Nonlinear optical trapping effect with reverse saturable absorption

Abstract. Nonlinear responses of nanoparticles induce enlightening phenomena in optical tweezers. With the gradual increase in optical intensity, effects from saturable absorption (SA) and reverse SA (RSA) arise in sequence and thereby modulate the nonlinear properties of materials. In current nonlinear optical traps, however, the underlying physical mechanism is mainly confined within the SA regime because threshold values required to excite the RSA regime are extremely high. Herein, we demonstrate, both in theory and experiment, nonlinear optical tweezing within the RSA regime, proving that a fascinating composite trapping state is achievable at ultrahigh intensities through an optical force reversal induced through nonlinear absorption. Integrated results help in perfecting the nonlinear optical trapping system, thereby providing beneficial guidance for wider applications of nonlinear optics.


Introduction
Optical tweezers can exert piconewton forces on small objects to manipulate their motion in a liquid surrounding medium. 1,2 A stable optical trapping is based on the balance between scattering and gradient forces. 3,4 This convenient and intuitive principle works in establishing and describing the normal configurations of optical tweezers and has been widely employed in physical, 5,6 biological, 7 and chemical research. 8 Currently, the nonlinear effects in optical trapping unveil new possibilities and opportunities previously unprecedented. Nonlinear interactions occur between the nano-objects and ultrahigh-peak pulses, from which comprehensive and novel phenomena emerge with the subverted balance of the original optical force. [9][10][11] For example, gold nanoparticles trapped in a linearly polarized beam present an unprecedented result that one trap site was split into two. 9 The adjustable distance between two spaced trapping positions can reach beyond the diffraction limit of the focus field without any other complex optical instruments. The nonlinear response of the nano-object gives rise to the significant impact on its physical parameters and properties, [12][13][14][15] thereby providing an additional regulatory dimension to the trapped objects.
However, the peak intensity of the trapping pulse employed in current nonlinear optical tweezers resides at relatively low excitation levels. 10 In these circumstances, the nonlinear interaction lies within the saturable absorption (SA) regime in which the coefficient of absorption for the nano-object decreases towards a saturation value. 16,17 For SA, the electrons of the absorptive medium are stimulated from the ground state to an excited state and leads to ground-state bleaching, in which the medium no longer is able to absorb photons and the transmittance of the material increases. In this process, the nonlinear absorption destroys the balance of the original forces and thus creates a split optical trap. [9][10][11] When the intensity of the pulse continues to increase through the accompanying transitions between energy levels of the excited states, the coefficient of absorption for the medium increases with higher excitation levels in a process known as reverse SA (RSA). [18][19][20][21][22][23][24] Within this RSA regime, the equilibration of forces is further modulated and extends the range of the nonlinear optical tweezing. However, the research on this topic at present is still comparatively immature.
In this work, we demonstrate a nonlinear optical trapping system operating within the RSA regime and develop a completed conceptually explicit mechanism of nonlinear optical tweezing. With this system, a nonlinear optical trapping of gold nanoparticles is achievable, demonstrated both in theory and experiment, by employing a femtosecond laser pulse of circularly polarized light. When the peak intensity of the pulse surpasses the threshold to excite the deep RSA regime, a remarkably composite state in the optical trap appears, comprising a circumgyration state of one particle encircling a central stationary state of another. The experimental results of the phenomena match well with theoretical predictions. The results help to perfect the nonlinear optical tweezer system for trap loading. Using universal materials with nonlinear responses, which make a substantial contribution, the nonlinear optical trapping mechanism that we reveal may be important in developing new modalities of manipulating nanoscale objects by choosing prescribed light intensities appropriate for a nonlinear response.

Theory
Gold nanoparticles possess evident nonlinear optical properties and feature a complex refractive index that depends on the applied field intensity in the presence of Kerr-type nonlinearity. [25][26][27] Numerous works have indicated that gold nanoparticles exhibit distinct responses in the SA and RSA regimes. [17][18][19][20][21][22] Physically, the rate of change in the coefficient of nonlinear absorption for a gold nanoparticle remains negative during SA, but switches to being positive within the RSA regime. This reversal of sign produces significant changes in physical properties. The dependence on intensity of the coefficient of nonlinear absorption κ for gold nanoparticle is calculated from Ref. 26, where I s denotes the saturable optical intensity, β the coefficient of two-photon absorption, and a 0 the coefficient of linear absorption for a gold nanoparticle. The complex refractive index of the gold nanoparticle can be obtained from n p ¼ n 0 þ n 2 I þ iκ; where n 0 denotes the real part of the linear refractive index and n 2 the real part of the nonlinear refractive index. It is known that the real part of the nonlinear refractive index is much smaller than the imaginary part. Thus, as an approximation, we set the real part equal to zero. 28 As the complex n p depends strongly on the coefficient of nonlinear absorption κ, the actual polarizability of the gold dipole varies with field intensity. In simulations of gold nanoparticles, we set I s ¼ 55 × 10 14 W∕m 2 (see Supplementary Note IV in the Supplementary Material), n 0 ¼ 0.41661 þ 5.2347i, and a 0 ¼ 7.8311 × 10 7 m −1 . 29 Figure 1 attributes the nonlinear response of gold nanoparticles to an increase in the field intensity of a femtosecond pulse. To stimulate the SA and RSA effects, a focused circularly polarized femtosecond pulsed laser is applied as an excitation source. The focused field is determined by Debye vectorial diffraction theory (see Supplementary Note II in the Supplementary Material). 30 Here, consistent with the experimental conditions, the NA of the objective lens is set to 0.65. Values for the characteristic parameters of the pulsed laser are pulse width τ ¼ 100 fs, repetition frequency ν ¼ 80 MHz, and central wavelength λ ¼ 840 nm.
Referring to the gold nanoparticles used in this work, Fig. 1(a) shows the distribution of coefficient of nonlinear absorption κ as a function of two-photon absorption coefficient and incident peak intensity. The white line marks the boundary of the SA and RSA regimes. By increasing the peak intensity from 0 to 14 TW∕cm 2 , the SA and RSA effects are clearly evoked in sequence, and the coefficient of two-photon absorption β is a crucial parameter in determining the transition threshold. From experimental measurements, we set β ¼ 29 cm∕GW. 18 Figure 1(b) shows a plot of the change in κ for a gold nanoparticle as a function of peak intensity under the specific condition. During SA, the coefficient of absorption decreases with a relatively low excitation power but starts increasing when entering the RSA regime, with the power overcoming the threshold value of 3.32 TW∕cm 2 . Figure 2(a) plots the relation between incident average power (P ave ) and peak intensity of the focusing field (I peak ); an average incident power range of 0 to 2.0 W is able to excite both SA and RSA effects. The sign reversal in the rate of change for the coefficient of nonlinear absorption in the SA and RSA regimes produces significant influences in basic optical properties. These changes play vital roles in the nonlinear optical trapping process, reflected most apparently through the polarizability α and extinction cross section σ ext . Here, the polarizability α of a gold nanoparticle under the dipole approximation is given as 31,32 where k denotes the wavenumber of the incident optical field in the host medium, ε the permittivity of the host medium, r the particle radius, and n h the refractive index of the host medium; the extinction cross section of a single particle σ ext is then obtained from σ ext ¼ k ImðαÞ∕ε. 32 Figures 2(b) and 2(c) plot the real polarizability of the gold nanoparticle, which determines the gradient force, and the extinction cross section, which depends on the imaginary part of α and contributes to the scattering force. With increasing incident power, two peaks are apparent for both parameters; the dashed lines indicate the switching threshold between the SA and RSA regimes. Consequently, by varying the incident power to modulate the final trapping forces, a fully well-equipped universal nonlinear optical tweezers system is constructed. It is noteworthy that the optical resonance effect (ORE) can affect optical forces exerted on the nanoparticle. 33,34 At the resonance wavelength of the gold nanoparticle, the imaginary part of polarizability ImðαÞ is amplified, indicating an increased scattering force. On the other hand, the real parts of polarizability ReðαÞ and the gradient force are close to their minimum values. 35 For this scenario, the gold nanoparticle suffers a large scattering force and is difficult to trap. Therefore, we use the near-infrared pulsed laser as a light source to keep away from the resonant condition.
The optical force depends on the polarizability of the dipole and the surrounding electric field. The equation for the timeaveraged forces is based on the dipole approximation theory. 36 For a femtosecond pulsed optical field, the force is written as (see Supplementary Note III in the Supplementary Material) where E 0 denotes the peak electronic field at the particle position, c the light speed in a vacuum, and ω 0 the central frequency of the pulse.

Results and Discussion
Nonlinear optical tweezers operating within the SA regime have been previously described in detail; [9][10][11]  In the former, the potential well is distributed similar to that within the SA regime; more interestingly, for the latter, a sharp sink appears at the center of the energy barrier surrounded by the still-maintained potential energy ravine. Consequently, a gold nanoparticle is constrained stably at the center while others are constrained in the outer ring and undergo circumgyration.
To confirm this assertion, various experiments were conducted; the experimental setup implemented is depicted in schematics given in Fig. S1 Figure 3(c) plots the trajectories of a trapped gold nanoparticle undergoing circumgyration in the potential well. Similarly, Fig. 3(d) presents the superimposed results of two trapped gold nanoparticles; red dots indicate the trajectory of a stably trapped nanoparticle at the center, whereas yellow dots mark that of a circumgyrating nanoparticle.
To understand the nonlinear physical mechanism more thoroughly, the evolution of the optical force and potential well within the RSA regime is elaborated. Typical values, labeled in Figs. 2(a) and 2(c), were selected for elucidation; points A to C are situated in the SA regime, whereas points D to F lie in the RSA regime. As to the switching process of the nonlinear optical trapping effect within the RSA regime, the nonlinear optical forces and trapping potential wells in the focal plane under the selected incident powers-corresponding to points D to F-were plotted [Figs. 4(a)-4(f)]. Combined with vectorial diffraction theory, the direction and distribution of the optical force were subsequently calculated based on the dipolar approximation method.
When the nonlinear effect begins to gain ground within the RSA regime, the extinction cross section displays a nonmonotonic change with increasing incident power. With this change, the original force equilibrium is perturbed to form a "neopotential well." For the three selected incident average power values in the RSA regime [D, E, and F in Fig. 2 Here, the trapping potential has been normalized using k B T, where k B denotes the Boltzmann constant and T (T ¼ 300 K) the absolute temperature of the environment. At the incident average power of 0.7 W (point D in Fig. 2), a potential barrier with a positive peak value of 20k B T is achieved to sweep the particles in this area away; meanwhile, a ring-shaped potential well having a negative extreme value of −19k B T remains almost unchanged from that found at the boundary of the SA regime [ Fig. S4(f)  With a further increase in the average power, an optical force with a sign reversal arises from the central point and expands gradually. Simultaneously, under this reversed optical force, the center of the convex positive potential barrier subsides, producing a "volcano-like" profile in the potential [Figs. 4(c) and 4(f)]. For the relatively high power of 1.54 W at point F, the original ring-shape potential depth remains nearly constant at −19k B T. The centrally generated neo-potential well possesses a relatively deeper value of −33k B T compared with the surroundings. Although the absolute minimum value of the central subsidence is positive, it is still encircled by a more powerful annular barrier. Ultimately, a strongly stable optical trap appears at the center, surrounded by a trap-functioning ravine-like potential.
To verify the physical phenomenon of nonlinear optical trapping within the RSA regime, two specific average powers (1.15 and 1.54 W) were selected for experimental demonstration. As there exists an annular energy flow in the longitudinal component of the focused field, the trapped nanoparticles are driven to rotate along the circular ravine. Figures 4(g) and 4(h) present successive frames of a video (Videos 1 and 2), from which trajectories were plotted [Figs. 3(c) and 3(d)]. In the early RSA regime, the optical force and trapping potential well retain a similarity to that of the SA regime (Fig. S4 in the Supplementary Material). A circumgyration also develops at P ave ¼ 1.15 W.
For the experiments at P ave ¼ 1.54 W, in addition to a circumgyrating nanoparticle, another particle is trapped at the center point inside the path of circumgyration. The bright flare at the center verifies the high stability of the central trapped nanoparticle, whereas the fall in brightness in the outer ring results from the particle being in a state of circumgyration of high speed. We conclude that circumgyration originally arising in the SA regime holds steady, and a second trap is formed through the deep RSA effect. Considering the small number of trapped nanoparticles and their size being significantly smaller than the incident wavelength, the optical binding effect 37 can be safely disregarded. After careful comparison with Ref. 10, the rotation rate of circumgyration is estimated as 200 to −300 Hz within the RSA regime (see Supplementary Note VII in the Supplementary Material).
The optical force originates from the physical properties of the trapped objects and its interactions with the optical field. The consensus is that stable optical trapping is achieved through the balance of optical forces. The nonlinear response of the gold material is excited in an ultrahigh electric field; however, it disturbs the established balance. Its transmittance increases as a consequence of SA and diminishes under RSA. With an incident femtosecond pulse of circularly polarized light, the optical forces exerted thereby on a gold nanoparticle reduce and reverse in the SA regime. The potential profile changes from concave to convex at the central position. Consequently, at relatively low intensities in the nonlinear optical tweezers, a ring-shaped potential well can be formed, and nanoparticles can be driven to circumgyrate along the orbital energy flux.
As input power increases, nonlinear RSA comes into effect, and the convex energy barrier strengthens gradually. When the incident intensity reaches a specific threshold value, the magnitude of the optical forces becomes smaller and another reversal in direction of the optical forces appears at the center. From a  potential energy perspective, a neo-subsidence appears at the original convex vertex within a larger excitation, indicating that an additional particle is trapped at the center, encircled by the original circumgyrating nanoparticle(s). Hereto, a completed nonlinear optical trapping mechanism is established in theory and demonstrated in experiments. It is noteworthy that the usage of femtosecond-pulsed illumination can lead to a sharp and transient increase in temperature of the nanoparticle. This effect can confine the heat within the close vicinity of the nanoparticle, which prevents extended heating of the surrounding environment to destabilize the stable optical trapping (see Supplementary Note VIII in the Supplementary Material).
In the nonlinear regime, it is possible to achieve subdiffraction-limit controllable rotation of gold nanoparticles. A quantitative analysis of the mechanism governing the nonlinear optical trapping effect improves our understanding of the physics of light-matter interactions. Implementing nonlinear composite optical manipulation phenomena now becomes possible, with the potential for an expansion in applications. For example, these optical tweezers can be utilized as optically driven sensors and machines for microfluidic environments. [38][39][40] Furthermore, the ability to control the speed, radius, and direction of rotation is of great value in metrology. 41,42

Conclusion
Using a circular polarization femtosecond beam, we performed intensive research on the evolution of nonlinear properties and responses of gold nanoparticles, as well as the nonlinear optical forces that accompany the process. Within the deep RSA regime, a composite trapping state, i.e., a state in which an additional static trap encircled by the original circumgyration appears at the very center, is demonstrated. The results from experiments match well with those from theory. The demonstration proves that the switching of trap states stems from a reversal of the optical forces, which is induced through the reversal in sign of the rate of change in the coefficient of nonlinear absorption. The results fill gaps in knowledge of the existing nonlinear optical tweezing, and help in perfecting a relatively complete physical system. Our understanding of the mechanism underscoring nonlinear optical trapping is improved, thus paving the way for a broader study of nonlinear metallic materials. Furthermore, the novel nonlinear optical trapping effect has the potential to synergize with diverse structured light beams and other nonlinear materials, including quantum dots and nonlinear nanocrystals. More novel trap phenomena binding with other nanotechnologies are expected to be discovered for the extension of the physical significance of nonlinear optical trapping, as well as being conducive to the further developments in practical applications.

Code, Data, and Materials Availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
Zheng Zhu is a PhD candidate at the optics research group, Delft University of Technology. He joined the joint training program of the Delft University of Technology and Shenzhen University. He is now under the supervision of Prof. Paul Urbach and Prof. Xiaocong Yuan. His research interests include nonlinear optical trapping, localized surface plasmons, and ultrafast optics. He now works as a research assistant at the Zhejiang lab.
Yuquan Zhang received his PhD in optical engineering from Nankai University, China, in 2015. Currently, he is an associate professor in the Nanophotonics Research Center of Shenzhen University. His research interests include singular optical field modulation, optical tweezers, surface/tip-enhanced Raman spectroscopy, and superresolution imaging. Recently, his research has focused on plasmonic trapping and nonlinear effects in optical traps.

Shuoshuo Zhang is a PhD candidate in Nanophotonics Research
Center of Shenzhen University. He received his master's degree in optical engineering from Shandong University of Technology, China, in 2020. His research interests include spatiotemporally structured light, ultrafast optics, and nonlinear optical tweezers.
Aurèle J. L. Adam is an assistant professor at Delft University of Technology. He joined the research team of Alain Kreisler at the Laboratoire de Génie Électrique de Paris (University Paris VI -CNRS) and the microelectronics group of Erik Kollberg at the Chalmers University of Technology (Sweden) for a coupled PhD thesis. The core of his research is the study of feasibility and utility of near-field techniques at terahertz frequencies.
Changjun Min received his BS and PhD degrees from the University of Science and Technology of China, followed by postdoctoral positions at Louisiana State University, and then Nanyang Technological University. In 2014, he joined the Nanophotonics Research Center in Shenzhen University as a professor. His research interests include plasmonics, optical tweezers, vector beams, surface-enhanced Raman scattering, and metasurfaces.