Effect of two-photon absorption on trapping of plasmonic nanoparticles

In this paper, we introduce a theoretical framework for optical trapping that integrates nonlinear polarization within the dipole approximation. This theory represents the most comprehensive analytic model to date capable of resolving the discrepancies between the observed and simulated trapping of plasmonic nanoparticles. Our theory elucidates how two-photon absorption can account for the stable trapping of gold nanoparticles, including their longitudinal stability, especially near their plasmon resonance. Furthermore, the experimentally observed split potential wells in the transverse plane, which are attributed to two-photon absorption, are in close agreement with our model’s predictions. Finally, this study provides new insights into the mechanism of optical trapping under conditions of intense light–matter interactions.


INTRODUCTION
In levitated optomechanics, plasmonic particles, particularly gold nanoparticles, are recognized for their capacity to enhance light-matter interactions via localized surface plasmon resonances [1].Contrary to predictions from linear trapping theory (which posits that highly absorbing particles should be unstable at the intensity maxima of a single laser beam due to overpowering repulsive forces) experimental work has shown that gold nanoparticles can indeed be stably trapped at these points [2][3][4].This stability has been attributed to surface creeping waves or non-spherical particle shapes reducing scattering force [5,7]; while useful, these explanations are insufficient for our purposes as micrographs of the particles in experiments we seek to explain how highly spherical morphologies [2][3][4] and creeping waves do not account for longitudinal stability [7].Furthermore, under femtosecond pulse illumination, gold nanoparticles exhibit unique trapping behaviors such as split potential wells in the transverse plane and circumgyration in potentials with radii smaller than the diffraction limit, which defy linear optical trapping theories [8][9][10][11].These phenomena, which hint at the involvement of two-photon absorption (TPA), point to the need for a more robust theoretical framework that can account for the full range of experimental observations, including the stable trapping of spherical gold nanoparticles in both the longitudinal and transverse planes.
In this paper, for the first time, we show that TPA accounts for the stable trapping of gold nanoparticles in the longitudinal direction, as well as the split traps in the transverse plane.To this end, we present a comprehensive theory for nonlinear optical trapping of all types of nanoparticles, including non-absorbing, weaklyabsorbing, and highly-absorbing nanoparticles.We investigate the effects of four-wave mixing (FWM) and TPA on optical trapping, separately.Our proposed theory not only aligns with prior experimental findings but also predicts new phenomena.Within the transverse plane, we illustrate how TPA creates a dual-split trap under the saturable absorption (SA) regime and a trisplit trap within the reverse saturable absorption (RSA) regime.Furthermore, we show that, unlike the transverse plane, TPA creates a stable asymmetric longitudinal potential in both SA and RSA regimes, where the stable point occurs in front of a focal point of a tightly focused Gaussian.Finally, we present the physical interpretation of the nonlinear trap system, likening it to a nonlinear harmonic oscillator and associating its parameters with linear and nonlinear susceptibilities.The theory findings are summarized in 1.
Gold nanoparticles exhibit strong third-order nonlinearity at plasmon resonances [13][14][15][16].To explore the effects of this nonlinearity on trapping, our analysis assumes a linearly polarized femtosecond laser (λ = 532 nm, repetition rate: ν = 80 MHz, pulse duration: 100 fs) that propagates in the z-direction to illuminate gold nanoparticles (R = 20 nm) immersed in water.Using the dipole approximation, the optical forces for a monotonic wave can be expressed as [19]: Where ) denotes the effective polarizability of the particle.Meanwhile, α 0 = 4πϵ 0 R 3 ϵp−ϵm ϵp+2ϵm stands for the static polarizability, with ϵ p and ϵ m being the relative permittivities of the particle and medium, respectively.The first term in equation 1 corresponds to the gradient force, whereas the second term signifies the scattering force.Under the influence of a pulsed laser or high-intensity continuouswave laser, the induced nonlinear polarizability becomes significant and must be taken into account in the computation of optical forces.Here we do not consider the nonlinear effect of the medium as it is much smaller than that of the particle.The relative permittivity of the particle by including the third-order nonlinearity is is the complex nonlinear susceptibility, in which the real part corresponds to FWM, the imaginary part relates to TPA, and E 2 is the amplitude of the electric field.The third-order nonlinearity of gold originates from interband transitions, intraband transitions, and hot electrons [17].Both intraband and interband transitions have an instantaneous response, while hot electrons exhibit a response time on the order of hundreds of femtoseconds.
Details of the calculations are presented in the Supplemental Information.Here, we present the final results.The real and imaginary parts of effective polarizability in terms of both linear and nonlinear static polarizabilities read where the real α and the real α and imaginary α (N L) 0,I components of nonlinear static polarizability are written as follows: where In low absorption limit (χ ′′ ≪ χ ′ , χ ′′ 3 ≪ χ ′ 3 ), more compact equations can be achieved as follows: where ) As shown, the FWM contributes to both gradient and scattering forces.Conversely, TPA solely contributes to the scattering force.The ratio of the FWM to TPA components in equation 9 is proportional to (kR) , where (kR) 3 ≪ 1 and As a result, in certain circumstances, the effect of FWM dominates, whereas in others, TPA primarily contributes to the scattering force.Depending on the values of χ ′ 3 and χ ′′ 3 , as well as the magnitude of the input power, the FWM and TPA effects can either increase or decrease the gradient and scattering forces.Hence, the new potential traps rather than the harmonic quadratic potential can be achieved.
From equations 8 to 10, both the real and imaginary components of effective polarizability can be formulated by a combination of linear and nonlinear components.Therefore, the gradient and scattering forces can also be depicted as the summation of their linear and nonlinear counterparts.This suggests that we can decompose the system into two coupled linear and nonlinear oscillators.Later, we provide a physical interpretation of the nonlinear oscillator by defining its parameters in terms of linear and nonlinear susceptibility.
In the subsequent sections, we explore the effects of TPA and FWM on the optical trapping of gold nanoparticles.Figure 2 presents the longitudinal potential at different powers, taking into account a complex value for the third-order susceptibility, i.e., considering both Four-Wave Mixing (FWM) and TPA.Here, the complex susceptibility of gold is chosen to be χ 3 = (3.9− 6.6j)×10 −21 (m 2 /V 2 ) [13].Additionally, the corresponding forces are detailed in the Supplemental Information.At low powers (P ave ≤ 150mW ), gold nanoparticles exhibit instability in the longitudinal direction, as shown in Figure 2(a).In this case, the effect of optical nonlinearities is not significant, and the trapping system behaves similarly to the linear regime.At higher powers (P ave > 150mW ), an asymmetric potential well emerges, with the stable point situated to the left of the focal point, as depicted in Figures 2(b)-(d).This stability trap occurs due to TPA, as we will demonstrate later.As the input power increases, the longitudinal potential energy maintains its asymmetric shape, and the position of the stable point progressively shifts to the left, as shown in Figure 4(a).Moreover, by increasing power, the width of the potential trap increases.These behaviors in the longitudinal direction are predictions of our theory that have yet to be demonstrated experimentally.
Figure 3 shows the transverse potential energies for different powers.Moreover, the corresponding forces are presented in the Supplemental Information.At lower powers (P ave ≤ 150mW ), the potential energy resembles a quadratic shape, similar to the linear potential energy.However, at higher powers (150mW ≤ P ave ≤ 1500mW ), a trap split occurs with two wells symmetrically positioned on opposing sides of the focal point.The SA absorption likely happens in this regime which results in a bistable potential trap.The depths of these wells significantly exceed the kinetic energy of Brownian motion, ensuring stable trapping.Within the SA regime, an increase in power increases the potential barrier between the split traps, further impeding particle transitions between the traps.At ultra-high powers (P ave > 1500mW ), an RSA regime is likely, characterized by a tri-stable potential trap with three wells: one central and two symmetrically positioned on either side.We established that the total potential trap can be viewed as a composite of linear and nonlinear potential traps.While the linear potential exhibits a single well at the origin, the nonlinear potential introduces two symmetrical offcenter wells.In the SA regime, the potential barrier separating these off-center wells surpasses the depth of the linear potential.Conversely, in the RSA domain, the linear potential depth overcomes the potential barrier, engendering a third central well.Compared to the central potential well, the off-center wells have a narrower width, indicating that the stiffness of the nonlinear trap is greater than that of the linear one.Interestingly, the depth of the off-center wells remains constant regardless of input power (Figure 3 (b)-(c)), yet depends on TPA, as we will show explicitly later.
The spacing between the two off-center split traps expands with increasing power as depicted in Figure 4 (b).Notably, the variance in distance is more pronounced in the SA domain compared to its RSA counterpart.The distance between these off-center potential points can surpass the diffraction limit.For instance, at an input power of P ave = 200mW , this distance is 95nm.These observations are in agreement with the experimental findings presented in [8].
In what follows, we only investigate the influence of FWM on the optical trapping of gold nanoparticles.Figure 5 shows the longitudinal potential energies for various powers when considering only the FWM process, i.e., (χ ′′ 3 = 0).Corresponding forces are again shown in the Supplemental Information.As shown, the gold nanoparticles are unstable in the longitudinal direction.Notably, by considering only FWM, gold nanoparticles remain unstable in the propagation direction.While the FWM introduces a slight perturbation to the longitudinal potential at high powers, the trap retains its instability.
Figure 6 shows the transverse potential energies, and, the corresponding forces are presented in the Supplemental Information.At low powers (P ave < 200mW ), the FWM impact of nonlinearity is minimal, rendering the potential energy as a simple quadratic form with a singular well.Within medium powers (200mW ≤ P ave ≤ 500mW ), the potential energy splits into two shallow wells symmetrically positioned on either side of the focal point.At higher powers (500mW ≤ P ave ≤ 2500mW ), the potential energy splits into three shallow wells: a central one at the focal point and two equidistant ones around the origin.In ultra-high powers (P ave ≥ 2500mW ), the central well deepens, while the depths of the off-central wells remain unaltered.These localized potential wells are superficial, preventing particles from remaining confined.Once trapped in one well, Brownian motion easily moves particles to transition between wells.Figures 5 and 6 reveal that the FWM process has little contribution to forming stable split traps observed experimentally.Even though it introduces shallow, split traps in the transverse plane, these traps lack stability.Moreover, it does not contribute to the longitudinal stability.These findings contradict experimental observations wherein particles are consistently trapped within split traps on the transverse plane or stably positioned longitudinally.These results show that the origin of split traps, as well as their longitudinal stability, is attributed to TPA.
The subsequent section provides a physical interpretation of the nonlinear trapping system.In the SA regime, the transverse potential traps exhibit behavior similar to the bi-stable potential of a non-harmonic oscillator.In contrast, within the RSA regime, the system behaves like a tri-stable potential.As previously demonstrated, the total trapping energy is a combination of linear and nonlinear potentials.Accordingly, the bi-stable and tri-stable potential energies can be presented by the equations , and Here, k 1 is linear stiffness and k 2 and k 3 denote nonlinear stiffnesses.To better understand the mechanism of nonlinear trapping, we make some simplifications.Specifically, we neglect the scattering force and self-induced back action, justifiable assumptions considering that gold nanoparticles trapped in water have a reduced scattering crosssection in the transverse plane.
After simplifications, the total potential can be expressed as a combination of linear (first part) and nonlinear (second part) counterparts (the details of the calculations are presented in the Supplemental Information).
As shown, the nonlinear potential is enveloped by the linear one.The magnitude of total potential at the focal point, i.e.U (ρ = 0) reads Here, the first term represents the depth of the linear potential, and the second term corresponds to the amplitude of the nonlinear potential barrier, with E 0 being the maximum amplitude of the electric field.As the power increases, the depth of the linear potential deepens, while the potential barrier of the nonlinear potential initially rises (in the SA regime) before converging to a fixed value (in the RSA regime).In the SA regime, the magnitude of the second term dominates the first, leading to positive amplitudes as confirmed in Figure 3(b).Conversely, in the RSA regime, the magnitude of the nonlinear term at the focal point remains unchanged with increasing power.In contrast, the depth of the linear potential continues to deepen.Thus, the magnitude of the linear component prevails over the nonlinear one, resulting in the emergence of a centered well, as depicted in Figures 3(c) and  (d).On the other hand, the off-center trap points happen at The condition for forming a split trap is that the argument of the logarithmic function must be positive; i.e.

3(χ
According to equation 13, the distance between off-center points is proportional to Ln(E 0 ) a behavior illustrated in Figure 4(a) through a more exact approach.Moreover, the depth of potential wells at off-centered wells donated U (ρ ± ) is given by As shown, the depth of the off-center trap points is independent of the input power.Moreover, the first part of the denominator is much smaller than the second part, which indicates that the depth of the off-center traps is primarily determined by TPA.These findings agree with the findings in Figures 3 and 6.
By comparing U bi and U , the coupling between linear and nonlinear stiffnesses can be calculated as follows: This equation shows that the coupling between the two oscillators is not constant; instead, it varies depending on the power, as well as linear and nonlinear susceptibility parameters.Furthermore, the value of ρ + is on the order of micrometers, indicating that k 2 is significantly larger than k 1 , as previously demonstrated.Similar analytical methods can also be applied to a tri-stable potential.

CONCLUSION
This paper presents a new theoretical approach to the nonlinear optical trapping of nanoparticles, emphasizing the pivotal roles of four-wave mixing and two-photon absorption.The research highlights TPA's key role in longitudinal trapping stability and reveals multiple split traps in diverse absorption settings.It suggests that the third-order nonlinearity of gold nanoparticles at plasmon resonances can explain the stable longitudinal trapping observed experimentally.By comparing the nonlinear trap system to a nonlinear harmonic oscillator, the paper deepens our understanding of dipole-regime trapping generally.Ultimately, this study fills existing gaps and advances the field of optical trapping, enriching insights into light-matter interactions.
We would like to thank Dr. Camacho for his useful discussions, as well as Spencer Duke for his contributions to this work.In this section, we develop the optical force calculations based on nonlinear polarization.
In the dipole regime, the total force acting on the particle has two components: gradient and scattering force.These force components depend on real and imaginary parts of effective polarizability, respectively.When considering the nonlinear susceptibility, the static polarizability can be written as follows: where ϵ l is linear permittivity of the particle .For simplicity in the calculations, we consider the medium to be air, thus: In general, the nonlinear parts are much smaller than their linear counterparts, i.e.
Thus, we can use the Taylor expansion of the denominator as follows: Moreover, E 4 terms are much smaller than the other terms.After removing these terms we get: The first term is the linear part, and the second term is the nonlinear part of static susceptibility (α 0 = α ).The linear static polarizability in terms of susceptibilities is written as And in term of real α (L) 0,r and imaginary α (L) 0,I components it can be written as Figure S2 shows the transverse forces at varying power levels when accounting for both four-wave mixing and two-photon absorption.At low average powers, the transverse force resembles the linear force.As the power increases, two zero-force points emerge on the left and right sides of the center, corresponding to the two split trapping wells (nonlinear traps).
At higher powers, three zero-force points become apparent: one is in the center (linear trap), and the other two are located on the left and right sides of the center, respectively.
At extremely high powers, the stiffness of the central trap is enhanced, while the stiffness of the side traps remains unchanged.Moreover, the stiffness of nonlinear traps is much stronger than that of the central linear trap.
Figure S3 shows the longitudinal forces at different powers when considering only a fourwave mixing process.As the power increases, the longitudinal force splits into two branches; however, no zero-force point occurs along the axial direction.Therefore, four-wave mixing does not lead to longitudinal trap stability.
Figure S4 shows the transverse forces at different powers when considering only four-wave mixing.As shown, at low average powers, the transverse force resembles the linear force.

II. PHYSICAL INTERPRETATION
In this section, we provide a physical interpretation of the nonlinear trapping system.
In the transverse plane, we model the nonlinear system within the saturable absorption regime using a bistable potential and illustrate the linear and nonlinear stiffnesses in terms of the linear and nonlinear susceptibilities.A similar approach can be used to model the potential trap within a reverse saturable absorption regime using a tri-stable potential well.
The nonlinear potential at distances between off-center points can be expressed by where k 1 and k 2 respectively are the linear and nonlinear stiffnesses.The minimum points of this potential (x = ± k 1 k 2 ) are equivalent to the off-center trap points.To have a better insight into the mechanism of nonlinear optical trapping, we make some further simplifications.Here, we ignore the scattering force and self-induced back-action.These are reasonable assumptions because the gold nanoparticles experience a smaller scattering force in the transverse plane when they are immersed in water.In this case, the gradient potential reads U = −ℜ( χ χ+3 )| ⃗ E| 2 where ⃗ E = ⃗ E 0 e −ρ 2 is the Gaussian electric field with

FIG. 1 .
FIG. 1.The comparison between linear (current theory) and nonlinear (this work) models for potential traps.(a) Shows nonlinear longitudinal and transverse potentials, and (b) shows linear longitudinal and transverse potentials.(c) The nonlinear trapping system behaves like a nonlinear harmonic oscillator, where, the total stiffness k(z) is a function of linear k1 and nonlinear k2 counterparts.
components of linear static polarizability are expressed by α

FIG. 4 .FIG. 5 .
FIG. 4. (a) The spacing between two split traps as a function of power in the transverse plane.(b) The positioning of trap points as a function of power in the longitudinal direction.