Virtual Optical Pulling Force

The tremendous progress in light scattering engineering made it feasible to develop optical tweezers allowing capture, hold, and controllable displacement of submicronsize particles and biological structures. However, the momentum conservation law imposes a fundamental restriction on the optical pressure to be repulsive in paraxial fields. Although different approaches to get around this restriction have been proposed, they are rather sophisticated and rely on either wavefront engineering or utilize active media. Herein, we revisit the issue of optical forces by their analytic continuation to the complex frequency plane and considering their behavior in transient. We show that the exponential excitation at the complex frequency offers an intriguing ability to achieve a pulling force for a passive resonator of any shape and composition even in the paraxial approximation, the remarkable effect which is not reduced to the Fourier transform. The approach is linked to the virtual gain effect when an appropriate transient decay of the excitation signal makes it weaker than the outgoing signal that carries away greater energy and momentum flux density. The approach is implemented for the Fabry-Perot cavity and a high refractive index dielectric nanoparticle, a fruitful platform for intracellular spectroscopy and lab-on-a-chip technologies where the proposed technique may found unprecedented capabilities.


Introduction
Light scattering is ubiquitous and plays an essential role in the study of nature and conquering light-matter interactions for modern technology. According to A. Einstein [1,2], the light quanta 2 (photons) carry the energy 0  and momentum k , where 0  and / c   k are the frequency and wavenumber of a photon, is the reduced Planck's constant, and c is the speed of light.
Hence, every act of light scattering, when a photon changes its direction (or disappears), is accompanied by a transition of a portion of its momentum to the object. If the object is light enough and the photon momentum flux ( N k , N is the photon density) is large enough, this momentum transition can be detected through a mechanical motion of the object, as has been done for the first time by Lebedev [3] and Nichols [4]. In the early years, this phenomenon helped to establish the theory of light, but after the work of Arthur Ashkin [5][6][7] (Nobel Prize in Physics, 2018), optical forces became a powerful catalyst for modern technologies. Namely, it has been shown that optical scattering allows controlling the position of small objects using purely electromagnetic forces. In such techniques, the intensity gradient force is used to trap the object in the lateral plane, whereas the radiation pressure allows controlling the object's position along the beam. Today, this laser trapping became routine for different kinds of optical spectroscopy of single particles and living cells, optical tweezers [8][9][10], optical binding [11], laser cooling, lab-on-a-chip technologies to name just a few [12].  In the paraxial approximation, when the size of the object is small enough, and the beam is unfocused, the intensity gradient over the object size vanishes, and the radiation pressure plays the key role. In this approximation, the momentum conservation law imposes a fundamental restriction on the optical pressure to be repulsive. Indeed, the momentum flux of the scattered light along the wave vector of incident light ( i k ) through the object's surface is smaller than the incident momentum flux, and hence the resulting radiation pressure is obliged to be repulsive. For example, consider the case of radiation pressure for a lossless dielectric slab, Figure 1( T and R are transmission and reflection coefficients. The total pressure on the slab is the vector sum of all these momentum flux densities [13]   The same equation can be rigorously derived from Maxwell's stress tensor approach [13]. For passive media, 1 RT , and hence the total pressure is always directed in the direction of beam propagation. Also, for a given reflection, reducing the transmission through dissipation leads to an increase of the pushing force. In the blackbody ( 0 R  , 0 T  ) and perfect reflection ( It can be rigorously shown that optical pulling force is forbidden not only in the particular case outlined above but also for a passive object of any shape or composition in a paraxial field [14]. Although different approaches to get around this fundamental restriction have been suggested [15], all of them rely on structuring the incident field [14,[16][17][18][19][20][21], utilizing gain media [22][23][24], or on modifying the surroundings of the manipulated object [25][26][27][28], and hence require either complex techniques or operate at certain conditions. In this work, we revisit the issue of optical forces by stepping out to the complex frequency plane ( i ) and considering 4 its dynamics upon Fourier untransformable excitations. We show that tailoring of the time evaluation of the light excitation field allows either enhancement of the repulsive force or achieving pulling force for a passive resonator of arbitrary shape and composition. This unusual response is caused by the "virtual absorption" and "virtual gain" effects described in the following.

Results and discussion
To begin with, let us consider the 2-port system and assume the incoming signal to be Figure 1 can be greater than the input one, Figure 1(c). In this case, the stored energy becomes effectively "negative", 10 T R A     , as one would have in a system with a real gain. We refer to this effect as "virtual gain", prove it in the following with analytical and numerical calculations, and demonstrate how it can give rise to negative optical pressure. Note that this complex excitation with negative imaginary frequency has been utilized for obtaining an improved resolution in flat imaging devices [31]. . As the complex frequency grows in the upper plane, the system can come into the zero transmission regime; as a result, the force reaches 0 /2 FF , the value we expect for perfect reflectors. In the lower plane, the difference between the poles of R and T gives the enlarged positive (repulsion) and negative (attraction) force.
As a first example, we consider the case of a dielectric slab with permittivity 40 Figure   2. The material is assumed to be dispersionless and all results are presented in dimensionless frequencies to make the discussion independent of the frequency range. The actual thickness of the slab in our calculations was 500 nm. The transmission and reflection coefficients in the complex plane, Figures 2(a,b), have nontrivial dependence with poles in the lower complex plane.
Such point-like exceptional points is an immutable attribute of any resonant structures and associated with the modes of the structure [32]. Interestingly, knowledge of these poles in the complex plane allows retrieving all (linear) electromagnetic properties of the structure via the 6 Weierstrass theorem [33]. The reflection coefficient also possesses zeros, associated with the tunneling effect [ 0 R  , 1 T  at the real axis] at the Fabry-Perot resonances. The results of the calculation of the virtual absorption parameter in the complex frequency plane are presented in Figure 2(c).
This quantity vanishes at the real axis [ ( , 0) 0 A   ] due to system losslessness. In the upper (lower) plane, virtual absorption ( , ) A    is positive (negative) that gives rise to the effective loss (gain) effect. This result is fair as long as the excitation has an exponentially increasing or exponentially declining character. Note also that due to the quasi-monochromatic character of the exponential excitation, the calculation results for (  . Figures 3(a,d) show the calculated transmitted ( T , yellow curve), and reflected ( R , red curve) signals. We note that the reflection and 8 transmission signals at the monochromatic region correspond to the expected values: ) for the Fabry-Perot resonance (a), and ( 0 T , 1 R ) in the intermediate case (d), which means good incident pulse quality and the absence of energy transfer to other frequencies. The virtual absorption/gain coefficient is nonzero only at the exponential excitation and exponential attenuation periods as expected for the lossless structure, Figures 3(b, e). The results of the numerical calculation of optical radiative force in transient for the chosen excitation are shown in Figures 3(c, f). We see that at the Fabry-Perot resonance, the force changes its sign, whereas in the intermediate case it always stays positive, in full agreement with analytical results, Figure 2(d).
Thus, these results demonstrate the possibility of getting around the restriction caused by the momentum conservation law and achieving negative optical pressure for exponentially decaying signals. We note that the required decay rate depends on the position of the reflection poles and can be made arbitrarily small with an appropriate chose of mode with a large Q-factor.
For instance, the recently introduced concept of optical bound states in the continuum (BICs) [34,35] supporting unboundedly large Q-factor (the pole is unboundedly close to the real axis), would be a promising platform for negative optical forces in paraxial beams slowly decaying in time.
As another example, important from the application viewpoint, we consider the case of a high-index dielectric nanoparticle. Recently, these particles have attracted a lot of interest from researches across many interdisciplinary fields, including quantum optics, nonlinear, and biosensing. For biological applications, these subwavelength dielectric particles are demonstrated to be a fruitful platform for intracellular spectroscopy and microscopy [36][37][38]. In our calculations, we have chosen permittivity 16   , which corresponds to c-Si in the visible, Ge in near-IR, and SiC in mid-IR [39].
The scattering cross-section ( sca Q ) of the dielectric nanoparticle in the complex frequency plane is presented in Figure 4(a). As in the previous example, we observe several poles in the lover complex plane that give rise to the corresponding resonances at the real axis [37,40], Figure 4(b).
The fundamental resonance is magnetic dipole (MD), whereas the resonance with the largest Qfactor is magnetic quadrupole (MQ). As the frequency grows, the higher-order resonant modes manifest themselves. Here, the anapole state, which corresponds to sca 0 Q  (it is not exactly zero because of MQ mode) regime is also shown. Recently, the optical radiation force for such a dielectric particle in the monochromatic excitation laser field has been investigated 9 theoretically [41,42] and experimentally [43]. The enhancement of the force around the resonances and the reducing of it at the zero-backscattering Kerker condition [37] have been reported.
Although the former is supposed to be used for optical force enhancement, the latter is suggested for stabilization in an optical trap [41]. In Ref. [44] the optical force acting on the Si particles has been utilized for 2D trapping over a substrate and printing onto the substrate by means of radiation pressure. However, the optical pressure in these works is reported to be always positive for the paraxial optical field because any act of scattering of the incident photons by the nanoparticle can reduce their forward momentum or, at best, leave it unchanged. In further consideration, we revisit this conclusion and demonstrate how the negative optical radiative force can be achieved in the complex excitation approach. section ( ext Q ) of the nanoparticle in the complex frequency plane. Despite that the particle is assumed to be lossless [ abs ( , 0) 0 Q   ], the results show finite virtual absorption in the upper plane and negative absorption (gain) in the lower plane. 12 cavities. An ultimate approach in this way is so-called bound states in the continuum or embedded eigenstates [46][47][48]. In contrast to conventional optical resonances (e.g., plasmonic, Mie, whispering gallery modes), these states are uncoupled from the continuum of radiative modes, and hence in a lossless scenario, their poles lie on the real frequency axis enabling negative optical pressure for slowly decaying fields. Next, the reported results of this work are rather general and remain fair in the microwave, THz, and optics spectral ranges. In microwaves, the generation of such pulses is rather established experimental technique. Finally, the exponentially decaying fields are "naturally" arise after abrupt turning-off of the excitation of a resonator, which starts releasing its stored energy exponentially with the decay rate of a particular resonant mode. An object situated in its vicinity is expected to manifest the negative optical radiative force.

Conclusions
In this work, we have revisited the issue of optical forces by stepping out to the complex frequency plane and considering its dynamics upon complex excitations. We have shown that tailoring of the time evaluation of the light excitation field allows either enhancement of the repulsive force or achieving pulling force for a passive resonator of arbitrary shape and composition. We have demonstrated how these effects are linked to virtual gain and virtual loss effects. Virtual gain can be achieved when an appropriate transient decay of the excitation signal makes it weaker than the outgoing signal that carries away greater energy and momentum flux density. In its turn, the virtual loss effect is achieved when the incoming signal exponentially grows in time. The approach has been implemented for the Fabry-Perot cavity and a high refractive index dielectric nanoparticle, a fruitful platform for intracellular spectroscopy and lab-on-a-chip technologies where the proposed technique may found unprecedented capabilities.