear effects. (See the Appendix for a discussion of why the time-averaged force differs.)

Therefore, it is of interest to determine the forces due to pulsed beams for wavelength-sized particles, and thus be able to evaluate the difference between the forces produced by pulsed and CW beams. First, we can answer the question of how well pulsed optical tweezers can be modelled by assuming the time-averaged forces are the same as for a CW beam of the same average power and beam shape. Assuming that any non-linear effects affect the trapping negligibly, if a pulsed laser is modelled as continuous, what error does that produce in calculated quantities of interest?

Second, if we observe a difference in the trapping of a particle by a pulsed beam compared with trapping by a CW beam, is the difference due to non-linear effects? If we know the difference expected due to linear effects, and the observed difference exceeds this, we can confidently attribute the difference to non-linear effects, or non-equivalence of the pulsed and CW beams.

2. Theory

If the effect of the particle on the incident field can be determined — that is, if the electromagnetic scattering problem can be solved — it is straightforward to determine the electromagnetic force acting on matter within a colume V, bounded by a surface S, by integration of the Maxwell stress tensor T:

F = \int_{S} T \cdot d A - ε μ \frac{d}{d t} \int_{V} S d V,

where ε and μ are the permittivity and permeability of the medium surrounding the particle, S is the Poynting vector, and A and dV are area and surface elements over which the integrations are performed. The momentum flux through the surface either increases the electromagnetic momentum within the volume (the last term in Eq. (1)), or is transferred to the particle, resulting in a force acting on the particle. The quantity of most interest is the impulse delivered by a pulse to the particle or, equivalently for a periodic sequence of pulses, the time-averaged force. This can be determined by integrating over a pulse, such that

\int_{t_{1}}^{t_{2}} F d t = \int_{t_{1}}^{t_{2}} \int_{S} T \cdot d A d t - ε μ \int_{V} S (t_{2}) d V + ε μ \int_{V} S (t_{1}) d V,

where t₁ is before the pulse, and t₂ after the pulse. If t₁ and t₂ are chosen so that the fields are negligible at these times, then the last two terms in Eq. (2) are zero. Alternatively, if t₂ − t₁ is the period of a sequence of pulses, then the sum of the last two terms is zero [14]. Therefore, it is sufficient to calculate

\int_{t_{1}}^{t_{2}} F d t = \int_{t_{1}}^{t_{2}} \int_{S} T \cdot d A d t .

It is prohibitively time-consuming to perform the surface integral numerically, when considering the repeated calculations required for modelling optical tweezers. For the trapping of a spherical particle by a CW beam, the best approach is to calculate the scattering problem using generalized Lorenz–Mie theory, which gives the fields as sums of vector spherical wavefunctions (VSWFs):

E_{inc} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} a_{n m} M_{n m}^{(3)} (k r) + b_{n m} N_{n m}^{(3)} (k r),

E_{scat} = \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} p_{n m} M_{n m}^{(1)} (k r) + q_{n m} N_{n m}^{(1)} (k r),

where

M_{n m}^{(3)} (k r)

and

N_{n m}^{(3)} (k r)

are the regular VSWFs,

M_{n m}^{(1)} (k r)

and

N_{n m}^{(1)} (k r)

are the outgoing VSWFs, and a_nm, b_nm, p_nm and q_nm are the beam shape coefficients [14–16].

The coefficients for the incident field are found [15, 17] and stored in a vector, a, containing all the a_nm and b_nm coefficients, up to some maximum n at which the calculation is truncated. The incident field coefficients can be related to the scattered field coefficients by a matrix, called the T-matrix [14, 18–20]:

p = T a,

where p is the vector of scattered field coefficients; and T is the T-matrix. In the event that the particle of interest is a sphere, the T-matrix becomes diagonal, and its entries are the Mie coefficients.

A surface can then be chosen in the far field, and the orthogononality of the VSWFs can be used to reduce the surface integral in (3) to a sum of products of the beam shape coefficients [17, 21, 22].

A similar procedure can be used for a pulsed beam. In this case, our fields are given by

E_{inc} = \sum_{j}^{N} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} a_{n m j} M_{n m}^{(3)} (k_{j} r) + b_{n m j} N_{n m}^{(3)} (k_{j} r)

and

E_{scat} = \sum_{j}^{N} \sum_{n = 1}^{\infty} \sum_{n = 1}^{\infty} \sum_{m = - n}^{n} p_{n m j} M_{n m}^{(1)} (k_{j} r) + q_{n m j} N_{n m}^{(1)} (k_{j} r),

where our VSWFs are now frequency-dependent since they are functions of the wavenumbers k_j. We can write the incident field coefficients as a_nmj = A_ja_nm and b_nmj = A_jb_nm, where A_j is the amplitude of the j-th frequency component. When we take the far field limit, we obtain a dependence on time and radial position of exp(ik_jr − iω_jt). Noting that the center of the incoming pulse will cross the far-field integration surface at some time in the past, t_in, we can rewrite this as exp(ik_jr − iω_j(t_in + Δt)), and since exp(ik_jr − iω_jt_in) = 0, we can perform the time integral in Eq. (3) from Δt = t₁ to Δt = t₂. Since the angular variation of the VSWFs does not depend on the frequency in the far field, we can combine the different frequency components into effective time-dependent amplitudes a′_nm and b′_nm for the incident field:

a_{n m}^{'} = a_{n m} \sum_{j = 0}^{N} A_{j} exp (- i ω_{j} t),

b_{n m}^{'} = b_{n m} \sum_{j = 0}^{N} A_{j} exp (- i ω_{j} t) .

For the scattered field, we have

p_{n m}^{'} = \sum_{j = 0}^{N} p_{n m j} exp (- i ω_{j} t),

q_{n m}^{'} = \sum_{j = 0}^{N} q_{n m j} exp (- i ω_{j} t) .

The usual analytical formula for the spatial integral [17,21,22] can then be used. This approach was tested by comparison with direct numerical integration of the spatial part of Eq. (3); the numerical integral converged to the result given by the effective time-dependent amplitudes as the number of spatial grid points was increased.

As is commonly done, we present the force in terms of the dimensionless force efficiency Q, which is related to the force by

F = \frac{n_{medium} P Q}{c},

where n_medium is the refractive index of the surrounding medium, and c is the speed of light in free space. If we use the time-average power for P in Eq. (13), we obtain the time-averaged force.

3. Computational parameters

We assume that the pulse has a Gaussian envelope, with a specified full-width half-maximum (FWHM). We assume that the pulse has a free-space wavelength of 790 nm, the beam is focussed with a convergence angle of 50° [15] (which corresponds to an optimally-filled objective of numerical aperture 1.02), and trapping takes place in water. We assume that the pulses are far enough apart in time so that they do not overlap. We tested the convergence of the time-averaged force as a function of pulse spacing and number of frequency components included; the results of this test are shown in Fig. 1. Based on these tests, we chose a pulse spacing of 500 fs and used 41 frequency components. Since our periodic sequence of pulses consists of the product of a carrier wave and the convolution of a Gaussian and a frequency comb, the spectrum is a comb, modulated by a Gaussian centered on the carrier frequency.

Fig. 1 Numerical convergence test for the calculated force for a pulsed beam as a function of pulse spacing and truncation frequency. The force acting on a particle at the focus is given in terms of the axial force efficiency. The asterisk shows the parameters used; this point is well within the region of good convergence. The smallest pulse spacing tested was 500 fs, with other pulse spacings being 500 fs times powers of two. The truncation frequency is given in terms of the 1/e half-width of the Gaussian envelope of the pulse.

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In addition, we tested whether it was necessary to include material dispersion. For common materials such as PMMA, polystyrene, and silica, the effect of dispersion is negligible, with variations in axial force efficiency Q_z of about 10⁻⁴ for polystyrene, and smaller than 0.5 × 10⁻⁴ for PMMA and silica, which is very small compared to |Q_z| which is about 0.1.

4. Results

Useful parameters of an optical trap to compare trapping by pulsed and CW beams are the equilibrium position of the trapped particle along the beam axis, and the radial spring constant. For both of these, we consider the time-average rather than the instantaneous values, since the time-average allows direct comparison with a CW beam. In addition, the time-averages of these quantities are both experimentally measurable and important (the instantaneous force can be important for some very specific applications where large forces are needed, such as attempting to unstick stuck particles [9]). We calculate the optical force as a function of position along the axis in order to determine the equilibrium position for a CW beam and a pulsed beam with pulses of FWHM 100 fs. The equilibrium positions for pulsed beams and CW beams is shown in Fig. 2, as a function of particle size and refractice index. In Fig. 2(c), we shown regions where the particle can be trapped with one type of beam, but not the other. In particular, for particular sizes of particles of relative refractive index of about 2, reflections from the front and back surface interfere destructively and backscattering is strongly reduced, allowing the gradient force to overcome the scattering force. Since the pulsed beam has a broad spectrum, this effect is reduced, and such high index particles cannot be trapped.

Fig. 2 (a) Axial equilibrium position of particle as a function of particle size and refractive index, for CW beam. The white region indicates that particles with the specified properties were untrappable. (b) Axial equilibrium position of particle as a function of particle size and refractive index, for pulsed beam. (c) Regions where particle can be trapped with one type of beam, but not the other.

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In order to see the quantitative results more clearly, we show in Fig. 3 the axial equilibrium positions and radial spring constants for three refractive indices typical of particles commonly trapped in optical tweezers: n = 1.45 (silica), n = 1.48 (PMMA), and n = 1.58 (polystyrene). These two quantities are of particular interest since they are relatively simple to measure experimentally. The CW trap shows the expected large variation with particle size [11–13]. This variation results from interference between light reflected (i.e., back-scattered) by the front and back surfaces of the trapped particle [11, 13]. When the light reflected by these surfaces interferes destructively, back-scattering is reduced, and hence the scattering force is smaller. Therefore, the particle is trapped closer to the focus. Where the interference is constructive, back-scattering is increased, the scattering force is larger, and the particle is trapped further from the focus. We can see a smiliar, but much smaller variation in the radial spring constant. This also results from the variation in reflectivity of the particle. In this case, only light that passes throug the particle contributes to the gradient force. Therefore, the gradient force is greater when the particle is less reflective, and smaller when the particle is more reflective. The variation is small, since the particle is not very reflective, even when the reflectivity is enhance by constructive interference.

Fig. 3 Equilibrium positions (left) and radial spring constants (right) for CW beams and pulses of different widths. The axial equilibrium positions and radial spring constants are shown for particles of refractive index (a)–(b) n = 1.45, (c)–(d) n = 1.48, and (e)–(f) n = 1.58.

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Variation in the equilibrium position is also present for the pulsed beams. For the longer pulses, the equilibrium positions are close to those for the CW beam, but the variation is smaller. For shorter pulses, the variation becomes smaller. The variation is also smaller for larger particles. This can be explained in terms of the width of the spectra of the pulsed beams and the variation of reflectivity with frequency. While we see in Fig. 3 the effects of the variation of reflectivity with particle size, the important parameter is the ratio of the particle size to the wavelength. Thus, similar variation will be seen if the particle size is held constant, and the frequency varied. A simple way to show the variation of reflectivity with frequency is to calculate the scattering force as a function of frequency. This can be done by calculating the force with the particle at z = 0, the focal plane. In Fig. 4(a), we show the variation of scattering force with frequency (green curve). We also show the frequencies of the spectral components in our calculations for the pulsed beam (blue circles), and the envelope of the spectrum (red curve). We can see that the pulse spectrum samples a small region of this variation with frequency. If the pulse is shorter, and the spectrum wider, the pulsed-beam force will approach the average over frequency, weighted by the spectral power, of this force. If the pulse becomes longer, the spectrum will be narrower, and the pulsed-beam force will approach the CW force. For pulsed beams, the force can be either higher or lower than the CW force, or can equal the CW force, depending on whether the peak of the pulse spectrum is located at a minimum or maximum of the force, or at an appropriate position in between. In Fig. 4(b), we show the effects of the width and position of the pulse spectrum; shifting the position of the pulse spectrum relative to the scattering force versus frequency curve is analogous to varying the particle size. These results are calculated for an idealised variation of force with frequency of sin². This idealised function was used for its perfectly periodic nature. As the width of the envelope of the spectrum increases, the variation with particle size becomes smaller. This is why the variation with particle size seen in Fig. 3 is smaller for shorter pulses. Since the variation of scattering force with frequency becomes more rapid with increasing particle size, we should see the variation with particle size also become smaller for larger particles. This can also be seen in Fig. 3.

Fig. 4 Variation of scattering force with frequency and effect on pulsed-beam force. (a) The variation with frequency of the scattering force due to a CW beam is shown (green). The frequencies of the spectral components of the pulsed beam are also shown (circles), and the envelope of the pulsed spectrum (red). The pulsed-beam force is approximated a weighted average of the CW curve, using this envelope as the weighting function at the frequency components. (b) The variation of the pulsed-beam force with position and width of the pulse spectrum; shifting the position of the pulse spectrum relative to the scattering force versus frequency curve is analogous to varying the particle size. This is calculated using an idealised sin² variation of scattering force with frequency, and pulses of the same duration, and half and double the durations of the pulse shown in (a). As the envelope of the spectrum broadens, the variation with particle size becomes smaller.

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While approximating the force as the weighted averages of the force due to individual frequency components provides a simple way to understand the origin of the effects discussed above, it must be kept in mind that this is an approximate model, and does not give the exact pulsed beam force. This is because the forces depend quadratically on the fields, rather than linearly; the forumulae for the forces depend on products of the VSWF coefficients of the fields [17, 21, 22].

In Fig. 5 we show the difference in equilibrium positions between the CW and pulsed beams of different pulse widths. Position differences of up to 100 nm are predicted for the 100 fs beam in Fig. 2, and up to 0.5μm for the shorter pulses shown in Fig. 5. We show differences in the radial spring constant in Fig. 6. The maximum difference is about 3% of the radial spring constant for the 100 fs pulses, and up to 8% for the shorter pulses.

Fig. 5 Difference in equilibrium position for pulses of different widths. The differences in equilibrium position between particles trapped in CW beams and pulsed beams are shown. The differences are shown for particles of refractive index (a) n = 1.45, (b) n = 1.48, and (c) n = 1.58. The RMS and maximum differences are shown in (d).

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Fig. 6 Difference in radial spring constant for pulses of different widths. The differences in time-averaged radial spring constant between particles trapped in CW beams and pulsed beams are shown. The absolute (left; (a), (c), (e)) and relative (right, (b), (d), (f)) differences are shown for particles of refractive index n = 1.45 ((a), (b)), n = 1.48 ((c), (d)), and n = 1.58 ((e), (f)). Legends are omitted where curves would be obscured, but note that legends for (a)–(f) are identical. The RMS and maximum differences are shown for (g) absolute and (h) relative differences.

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Another quantity of practical interest is the trap strength, which is the smallest maximum restoring force keeping the particle within the trap [14]. This weakest force usually occurs along the beam axis in the direction of propagation (“downstream” along the beam), since the axial gradient forces are usually weaker than the radial gradient forces and, in this direction, the scattering force acts to push the particle out of the trap. We show the trap strength for CW beams and pulsed beams of different pulse widths in Fig. 7. We also show the differences between the CW trap strength and pulsed beam trap strengths. The variation in scattering force with particle size results in a relatively large variation in the trap strength for higher refractive indices. This can result in differences in trap strength between the CW beam and short pulsed beams of about 10%. From Fig. 7(g), it can be seen that this results from the variation in trap strength with size being much smaller for a short pulsed beam than the CW beam. Thus, particles should be much more consistent in their trapping behaviour in short pulsed beams, with the strong size dependence seen with CW beams being greatly reduced.

Fig. 7 Trap strengths and differences in trap strength for pulses of different widths. The trap strength is the maximum axial reverse restoring force. We show the trap strength (left; (a), (c), (e)) and the differences in trap strength between CW beams and pulsed beams, for particles of refractive index n = 1.45 ((a), (b)), n = 1.48 ((c), (d)), and n = 1.58 ((e), (f)). The RMS and maximum differences are shown for (g) absolute and (h) relative differences.

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For completeness, we also show the axial spring constant differences in the axial spring constant between CW and pulsed beams in Fig. 8.

Fig. 8 Axial spring constants and differences in axial spring constant for pulses of different widths. We show the axial spring constant (left; (a), (c), (e)) and the differences in axial spring constant between CW beams and pulsed beams, for particles of refractive index n = 1.45 ((a), (b)), n = 1.48 ((c), (d)), and n = 1.58 ((e), (f)). The RMS and maximum differences are shown for (g) absolute and (h) relative differences.

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In Figs. 5 – 8, we have summarized the differences between the CW beam and the pulsed beams by giving the maximum and root mean square (RMS) differences between the CW results and the results for the various pulse widths. From these, we can estimate the error we would incur if we were to approximate the pulsed beam by the equivalent CW beam. The difference between the forces produced by 100 fs pulsed beams and CW beams is small, but not vanishingly small. If the difference does not introduce an error larger than can be tolerated, the CW beam is a sufficient approximation for the pulsed beam. If higher precision results are desired, or if the pulses are shorter, it is necessary to perform the calculation for a pulsed beam.

4.1. Approximation of time-averaged force as sum of spectral components

As noted above, it should be possible to obtain an approximate calculation of the time-averaged forces by calculating the force due to each spectral component, and adding these forces together. Finding the force due to each individual spectral component is a straightforward CW problem.

These can then be added to give the total force:

f = \sum_{j}^{N} {| A_{j} |}^{2} f_{j},

where f_j is the force due to the j-th spectral component, calculated from the incident and scattered field amplitudes a_nmj, b_nmj, p_nmj, and q_nmj. The results are shown in Fig. 9. For the shortest pulses considered here (25 fs), the error in equilibrium position is smaller than an order or magnitude, compared with approximating the pulsed beam by a single CW beam, and the error in radial spring constant smaller by a factor of about 2 for refractive index n = 1.58. However, for the longer pulses, this weighted average of CW forces performs worse than approximating the pulsed beam by a single CW beam. Thus, this method may be of use for ultra-short pulses, but it is of limited practicality. It is worth noting that this weighted CW approach is not much faster than our pulsed method, but it is conceptually easier.

Fig. 9 Error in equilibrium position and radial spring constant due to approximating force of pulsed beam as sum of forces of spectral components. (a) Error in equilibrium position. (b) Absolute error in radial spring constant. (c) Relative error in spring constant. The RMS and maximum differences between the pulsed beam calculation and the sum of forces of spectral components are shown.

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This behavior, where this method performs worse as an approximation for longer pulses, results from number of significant spectral components. For a shorter pulse, many spectral components are important, and there will be a large number of cross-terms of their amplitudes when products of the fields are taken. On average, the sum of these cross-terms will tend to zero. For example, if we wished to calculate (a + b + c + d + e)², for random values of a, b, c, d, and e, our terms a², b², etc. will all be positive, but our cross-terms ab, ac, bc, etc., will have random signs, and can be expected to partially cancel. Thus, (a + b + c + d + e)² ≈ a² + b² + c² + d² + e² can be a reasonable approximation. With fewer significant spectral components for longer pulses, there is less opportunity for cross-terms to cancel, so the approximation is worse—if we approximate (a + b)² ≈ a² + b², nothing will cancel our cross-term ab, and the approximation will be poor.

What we are missing in this simple approximation is information about the phase of the spectral components. Clearly, from the results above, the phase matters. This suggests that it is important to consider the effects of dispersion, since dispersion can alter the relative phase of the spectral components.

4.2. Pulse stretching due to dispersion

In practice, it is difficult to know the width of the pulse at the focus, because we can expect stretching of the pulse due to dispersion in the objective, and in the rest of the optical system. While the width of the pulse is known at the laser, the width of the pulse at the focus can be quite different. Clearly, if the pulse is stretched by dispersion, there will be a large effect on the instantaneous force acting on a particle in the trap, since the instantaneous power will be reduced. It is much less clear what the effect will be on the time-averaged force.

Therefore, we investigated the effect of stretching by dispersion. The effect of dispersion on the spectrum is a change in the relative phase of the spectral components. A linear change in the relative phase, proportional to ω₀ − ω_j, where ω₀ is the carrier frequency and ω_j the frequency of the j-th spectral component, will shift the pulse temporally. While this will change the force as a function of time, it will have no effect on the time-averaged forces. Therefore, we are interested in phase shifts proportional to the square, and higher powers, of ω₀ − ω_j. We modelled the effect of dispersion on the spectrum as a phase shift Δϕ_j = α(ω₀ − ω_j)² of the j-th spectral component, for a quadrative phase shift, and Δϕ_j = β(ω₀ − ω_j)³ for a cubic phase shift. The quadratic phase shift, which corresponds to linear dispersion, stretches the envelope of the pulse, but leaves the shape of the envelope otherwise unchanged. The cubic phase shift will distort the shape of the envelope. The value of α was chosen to give the desired FWHM of the stretched pulse, and the value of β to give the same spectral power weighted phase shift, i.e., so that ∑_j |A_j|² Δ_j was the same for both the quadratic and cubic phase shifts. We began with the spectrum of an unstretched pulse with a FWHM of 25 fs, and calculated the forces, equilibrium positions and radial spring constants for various degrees of stretching.

The results are shown in Fig. 10. It is clear that stretching of the pulse by linear dispersion has almost no effect on the equilibrium position or the time-averaged forces. In theory, stretching by linear dispersion should have no effect, and the very small difference in Fig. 10(b) can be attributed to numerical error. Distortion of the pulse does result in larger changes, but in the case of cubic phase shift considered here, the change is well under 1%. This difference is many orders of magnitude larger than that for linear dispersion, and we can safely state that it is not due to numerical error.

Fig. 10 Difference in equilibrium position and radial spring constant for pulses stretched by dispersion. The differences between the initial unstretched and undistorted pulse, of pulse width 25 fs, and the pulses stretched or distorted by dispersion are shown. We show the differences in equilibrium position (left) and radial spring constant (right) for quadratic phase shifts (i.e., linear dispersion), (top, (a) and (b)), and cubic phase shifts (bottom, (c) and (d)).

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This has three important consequences. First, we can ignore the effect of dispersion when estimating the likely error resulting from approximating the pulsed beam by a CW beam. Second, if we decide that we need to calculate the forces using the pulsed beam, we do not need to know how much the pulse is stretched by dispersion.

Third, this provides a useful probe for nonlinear effects. If the pulse is stretched by dispersion, the instantaneous power will change, and hence the time-averaged nonlinear effects will change, while the time-averaged linear effects remain almost exactly the same.

5. Conclusion

We have introduced a method to calculate the forces due to linear effects of pulsed-beam optical tweezers. This method has been verified against a numerical integration of the Maxwell stress tensor. This method can be used to detect the presence of non-linear effects in experimental results. Our pulsed beam method can also be used to calculate the fields as a function of time, using Eqs. (7) and (8), allowing for approximate calculation of weak non-linear effects, or as a baseline for full calculation of non-linear effects. E.g., it can also be used to determine where non-linear effects (such as two-photon fluorescence) are likely to occur if the main contribution to the energy density is linear. Finally, this method can be used as a basis for testing potential future methods that include the non-linear effects in the force calculation.

Our results show the limits of using a CW beam of the pulsed-beam carrier frequency as an approximation for finding the time-averaged forces due to a pulsed beam. For pulses of FWHM 100 fs, a pulsed beam can be approximated as a continuous wave beam to within 3%. The results in Fig. 5 and Fig. 6 can be used to determine whether the CW beam should be an adequate approximation.

We have also shown why the time-averaged forces depend on pulse duration. As noted above, the time-averaged force can be closely approximated by a weighted average of CW forces for each spectral component. Since the force (especially the scattering force) varies with frequency, this weighted average depends on the pulse spectral width and position relative to this variation with frequency. As the pulse becomes longer, the spectral width narrows, and this weight average approaches a delta-function sampling of the variation with frequency. As the particle becomes shorter, the spectral width increases, and the pulsed beam force approaches an average over frequencies of the force versus frequency curve.

As the particle becomes larger, the period of the variation of force with frequency decreases, giving the same effect as shortening of the pulses. Noting that the average over frequencies for large particles is the ray optics result for the force [12], we can see that the pulsed beam force approaches the ray optics force for large particles, as expected [10]. For Rayleigh particles, the variation of force with frequency becomes featureless, and, again, the average over frequencies approaches the CW force. Thus, for very small and very large particles, the pulsed beam and CW beam give the same time-averaged force, assuming only linear effects. For wavelength-scale particles, the forces can differ significantly. This can be important if we are trying to detect the presence of non-linear effects. If observed differences exceed the predicted differences shown above, it can be concluded that either non-linear effects are present, or the pulsed and CW beams being compared are not equivalent (e.g., different beam profiles).

If one requires the force as a function of time, rather than the time-averaged force, then the pulsed beam calculation is required.

Finally, we can briefly consider some experimental reports in light of our results. Where pulses lengths have been changed through dispersion [3], no effect of stretching of the pulses was seen; our results agree with this observation. Where nanoparticles, or other small particles are trapped [7, 8], linear effects on the forces due to a pulsed beam instead of a CW beam, all else being the same, are very small. surroundings are usually small. In such cases, large observed differences are certainly not due to linear effects, and if it is shown that the beam profile is unchanged as the beam is switched between pulsed and CW, it is reasonable to assert that nonlinear effects are responsible. If the proportional difference between the CW and pulsed cases is power-dependent, then the effect is clearly nonlinear.

Experiments where larger objects are trapped often use biological specimens [5,6]. For example, Im et al. [6] trapped human red blood cells, and compared 200 fs pulses with the equivalent CW beam, and found axial trap strengths increased by 10–20%. As the linear differences are expected to be below 1% for such pulse lengths, nonlinear effects appear to be responsible for the observed differences. Im et al. believed that differences in the focal spot due to self-focusing caused the difference, i.e., nonlinear effects due to the surrounding medium rather than the trapped particle. Similarly, Mao et al. [5] observed differences in radial trap strength exceeding 10%. They also demonstrated power-dependence of the relative difference, so, again, nonlinear effects appear likely. Since they observed nonlinear effects in the trapped particle (two-photon fluorescence), this is a safe conclusion.

It is also possible to encounter cases where the difference is qualitative rather than quantitative, for both linear and nonlinear effects. That is, where the particle can be trapped using a pulsed beam, but not with a CW beam, or the other way around. This is most likely for high-index particles, where reflection can make trapping difficult, or traps where aberration makes trapping difficult. Since the effect of the pulsed beam is to smooth out the variation in reflectivity with particle size, whether or not an individual particle can be trapped will depend on the type of beam (and the particle size). If the average particle can be trapped, and only the more reflective particles cannot be trapped, then trapping can be easier with the pulsed beam. If the average particle cannot be trapped, and only the least reflective particle trapped in the CW beam, then it is likely that no particles will be trapped in the pulsed beam.

6. Appendix: Why the mean optical force depends on the spectrum

If we represent a periodic series of pulses as a sum of N frequency components of amplitude A_j, the time-averaged energy is simply

P = \sum_{j}^{N} {| A_{j} |}^{2}

(or including a normalisation constant, depending on the units of A_j). Since there is no fundamental reason why all of the spectral components need to have the same spatial distribution, we can represent the spatial variation as a sum of spatial modes, such as vector spherical wave-functions (VSWFs) as we have done in this paper. The VSWFs are orthogonal with respect to power, and the time-averaged power is

P = \sum_{j}^{N} \sum_{n}^{\infty} \sum_{m = - n}^{n} {| A_{j} |}^{2} ({| a_{m n} |}^{2} + {| b_{n m} |}^{2}) .

Notably, the time-averaged power does not depend on the complex phase of A_j, but only on the magnitudes |A_j|. If this were also the case for the time-averaged momentum flux, the approximation tested in section 4.1, Eq. (14), would be exact, rather than an approximation. Therefore, a brief discussion of why the momentum flux does depend on the phase of the amplitudes A_j is warranted.

For a ray of light (or, equivalently, a parallel beam or a section of a plane wave), the magnitude |p| of the momentum flux p is proportional to the energy flux:

| p | = n_{medium} P / c,

where n_medium is the refractive index of the medium, and c is the speed of light in free space. However, it must be recognized that this is a special case. More generally, we cannot say that the momentum flux is proportional to the energy flux, since the former is a vector quantity and the latter is a scalar quantity. In the case of a ray, the direction of propagation of the ray provides a suitable direction for the vector, but in the general case, there will not be a unique direction in which energy travels. For example, if we consider a spherical wave, energy is moving outwards in almost all directions. It is this directional distribution of energy that determines the momentum flux. This directional distribution of energy depends on the phases of the spatial amplitudes a_nm and b_nm, and hence on the products A_ja_nm and A_jb_nm. The energy flux does not depend on these phases, since the energy flux doesn’t depend on the directional distribution of the energy. If we consider a focused Gaussian beam, we might have a momentum flux of about 0.6n_mediumP/c in the direction of propagation, which is less than the momentum flux for a ray or plane wave given in Eq. (17), since the energy is moving in a range of directions, rather than a single direction. By appropriately changing the phase of the amplitudes a_nm and b_nm of the spatial modes, without changing their magnitudes, we can reverse the direction of the beam. This produces a change of 1.2n_mediumP/c in the momentum flux, while leaving the energy flux unchanged.

We can demonstrate the dependence of the directional distribution of the energy on the phases of the amplitudes by direct calculation. In Fig. 11, we show the intensity in different directions over time, for scattering of 25 fs pulses, spaced 50 fs apart, by a 3s μm diameter sphere of refractive index 1.58, in water, located at 0.2 μm past the focus, illuminated by a 790 nm beam with a convergence angle of 50°. This beam can be approximated by three frequency components, of wavelengths 750 nm, 790 nm, and 834 nm. 75% of the power is in the central frequency component. If we change the phase of the high frequency sideband (750 nm) by a quarter-wave. we can see that the directional distribution changes (compare dashed and solid lines in Fig. 11).

Fig. 11 Change in time-averaged angular distribution of scattered light. Solid blue lines show the scattered light from the original beam, and dashed red lines with the upper sideband phase-shifted by a quarter-wave. Note that the forward scattering is almost the same in both cases (b), but the difference in backscattering is clearly visible (c).

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The difference in the angular distribution of the radiation due to the phase change of one frequency component is small. In this case, the relative difference in axial force is about 5%. This is similar to the error in approximating the time-averaged force by the sum of forces due to individual spectral components, shown in Fig. 9. Where phase shifts are smaller, such as when we show the effects of dispersion on the force in Fig. 10, the difference in force due to the phase shift is smaller. While this difference of 5% is small, it is much larger than our numerical error, which can be estimated from Fig. 10(b), which should be zero but shows relative differences approaching 10⁻⁹.

Acknowledgments

This research was supported under Australian Research Council’s Discovery Projects funding scheme (project number DP1095880).

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Forces due to pulsed beams in optical tweezers: linear effects

Abstract

1. Introduction

2. Theory

3. Computational parameters

4. Results

4.1. Approximation of time-averaged force as sum of spectral components

4.2. Pulse stretching due to dispersion

5. Conclusion

6. Appendix: Why the mean optical force depends on the spectrum

Acknowledgments

References and links

Cited By

Figures (11)

Equations (17)

Optics Express