Theoretical investigation on nonlinear optical effects in laser trapping of dielectric nanoparticles with ultrafast pulsed excitation

Anita Devi; Arijit K. De

doi:10.1364/OE.24.021485

1. Introduction

In spite of its vast application in measuring nanoscale mechanical forces driving fundamental life processes in nature [1–5], laser tweezers (i.e. gradient optical trap created by a tightly focused laser beam with Gaussian transverse intensity profile [6, 7]) so far have very limited application in trapping and manipulation of dielectric nanoparticles [8–11] compared with metallic nanoparticles [12–16]. Direct optical trapping of a single macromolecule (as opposed to widely used indirect optical trapping of a micron-sized dielectric particle tethered to the single molecule [1–5]) is yet to be achieved.

In recent years the role of low-power high-repetition-rate ultrafast pulsed excitation on trapping efficiency of larger dielectric particles with diameter ≥ 0.5µm has been investigated [17–23] and it has been demonstrated that such excitation results in stable trapping of individual dielectric nanoparticles with diameter ≤ 0.1µm (which is confirmed by sensitive measurement of precise step-wise rise in two-photon fluorescence and/or scattering signal), for example, trapping of 100nm diameter latex nanospheres [24,25] and 10-20nm diameter Cd-Se quantum dots [25,26]. The high pulse peak power leads to a fleeting but steep trapping potential required to circumvent the difficulties associated with small dimension of nanoparticles leading to more erratic Brownian motion as well as with low polarizability of dielectrics (compared with metals) whereas the high pulse repetition-rate restricts diffusion of the trapped particle out of focus during the dead-time between consecutive pulses [24–26]. A phenomenological description has been put forward by one of us which captures the underlying physics of the origin of enhanced trapping efficiency with such excitation [26]: Stable trapping is facilitated due to repetitive instantaneous momentum transfer by a train of pulses with typically 100fs pulse-width at 100MHz repetition-rate (i.e. with 10ns time lag between consecutive pulses) but the trapped particle’s inertial time (which decreases with decreasing particle size) dictates the response of the particle to an instantaneous momentum transfer. A smaller particle with diameter ≤ 0.1µm can respond to this impulsive force. However, for a larger particle with diameter ≥ 0.5µm the typical inertial time is several tens of nano-seconds; therefore, 1) the particle always responds to cumulative momentum transfer by several pulses in the pulse train such that the trapping efficiencies turn out to be the same under pulsed and continuous-wave (CW) excitation [17,22] and 2) even after chirping the pulse-width remains too small to have any effect such that the trapping efficiency turns out to be independent of femto- to pico-second pulse-chirping [18].

In spite of the growing body of work along this direction, nonlinear optical effects on trapping efficiency under pulsed excitation has hitherto been explored in detail although these effects are non-negligible for such high peak-power pulsed excitation combined with tight-focusing condition. In this paper, we theoretically show that for a dipolar particle (having diameter much less that the wavelength of the trapping beam, i.e. $d ≪ λ$ ) optical Kerr effect always has a stabilizing effect on the radial component of trapping potential but it significantly modulates the axial component. At lower power level the fine balance between the gradient force and the scattering force along axial direction renders the trap progressively more stable upon increasing the power; however, above a critical power level the trend is reversed leading to an unstable trap along axial direction. With increase in power the trapping potential along axial direction becomes more asymmetric and in order to quantify the trapping efficiency the appropriate quantity is the height of the potential barrier along beam propagation direction and not the absolute depth of the trapping potential which is contrary to the common understanding in exsisting literature regarding the stability of an optical trap. We also discuss under what optimal excitation parameters the dipole trap is most stable.

Finally, although the explicit nature of the force (and potential) differs with particle size the essential physics remains unchanged and, our model nicely explains few puzzling observations that were found experimentally: 1) for larger colloidal particles (diameter ≥ 0.5µm), if we keep on increasing the power under pulsed excitation trapping is destabilized around few tens of milli-Watt power level (the particle first moves laterally toward trap center and then is ejected out of the trap axially, as observed by video microscopy) but this is not observed under CW excitation at similar power level [18,22,23] and 2) 100nm polystyrene beads were reported to be optically trapped for few seconds under pulsed excitation at power levels slightly over 30mW but no stable trapping was observed at higher power level [24,25].

2. General discussion

In this paper, we discuss the trapping of polystyrene nanoparticles in water. For a focused Gaussian beam, the intensity is given by [27]:

I (r, z) = (\frac{2 P}{π w_{0}^{2}}) \frac{1}{1 + {(2 \tilde{z})}^{2}} \exp [- \frac{2 {\tilde{r}}^{2}}{1 + {(2 \tilde{z})}^{2}}]

where P is the average power (for CW excitation) or the peak power (for pulsed excitation),

w_{0} = \frac{0.82 * λ}{N A}

is the spot-size (i.e. Gaussian beam radius at focus) with

N A

the numerical aperture and the reduced coordinates are given by:

\tilde{r} = \frac{r}{w_{0}}, \tilde{z} = \frac{z}{k * w_{0}^{2}}

. Under tight focusing, the radial component of the total force is contributed only by (radial component of) the gradient force while the axial component of the total force is contributed by both (axial component of) the gradient force as well as the scattering force [6,7]. We compute the magnitudes of the total force around the geometrical focus (

r = z = 0

) which, for dipolar particles (

d ≪ λ

), can be analytically expressed as [27]:

\begin{matrix} F_{r a d i a l} (r; z = 0) = - \frac{2 π n^{w} a^{3}}{c} (\frac{m^{2} - 1}{m^{2} + 2}) \frac{4 \tilde{r}}{w_{0}} (\frac{P}{π w_{0}^{2}}) \exp [- 2 ({\tilde{r}}^{2})] \\ F_{a x i a l, g r a d i e n t} (z; r = 0) = - \frac{2 π n^{w} a^{3}}{c} (\frac{m^{2} - 1}{m^{2} + 2}) \frac{8 \tilde{z} / (k w_{0}^{2})}{1 + {(2 \tilde{z})}^{2}} (\frac{2 P}{π w_{0}^{2}}) \frac{1}{1 + {(2 \tilde{z})}^{2}} \\ F_{a x i a l, s c a t t e r} (z; r = 0) = \frac{8 π n^{w} {(k a)}^{4} a^{2}}{3 c} {(\frac{m^{2} - 1}{m^{2} + 2})}^{2} (\frac{2 P}{π w_{0}^{2}}) \frac{1}{1 + {(2 \tilde{z})}^{2}} \end{matrix}

where

n^{w}

is the refractive index of medium (here, water),

a = \frac{d}{2}

is the radius of the trapped particle,

k = \frac{2 π * n^{w}}{λ}

is wave-vector inside the medium, c is the speed of light in vacuum,

m = \frac{n^{p}}{n^{w}}

is the ratio of refractive index of the material the particle is made of (here, polystyrene) to that of surrounding medium (here, water). Numerical integration of the analytical expressions for force yields the corresponding expressions for potential:

\begin{matrix} U_{r a d i a l} (r; z = 0) = - \int^{​} F_{r a d i a l} (r; z = 0) d r \\ U_{a x i a l, g r a d i e n t / s c a t t e r} (z; r = 0) = - \int^{​} F_{a x i a l, g r a d i e n t / s c a t t e r} (z; r = 0) d z \end{matrix}

2.1 Excitation parameters

Here, we take wavelength of trapping to be 800nm and, for pulsed excitation, the pulse-width is chosen as 120fs ( $τ$ ) while repetition-rate ( $f$ , i.e. the inverse of the time lag between consecutive pulses, $T$ ) is chosen as 76MHz; these are typical values for a commercial Titanium:sapphire oscillator for which the peak-power/-intensity to average-power/-intensity ratio turns out to be:

\frac{P_{p e a k}}{P_{a v e r a g e}} \equiv \frac{I_{p e a k}}{I_{a v e r a g e}} = \frac{1}{f * τ} \approx 10^{5}

Note that such a definition assumes the peak-power/-intensity to be constant over the duration of the pulse, neglecting the explicit time dependence of the envelope of the pulse (which is equivalent to saying that the pulse has a rectangular temporal envelope).

2.2 Dipole approximation

Harada and Asakura [27] discussed the conditions for the validity of the dipole limit; they compared the numerical simulation results using the Rayleigh scattering expression with that using the generalized Lorentz-Mie theorem (GLMT) and found excellent agreement for particles with diameters an order of magnitude smaller than the wavelength. Following their work, we chose the particle sizes to be 80nm and 70nm (i.e. $d = 2 a \leq \frac{λ}{10}$ ). Although GLMT is a more rigorous method for estimating trapping force (or potential), in what follows we show that the simpler theoretical treatment under dipole approximation yields very accurate estimates of trapping force (or potential) reported in earlier experiments. Also, in this size limit, Rayleigh scattering expression is more useful as there is, in principle, no restriction on the $N A$ of the focusing objective which is often a problem to deal with in GLMT. For realistic estimation, we consider typical values for commercial oil-immersion objectives as $N A = 1.4, 1.3 & 1.45$ (with typical refractive index of immersion oil being 1.515); these values are 0.924, 0.858 & 0.957 and 1.228, 1.140, & 1.272 in air and water, respectively.

2.3 Kerr effect

The (linear) refractive indices of water and polystyrene are taken to be 1.329 and 1.578, respectively [28]. Taking into account for the intensity-dependent nonlinear refractive indices of both polystyrene and water, we have the expression:

n^{w / p} = n_{0}^{w / p} + n_{2}^{w / p} * I (r, z)

The values of

n_{2}^{w}

and

n_{2}^{p}

are taken as 2.7x10⁻²⁰m²W⁻¹ [29] and 5.9x10⁻¹⁷m²W⁻¹ [30–32], respectively. Note that these values were reported at 532nm and we ignore the wavelength dependence of these quantities since, at 800nm, these values are expected to change but only slightly (and certainly not by an order of magnitude); so this is an reasonable approximation.

Let us now carefully examine the magnitudes of the nonlinear terms ( $n_{2}^{w / p} * I$ ) with respect to the linear terms ( $n_{0}^{w / p}$ ) considering intensity around geometric focus, $I (r = z = 0) = \frac{2 P}{π w_{0}^{2}}$ under tight-focusing condition ( $N A = 1.4$ ). Under CW excitation, the terms $n_{2}^{p / w} * I_{a v e r a g e}$ are much smaller than $n_{0}^{w / p}$ such that $n^{w / p} \approx n_{0}^{w / p}$ ; for example, even at average power level as high as 100mW the values for $n_{2}^{p / w} * I_{a v e r a g e}$ are only 1.3x10⁻⁵ and 6.0x10⁻⁹, respectively, for polystyrene and water. Contrary to this, under pulsed excitation, $n_{2}^{p} * I_{p e a k}$ is non-negligible compared with $n_{0}^{p}$ ; for example, at 100mW average power $n_{2}^{p} * I_{p e a k} \approx 1.4$ which is comparable with $n_{0}^{p} = 1.578$ . However, even at this high peak intensity $n_{2}^{w} * I_{p e a k}$ is only about 6.6x10⁻⁴. So $n^{w} \approx n_{0}^{w}$ for all excitation conditions and non-linear beam propagation effects inside water can be ignored ( $k = \frac{2 π * n^{w}}{λ} \approx \frac{2 π * n_{0}^{w}}{λ}$ ).

2.4 Time averaged force (or potential)

Under pulsed excitation, the force (or potential) corresponds to the instantaneous force (or potential) acting on the particle for which time-averaging over one duty cycle yields:

〈 F_{p u l s e d} 〉 = \frac{1}{T} \int_{- \frac{τ}{2}}^{\frac{τ}{2}} F_{p u l s e d} d t = F_{p u l s e d} \frac{1}{T} \int_{- \frac{τ}{2}}^{\frac{τ}{2}} d t = F_{p u l s e d} * (\frac{1}{T} * τ) = F_{p u l s e d} * (f * τ)

Here, the instantaneous force is independent of time since the peak intensity is assumed to be constant over the duration of the pulse. If we ignore Kerr effect the instantaneous force is only linearly proportionl to the peak intensity (as the terms involving

m = \frac{n^{p}}{n^{w}} = \frac{n_{0}^{p}}{n_{0}^{w}}

do not depend on peak intensity) for which we have:

〈 F_{p u l s e d} 〉 = c o n s t a n t * I_{p e a k} * (f * τ) = c o n s t a n t * I_{a v e r a g e} \equiv F_{C W}

which means trapping efficiency would be the same under pulsed and CW excitation; this has been observed under low power excitation [17,22] for which the Kerr effect may be neglected. However, the same result does not hold if we take into account of Kerr effect because the instantaneous force has now nonlinear dependence on peak power (as the terms involving

m = \frac{n^{p}}{n^{w}} = \frac{(n_{0}^{p} + n_{2}^{p} * I_{p}_{e a k})}{(n_{0}^{w} + n_{2}^{w} * I_{p}_{e a k})} depend on peak intensity

) for which we have:

〈 F_{p u l s e d} 〉 \neq F_{C W}

and very different results are obtained; the pulse-chirping experiment on silica microspheres [18] did not reveal this owing to the very low Kerr nonlinearity of silica (2x10⁻²⁰m²W⁻¹ [33]).

3. Results and discussion

Since nonlinear optical effects are instantaneous, to include the Kerr effect under pulsed excitation we first calculate the instantaneous force (or potential) which readily yields the time averaged force (or potential) just after multiplication by $(f * τ)$ . The NA is chosen as 1.4 and the particle diameter as 80nm unless otherwise mentioned.

3.1 Radial force and potential

In Fig. 1, the radial components of the trapping force are plotted. Introduction of Kerr effect increases the maximum of (radial component of) the gradient force under pulsed excitation by a factor of ~4. As a result of Kerr effect, the maxima shift a bit from the classical turning points ( $r = \pm \frac{w_{0}}{2}$ ). This results in more stable (~3 times) lateral trapping which is evident from the corresponding plots of trapping potential along radial direction as shown in Fig. 2.

Fig. 1 Plots of trapping force along radial direction at 100mW average power under CW and pulsed excitation ignoring and including Kerr effect.

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Fig. 2 Plots of trapping potential along radial direction at 100mW average power under CW and pulsed excitation ignoring and including Kerr effect.

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3.2 Axial force and potential

In Fig. 3, the axial components of the trapping force are plotted. As before, introduction of Kerr effect increases the maximum of (axial component of) the gradient force under pulsed excitation by a factor of ~4.2; however, maximum of the scattering force is increased by a factor of ~23.4. Here also the maxima shift a bit from the classical turning points for the gradient force ( $z = \pm \frac{k w_{0}^{2}}{2 \sqrt{3}}$ ) and away from the focal plane for the scattering force ( $z = 0$ ). The corresponding plots of trapping potential along axial direction are shown in Fig. 4.

Fig. 3 Plots of trapping force along axial direction at 100mW average power under CW and pulsed excitation ignoring and including Kerr effect. The blue lines correspond to scattering force, the green to gradient force and the red to total force.

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Fig. 4 Plots of trapping potential along axial direction at 100mW average power under CW and pulsed excitation ignoring and including Kerr effect. The blue lines correspond to scattering potential, the green to gradient potential and the red to total potential.

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3.3 Dependence on power

From now on, let us restrict our discussion on trapping along axial direction under pulsed excitation only for which Kerr effect significantly modulates the potential. The influence of Kerr effect on total trapping potential needs a special attention. As shown in Fig. 4, there is an absolute depth of the global potential minimum, $U_{m i n}$ (longer double-headed arrow in Fig. 4). However, the potential well is highly asymmetric and there is a potential barrier, $U_{e s c}$ (smaller double-headed arrow in Fig. 4), along beam propagation direction (i.e. positive z direction) to come out of this potential well. Introduction of Kerr effect increases $U_{m i n}$ by a factor of ~15.7 while it increases $U_{e s c}$ by a factor of ~1.1 only.

Upon increasing the power, a point is reached when there is no restoring force along beam propagation direction; as shown is Fig. 5, this condition is already reached at 500mW power. At this power level, although $U_{m i n}$ is very high, $U_{e s c} = 0$ i.e. the axial trapping potential becomes unbound; this is depicted in Fig. 6. So the particle is initially dragged toward the trap center but eventually propelled along axial direction which is consistent with the well known observation of axial ejection for larger colloidal particles (diameter ≥ 0.5µm) upon increasing the power under pulsed excitation only [18,22,23]. Therefore, the more relevant parameter to quantify the trapping efficiency is $U_{e s c}$ and not $U_{m i n}$ .

Fig. 5 Plots of trapping force along axial direction at different average power levels. The blue lines correspond to scattering force, the green to gradient force and the red to total force.

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Fig. 6 Plots of trapping potential along axial direction at different average power levels. The blue lines correspond to scattering potential, the green to gradient potential and the red to total potential.

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To have a better insight into the nature of these two types of potential, $U_{m i n}$ and $U_{e s c}$ , we plotted them separately with increasing power. Quite interestingly, as shown in Fig. 7, we see that although $U_{m i n}$ monotonically increases with power, $U_{e s c}$ increases reaching a maximum before dropping off. This means there is an optimal average power for which the dipolar trap is most stable which is about 150mW for the 80nm particle using 1.4 NA objective. This is a very important finding as we can theoretically predict the optimal excitation parameters for having most stable dipole trap.

Fig. 7 Plots of absolute depth of the trapping potential (left) and escape potential (right) along axial direction at different average power levels.

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When we further investigate the shift in axial equilibrium position with increasing power, we note that the most stable trap around 150mW corresponds to a trap center away from the geometrical focus ( $z = 0$ ); this is shown in left panel of Fig. 8. As shown in middle panel of Fig. 8, when we plot the gradient and the scattering potential at the location corresponding to this axial equilibrium position, we notice an initial dominance of the gradient potential which, upon increasing power, is reversed around 40mW (zoomed within inset); also note that the gradient potential goes through a maximum around 250mW. So, contrary to the common understanding, the trap is neither destabilized due to pushing the particle away from focus by scattering potential nor it is stabilized by maximizing gradient potential alone.

Fig. 8 Plots of position of the minima for absolute trapping potential along axial direction (Left) and the gradient and the scattering potential (green and blue lines, respectively) corresponding to this minima (Middle) at different average power levels. Right: Plots of the absolute depth of the trapping potential and the asymptotic scattering potential (red and blue lines, respectively) for the same.

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In right panel of Fig. 8, we plot sum of these two potential at axial equilibrium position, i.e. $U_{m i n}$ , alongwith its asymptotic value, i.e. $U_{a s y m p} = U_{} (z \to \infty) \equiv U_{s c a t t e r} (z \to \infty)$ since $U_{g r a d i e n t} (z \to \infty) = 0$ . We notice the separation between these two potential, i.e. $U_{e s c}$ , is maximum around 150mW when the trap is most stable (zoomed within inset) and there is a merging near 400mW when the potential is unbound, i.e. $U_{e s c} = 0$ . So trapping efficiency is in fact subtly controlled by a delicate balance between $U_{m i n}$ and $U_{a s y m p}$ .

Quite interestingly, the above results indicate that for particles having linear refractive indices less than the surrounding medium, which are repelled by single beam trapping potential, there is a possibility of observing Kerr effect assisted stable trapping (depending on the strength of Kerr nonlinearity). We note an earlier theoretical work in literature [34] that discussed the role of Kerr effect resulting only in slight increase in trapping force under CW excitation, however for an idealized nanoparticle with quite high Kerr nonlinearity (1.8x10⁻¹² m²W⁻¹) and without any detailed analysis on the exact nature of trapping force and potential.

3.4 Dependence on numerical aperture

Let us now compare the trapping efficiencies under different focusing conditions. From Fig. 9 we see that $U_{e s c}$ is decreased/increased by a factor of about 1.5/1.2 when the $N A$ is switched from 1.4 to 1.3/1.45; the power corresponding to maximum $U_{e s c}$ is also changed but only slightly. Tighter focusing leads to more stable trap as the relative contributions of gradient and scattering potential are changed. For stable trapping a ‘rule of thumb’ is that a minimum of 10 k_BT barrier height is required; so the trap is thermally unstable when $N A$ is changed to 1.3.

Fig. 9 Plots of escape potential with NA = 1.3, 1.4 and 1.45 (black, red and grey lines respectively) at different average power levels.

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3.5 Dependence on particle size

It is interesting to explore the optimal average power and the corresponding $U_{e s c}$ for particles with different diameters (each satisfying the criterion, $d = 2 a \leq \frac{λ}{10}$ ), for example, 80nm and 70nm without changing the focusing condition ( $N A = 1.4$ ). As shown in the left panel of Fig. 10, while the optimal average power is 150mW for the 80nm particle, it is around 500mW for the 70nm particle which means it requires more power for stable trapping of the smaller particle. The maximum height of the potential barrier is also increased by a factor of about 2.6 for the smaller particle which means the smaller particle can be trapped with more stability. This is a very promising finding considering the direct trapping of dielectric nanoparticles. Also, the threshold power ( $U_{e s c} = 0$ ) decreses with increasing particle size. A small change in particle size results in a significant change in the optimal average power level which is a consequence of third-/sixth-power dependence of force on radius of the particle.

Fig. 10 Left: Plots of escape potential on an 80nm particle (red line) and on a 70nm particle (black line) at different average power levels. Right: The same for a 100nm particle.

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Although 100nm polystyrene beads do not strictly satisfy the dipole limit, we are still intrigued to find the optimal power level for such particles. As shown in right panel of Fig. 10, the optimal power is ~40mW for which $U_{e s c}$ is just about 2.7k_BT; this means the particle is trapped but only for a while (considering the ‘rule of thumb’ of minimum 10k_BT barrier height for stable trap). All these quantitatively agree with the experimental findings [24,25].

4. Conclusion

We have shown how the efficiency of laser trapping of dielectric nanoparticles under high-repetition-rate ultrafast pulsed excitation is dramatically modulated by optical Kerr effect. We have thoroughly analyzed the results of such effect. Considering trapping efficiency, the height of the potential barrier along beam propagation direction has been identified to be the most relevant quantity which nicely explains previous experimental results. Our analysis have shown how to estimate optimal average power level for most stable dipole trap which is extremely important in direct trapping of individual nanoparticles. We have also predicted Kerr effect assisted stable trapping of nanoparticles that are difficult to trap otherwise.

Funding

This work is supported by SERB, DST (Grant No: ECR/2016/000467) and IISER Mohali (Start-up Grant). We thank IISER mohali for providing graduate fellowship to AD.

Acknowledgment

This paper is dedicated to Prof. N. Sathyamurthy on 65.

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Theoretical investigation on nonlinear optical effects in laser trapping of dielectric nanoparticles with ultrafast pulsed excitation

Abstract

1. Introduction

2. General discussion

2.1 Excitation parameters

2.2 Dipole approximation

2.3 Kerr effect

2.4 Time averaged force (or potential)

3. Results and discussion

3.1 Radial force and potential

3.2 Axial force and potential

3.3 Dependence on power

3.4 Dependence on numerical aperture

3.5 Dependence on particle size

4. Conclusion

Funding

Acknowledgment

References and links

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Figures (10)

Equations (8)

Optics Express