- Advanced Photonics
- Vol. 5, Issue 4, 046006 (2023)
Abstract
1 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
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
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.
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2 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
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 , repetition frequency , and central wavelength .
Figure 1.Nonlinear absorption of gold nanoparticles. (a) Coefficient of nonlinear absorption
Referring to the gold nanoparticles used in this work, Fig. 1(a) shows the distribution of coefficient of nonlinear absorption 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 , the SA and RSA effects are clearly evoked in sequence, and the coefficient of two-photon absorption is a crucial parameter in determining the transition threshold. From experimental measurements, we set .18Figure 1(b) shows a plot of the change in 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 .
Figure 2(a) plots the relation between incident average power () and peak intensity of the focusing field (); 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 and extinction cross section . Here, the polarizability of a gold nanoparticle under the dipole approximation is given as31,32
Figure 2.Nonlinear polarizability and extinction cross section of gold nanoparticles. (a) Peak intensity of the focused excitation field as a function of averaged incident power
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 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 is amplified, indicating an increased scattering force. On the other hand, the real parts of polarizability 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 time-averaged 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)
3 Results and Discussion
Nonlinear optical tweezers operating within the SA regime have been previously described in detail;9
Figure 3.Nonlinear optical tweezers within the RSA regime. (a) and (b) Schematics of the trapping states of gold nanoparticles within the early and deep RSA regimes. (c) and (d) Recorded particle trajectories obtained from experiments operating in the early and deep RSA regimes, respectively.
To confirm this assertion, various experiments were conducted; the experimental setup implemented is depicted in schematics given in Fig. S1 in the Supplementary Material (Supplementary Note I in the Supplementary Material). Incident powers of 1.15 and 1.54 W were employed to excite the early and deep RSA responses. 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.
Figure 4.Switching of the nonlinear optical tweezers from the early to the deep RSA regime. (a)–(c) Radial optical force and (d)–(f) trapping potential distributions in the focal plane acting on the gold nanoparticle, for averaged incident power
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 “neo-potential well.” For the three selected incident average power values in the RSA regime [D, E, and F in Fig. 2(c)], the optical force and potential well were plotted [Figs. 4(a)–4(c) and 4(d)–4(f), respectively]. Both the magnitude and region of the reversed optical force are enlarged. The detailed description of the force variation tendency in the central region from point D to F is included in Supplementary Note VI in the Supplementary Material. By increasing the incident power, the size of the ring-shaped nonlinear potential well slightly expands in the early RSA regime [Figs. 4(a) and 4(b)]. Here, the trapping potential has been normalized using , where denotes the Boltzmann constant and () 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 is achieved to sweep the particles in this area away; meanwhile, a ring-shaped potential well having a negative extreme value of remains almost unchanged from that found at the boundary of the SA regime [Fig. S4(f) in the Supplementary Material]. At the peak point of the extinction cross section in the RSA regime, i.e., point E with an average power of 1.15 W, the optical force is close to zero at the focal center point, signifying that a subsequent reversal emerges [Figs. 4(b) and 4(e)]. In this circumstance, the central potential barrier flattens out, and a maximum positive potential barrier of forms.
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 . The centrally generated neo-potential well possesses a relatively deeper value of 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 . For the experiments at , 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 effect37 can be safely disregarded. After careful comparison with Ref. 10, the rotation rate of circumgyration is estimated as 200 to 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
4 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.
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.
Hendrik Paul Urbach is a professor at Delft University of Technology. He joined Philips Research Laboratory in Eindhoven (“Nat.Lab” as it was known at the time), where he was appointed Principal Scientist. In 2000, he became part-time professor of diffraction optics at Delft University of Technology. In 2008, he became group leader of the Optics Research Group of Delft University of Technology. His research interests include direct and inverse problems of high-resolution optical imaging, diffraction theory, and the application of asymptotic methods to optics.
Xiaocong Yuan is Changjiang Scholar Professor and director of the Nanophotonics Research Center in Shenzhen University. He received his PhD in physics from King’s College London in 1994. He is now a fellow of SPIE and Optica, and a board member of the Chinese Optical Society and the Chinese Optical Engineering Society. His research interests include singular optics, nanophotonics, plasmonics, advanced light manipulations, and their applications in communications, imaging, and optical tweezers.
References
[4] K. C. Neuman, S. M. Block. Optical trapping. Rev. Sci. Instrum., 75, 2787-2809(2004).
[14] M. Kauranen, A. V. Zayats. Nonlinear plasmonics. Nat. Photonics, 6, 737-748(2012).
[32] L. Novotny, B. Hecht. Principles of Nano-Optics(2007).
[39] M. Padgett, R. Bowman. Tweezers with a twist. Nat. Photonics, 5, 343-348(2011).
[43] R. Paschotta. Field Guide to Laser Pulse Generation(2008).
[45] R. W. Boyd. Nonlinear Optics(2020).
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