## Abstract

Progress in light scattering engineering made it feasible to develop optical tweezers allowing capture, hold, and controllable displacement of submicrometer-size particles and biological structures. However, the momentum conservation law imposes a fundamental restriction on the optical pressure to be repulsive in paraxial fields, which severely limits the capabilities of optomechanical control, e.g., preventing attractive force acting on sufficiently subwavelength particles and molecules. Herein, we revisit the issue of optical forces by their analytic continuation to the complex frequency plane and considering their behavior in the transient regime. We show that the exponential excitation at the complex frequency offers an intriguing ability to achieve a pulling force for a passive resonant object of any shape and composition, even in the paraxial approximation. The approach is elucidated on a dielectric 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 find unprecedented capabilities.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. INTRODUCTION

Light scattering is ubiquitous and plays an essential role in the study of nature and conquering light–matter interactions for modern technologies. According to Einstein [1,2], the light quanta (photons) carry the energy $\hbar {\omega _0}$ and momentum $\hbar {\bf k}$, where ${\omega _0}$ and $| {\bf k} | = \omega /c$ are the frequency and wavenumber of a photon, $\hbar$ 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 by absorption), 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 ($N\hbar {\bf k}$, $N$ is the photon density) is large enough, this momentum transition can be detected through a mechanical motion of the object as was 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 [5–7], optical forces became a powerful catalyst for modern technologies. Namely, it has been shown that optical scattering allows control of the position of small objects using only electromagnetic forces. In such techniques, the intensity gradient force is applied to trap an object in the lateral plane, whereas the radiation pressure allows control of the object’s position along the beam. Today, this laser trapping has become routine for different kinds of optical spectroscopy of single particles and living cells, optical tweezers [8–10], optical binding [11], laser cooling, and 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. This case is very important because, for example, for essentially subwave objects (molecules, quantum dots, nanoparticles), it is challenging to achieve strong gradients in freely propagating fields, and the paraxial approximation is inevitable. 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 wavevector of incident light (${{\bf k}_i}$) in a passive system is always smaller than the incident momentum flux through the object’s surface. In fact, it can be rigorously shown that the optical pulling force is forbidden for a passive object of any shape or composition in a paraxial field [13]. Although different approaches to get around this fundamental restriction have been suggested [14], all of them rely on structuring the incident field [13,15–20], utilizing gain media [21–24], or on modifying the surroundings of the manipulated object [25–28], and hence they require either complex techniques or operate at certain conditions. Note also that negative optical torque is another related area of significant interest today [29,30].

In this work, we revisit the issue of optical forces by stepping out to the complex frequency plane ($\omega = \omega ^\prime + i\omega ^{\prime \prime}$) and considering its dynamics upon excitations not amenable to Fourier transformation. We show that tailoring of the time evolution of the light excitation field allows either *enhancement of the repulsive force* or *achievement of 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. Recently, a similar approach has been suggested to achieve perfect light storage [31], virtual critical coupling [32], virtual parity-time symmetry (PT-symmetry) and associated effects [33], and it has been demonstrated experimentally for acoustic waves [34].

## 2. CONCEPT

The radiation pressure is obliged to be repulsive in any passive system in the paraxial approximation [13]. For example, consider the case of radiation pressure acting on a dielectric slab [Fig. 1(A)] lit by a monochromatic incident wave with the photon density per unit square and time duration ${{N}_{i}}=c{{\varepsilon }_{0}}E_{0}^{2}\!/\!2\hbar \omega $, where $E_0^2$ is the electric field intensity and ${\varepsilon _0}$ is the electric constant. This incident light carries the momentum density ${N_i}\hbar k = {N_i}\hbar \omega /c$. A part of this momentum gets transmitted through the slab (${\textit{TN}_i}\hbar k$), and another part gets reflected towards the source (${\textit{RN}_i}\hbar k$), with $T$ and $R$ being the transmission and reflection coefficients, respectively. The total pressure on the slab is the vector sum of all these momentum flux densities [35]:

This equation can be rigorously derived from Maxwell’s stress tensor approach [35]. For passive media ($R,T \le 1$), $1 + R \gt T$, and hence the total pressure is always directed towards beam propagation (positive). Also, for a given reflection, reducing the transmission through dissipation leads to an increase of the pushing force. In the blackbody ($\!R = 0$, $T = 0$) and perfect reflection ($\!R = 1$, $T = 0$) cases, this formula gives $F = {N_i}\hbar k$ and $F = 2{N_i}\hbar k$, respectively.

We stress that Eq. (1) is a definition of an optical pressure force as a total momentum flow through the surface of a body requires knowledge only $T$ and $R$. All the processes that take place inside the body are taken into account automatically via transmission and reflection coefficients. This approach is fair not only in the frequency domain but also in the transient regime at any time instant and does not require time averaging. Hence, it can be analytically continued to the complex plane (a special case of a transient regime), which is essential for the following discussion.

Unlike the real frequency axis, where the scattering matrix ($\hat S$) of a passive structure and its eigenvalues (${\lambda _i}$) are bound to $|{\lambda _i}{|^2} \le 1$ ($|{\lambda _i}{|^2} = 1$ in the Hermitian lossless case), in the complex frequency plane ($\omega = \omega ^\prime + i\omega^{\prime \prime}$), they can take any value between 0 (scattering zero) and $\infty$ (scattering pole) [36]. Interestingly, knowledge of these poles in the complex plane allows retrieval of all (linear) electromagnetic properties of the structure via the Weierstrass theorem [37]. As a result, the complex continuation of scattering amplitudes (for example, $T$ and $R$, in the 1D scenario) can also take any value ($|R,T| \gt 1$). This complex continuation lifts the discussed restriction on optical forces and makes the structure dynamics less trivial, which opens up many opportunities as discussed in what follows.

For example, let us consider this two-port system (Fig. 1) under exponentially growing excitation [$E_i (t) \propto E_0 \exp (\omega ^{\prime \prime} t)\exp (i\omega ^\prime t)$] [Fig. 1(B)]. In this case, as the incoming signal grows in time faster than the decay rate of the cavity, the incoming signal $|{E_i}|$ and the momentum flux density ($c{\varepsilon _0}\hbar k|E_i {|^2}/2\hbar \omega$) can be greater than the corresponding outgoing signal ($|{E_t}| + |{E_r}|$) and momentum [$c\!{\varepsilon _0}\hbar k(|E_t {|^2} + |E_r {|^2})/2\hbar \omega$], even in the lossless case. The remaining part of the signal, which is neither reflected nor transmitted, gets stored in the cavity (virtual loss) until the incoming signal keeps the exponential growth. In this case $1 - T(\omega ^\prime \!,\omega ^{\prime \prime}) - R(\omega ^\prime \!,\omega ^{\prime \prime}) = A \gt 0$, and the optical pressure can be increased. Here $A$ denotes the stored energy in the object (cavity). Note that this idea of virtual absorption has been suggested in Ref. [31] for perfect energy capture in a resonant cavity for a long time and for release on-demand. This effect turned out to be very promising and found application in many areas [31–33]; it was experimentally realized in Ref. [34] for elastodynamic waves. In the present work, we show that the virtual absorption effect gives a real contribution to the optical radiation pressure and can be utilized for all-optical manipulation.

Further, if the incoming signal exponentially decays faster than the cavity decay rate, the output signal $|{E_t}| + |{E_r}|$ [along with the momentum flux density $(c{\varepsilon _0}\hbar k(|{E_t}{|^2} + |{E_r}{|^2})/2\hbar \omega)$] can be *greater than the input one* [Fig. 1(C)]. In this case, the stored energy becomes effectively “negative,” and $1 - T - R = A \lt 0$, 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 recently utilized for imaging devices [38] and a virtual PT-symmetry effect [33]. It worth noting that, technically, the negative virtual pulling force is equivalent to the appearance of an additional gradient force pointing backward and caused by the exponential decay of the plane wave in time.

## 3. RESULTS AND DISCUSSION

To elucidate this concept, let us first consider the case of a dielectric slab with dispersionless permittivity $\varepsilon = 40$ (Fig. 2). 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 [Figs. 2(A) and 2(B)], have nontrivial dependence with poles in the lower half-plane. Such poles are an immutable attribute of any resonant structures and are associated with the modes of the structure [39]. It worth noting that the characteristic decay time of a mode is determined by the position of the corresponding pole, $\tau = 1/\omega ^{\prime\prime}$. The reflection coefficient also possesses zeros, associated with the tunneling effect [$R = 0$, $T = 1$ at the real axis] at the Fabry–Perot resonances.

The results of the calculation of the virtual absorption parameter $A(\omega ^\prime ,\omega ^{\prime \prime}) = 1 - R(\omega ^\prime ,\omega ^{\prime \prime}) - T(\omega ^\prime ,\omega ^{\prime \prime})$ in the complex frequency plane are presented in Fig. 2(C). This quantity vanishes at the real axis [$A(\omega ^\prime ,0) = 0$] due to the system hermicity [36]. In the upper (lower) plane, virtual absorption $A(\omega ^\prime ,\omega ^{\prime \prime})$ 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 (exponentially decaying) character. Note also that due to the quasi-monochromatic character of the exponential excitation, the calculation results for $A(\omega ^\prime ,\omega ^{\prime \prime})$, $R(\omega ^\prime ,\omega ^{\prime \prime})$, and $T(\omega ^\prime ,\omega ^{\prime \prime})$ are *fair for any time instant*.

The corresponding results of the calculation of the optical pressure are presented in Fig. 2(D). First, we note that in the upper plane [exponentially growing excitation, $E_i (t) \propto E_0 \exp (|\omega ^{\prime \prime} |t)$], as the complex frequency gets increased, the system can come into the zero transmission regime [$T = 0$, Figs. 2(A) and 2(B)]. As a result, the force reaches $F = 2{N_i}\hbar k$, i.e., the expected value for the perfect reflectors. More remarkably, in the lower plane [exponentially attenuating excitation, $E_i (t) \propto E_0 \exp (- |\omega ^{\prime \prime} |t)$], at a fixed real frequency, the optical pressure experiences the abrupt transition from large positive to large negative values, where the system is expected to experience pulling action from the laser beam [Fig. 2(D)]. This result can be explained as follows. Reflection and transmission coefficients contribute to the resulting radiation force with the opposite sign, and one can expect that their poles compensate each other. However, the presence of reflection zeros near the corresponding poles “deform” the poles. As a result, the optical pressure in the lower complex frequency plane has a nontrivial character with increased and decreased values.

To investigate the effect of complex excitation in the transient regime, we use full-wave FDTD simulations in CST Microwave Studio. To this end, we tailor two incident signals ($I$, blue curve) corresponding to the Fabry–Perot resonance [${\text{Re}} (\omega d\!/\!c) = 0.5$] and the middle frequency between two Fabry–Perot resonances [${\text{Re}} (\omega d/c) = 0.75$] [Figs. 3(A) and 3(D)]. During the first period (0/ 1.3 ps), the oscillation amplitude of both incident signals is increasing exponentially with ${\text{Im}} (\omega d/c) = 0.025$, then it reaches monochromatic excitation with the same real frequency (1.3/ 2.5 ps), and then it decays exponentially with ${\text{Im}} (\omega d/c) = - 0.025$. The shape of the chosen excitation signals allows consideration of both exponential growth and exponential decay regimes along with a quite long period of monochromatic excitation for revealing the steady-state regime.

Figures 3(A) and 3(D) show the calculated transmitted ($T$, yellow curve), and reflected ($R$, red curve) signals. We note that the reflection and transmission signals at the monochromatic region correspond to the expected values: ($T \sim 1$, $R \sim 0$) for the Fabry–Perot resonance [Fig. 3(A)], and ($T \sim 0$, $R \sim 1$) in the intermediate case [Fig. 3(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 [Figs. 3(B) and 3(E)]. The results of the numerical calculation of optical radiative force in the transient regime for the chosen excitation are shown in Figs. 3(C) and 3(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 [Fig. 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 choice of a mode with a large Q factor. For instance, the recently introduced concept of optical bound states in the continuum (BICs) [40,41] 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 researchers across many interdisciplinary fields, including quantum optics, nonlinear photonics, and biosensing. For biological applications, these subwavelength dielectric particles are demonstrated to be a fruitful platform for *intracellular* spectroscopy and microscopy [42–44]. In our calculations, we have chosen permittivity $\varepsilon = 16$, which corresponds to c-Si in the visible, Ge in near-IR, and SiC in mid-IR [45].

The scattering cross section ($\!{Q_{\text{sca}}}$) of the dielectric nanoparticle in the complex frequency plane is presented in Fig. 4(A). As in the previous example, we observe several poles in the lower complex plane that give rise to the corresponding resonances at the real axis [43,46] [Fig. 4(D)]. The fundamental resonance is magnetic dipole (MD), whereas the resonance with the largest ${Q}$ factor is magnetic quadrupole (MQ). As the frequency grows, the higher-order resonant modes manifest themselves. Here, the anapole state, which corresponds to the ${Q_{\text{sca}}} \approx 0$ regime (it is not exactly zero because of the MQ mode), is also shown. Recently, the optical radiation force for such a dielectric particle in the monochromatic excitation laser field has been investigated theoretically [47,48] and experimentally [49]. The enhancement of the force around the resonances and its reduction at the zero-backscattering Kerker condition [43] have been reported. While the former is supposed to be used for optical force enhancement, the latter is suggested for stabilization in an optical trap [47]. In Ref. [50] 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.

The radiative optical force in this case of a spherical particle can be calculated analytically by using the method rigorously derived based on the time-averaged Maxwell stress tensor [13,48,51] and found to be consistent with experimental results [49]. According to this method, the time-averaged radiative optical force equals

The results of the calculation of the optical radiation pressure acting on the dielectric particle with permittivity $\varepsilon = 16$ are presented in Fig. 5(A). Here, by points “1,” “2,” and “3,” we denote the virtual repulsive, repulsive, and virtual pulling regimes, respectively [Fig. 5(C)]. The optical force in the complex frequency plane normalized by its value at the real axis is presented in Fig. 5(D). For comparison, Fig. 5(D) shows the value of optical radiation pressure at the real frequency axis [${ \text{Im} } (\omega R/c) = 0$]. This result coincides with the reported theoretical works [47] and existing experimental results [49]. Figure 5(E) demonstrates the optical radiation pressure as a function of imaginary frequency ${ \text{Im} } (\omega R/c)$ at the fixed real frequency [${ \text{Re} } (\omega R/c) = 0.75$], which corresponds to the MD mode. We observe the characteristic Fano-like transition from enhanced negative to enhanced positive values. Note that the intensity of light in our calculations $I = {1}\;{\text{W/mm}^2}$ coincides with that used in recent experimental works on similar Si nanoparticles [49] in liquids and gives the same absolute value of optical force (${\sim}{10}\;\text{fN}$).

Lastly, we discuss possible experimental approaches to achieve the reported effects. First, as we mentioned above, the rate of exponential growth or decay required to archive enhanced positive or negative radiation force depends on the position of the poles in the complex plane. These poles may lie close to the real frequency axis in high-${Q}$ cavities. An ultimate approach is so-called bound states in the continuum or embedded eigenstates [36,52–54]. In contrast to conventional optical resonances (e.g., plasmonic, Mie, whispering gallery modes), these states are uncoupled from the continuum of radiative modes; 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, terahertz (THz), and optics spectral ranges. In microwaves, the generation of such pulses is an established experimental technique. Moreover, since the effect of virtual absorption has been demonstrated experimentally for acoustic waves [34], it raises an intriguing question on the applicability of our findings in acoustics, where unusual acousto-mechanical effects are also of great interest [55]. Finally, it worth mentioning that the total force over the entire *physically real pulse* (i.e., integrable and amenable to Fourier decomposition) is always positive. Nevertheless, exponentially growing/decaying signals can be a good approximation to some regions of realistic pulses. For example, the decaying field of a leaky mode of a microcavity after an abrupt excitation turn-off is well described by an exponentially decaying signal (see Supplement 1, Fig. S3). An object situated in such a field is expected to manifest the negative optical radiative force. We can also suggest several other ways to achieve this effect. One way is to put a particle onto a dielectric substrate, preventing its movement in the forward direction (see Supplement 1, Fig. S3). Another way is to consider a particle that appears in the laser beam in its decaying phase and, therefore, unaware of the pulse back history.

## 4. 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 achievement of 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 turn, the virtual loss effect is achieved when the incoming signal exponentially grows in time. The approach has been demonstrated for the Fabry–Perot cavity and a high-refractive-index dielectric nanoparticle.

## Funding

Russian Science Foundation (18-72-10140); Foundation for the Advancement of Theoretical Physics and Mathematics

## Acknowledgment

The authors thank Prof. Andrea Alú for fruitful discussions. We acknowledge support from the Russian Science Foundation (Project No. 18-72-10140) and Foundation for the Advancement of Theoretical Physics and Mathematics.

## Disclosures

The authors declare no conflicts of interest.

See Supplement 1 for supporting content.

## REFERENCES

**1. **A. Einstein, “Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt,” Ann. Phys. **322**, 132–148 (1905). [CrossRef]

**2. **A. B. Arons and M. B. Peppard, “Einstein’s proposal of the photon concept—a translation of the Annalen der Physik paper of 1905,” Am. J. Phys. **33**, 367–374 (1965). [CrossRef]

**3. **P. Lebedev, “Untersuchungen über die druckkräfe des lichtes,” Ann. Phys. **311**, 433–458 (1901). [CrossRef]

**4. **E. F. Nichols and G. F. Hull, “A preliminary communication on the pressure of heat and light radiation,” Phys. Rev. **13**, 307–320 (1901). [CrossRef]

**5. **A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**, 288–290 (1986). [CrossRef]

**6. **A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**, 156–159 (1970). [CrossRef]

**7. **A. Ashkin, “History of optical trapping and manipulation of small-neutral particle, atoms, and molecules,” IEEE J. Sel. Top. Quantum Electron. **6**, 841–856 (2000). [CrossRef]

**8. **M. L. Juan, M. Righini, and R. Quidant, “Plasmon nano-optical tweezers,” Nat. Photonics **5**, 349–356 (2011). [CrossRef]

**9. **M.-C. C. Zhong, X.-B. Bin Wei, J.-H. H. Zhou, Z.-Q. Q. Wang, and Y.-M. M. Li, “Trapping red blood cells in living animals using optical tweezers,” Nat. Commun. **4**, 1768 (2013). [CrossRef]

**10. **D. G. Grier, “A revolution in optical manipulation,” Nature **424**, 810–816 (2003). [CrossRef]

**11. **K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. **82**, 1767–1791 (2010). [CrossRef]

**12. **O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. **8**, 807–819 (2013). [CrossRef]

**13. **J. Chen, J. Ng, Z. Lin, and C. T. Chan, “Optical pulling force,” Nat. Photonics **5**, 531–534 (2011). [CrossRef]

**14. **A. Dogariu, S. Sukhov, and J. Sáenz, “Optically induced ‘negative forces’,” Nat. Photonics **7**, 24–27 (2013). [CrossRef]

**15. **H. Chen, S. Liu, J. Zi, and Z. Lin, “Fano resonance-induced negative optical scattering force on plasmonic nanoparticles,” ACS Nano **9**, 1926–1935 (2015). [CrossRef]

**16. **D. E. Fernandes and M. G. Silveirinha, “Optical tractor beam with chiral light,” Phys. Rev. A **91**, 1–6 (2015). [CrossRef]

**17. **S. Sukhov and A. Dogariu, “Negative nonconservative forces: optical ‘tractor beams’ for arbitrary objects,” Phys. Rev. Lett. **107**, 203602 (2011). [CrossRef]

**18. **A. Novitsky, C.-W. Qiu, and H. Wang, “Single gradientless light beam drags particles as tractor beams,” Phys. Rev. Lett. **107**, 203601 (2011). [CrossRef]

**19. **O. Brzobohatý, V. Karásek, M. Šiler, L. Chvátal, T. Čižmár, and P. Zemánek, “Experimental demonstration of optical transport, sorting and self-arrangement using a ‘tractor beam’,” Nat. Photonics **7**, 123–127 (2013). [CrossRef]

**20. **D. Gao, A. Novitsky, T. Zhang, F. C. Cheong, L. Gao, C. T. Lim, B. Luk’yanchuk, and C.-W. Qiu, “Unveiling the correlation between non-diffracting tractor beam and its singularity in Poynting vector,” Laser Photon. Rev. **9**, 75–82 (2015). [CrossRef]

**21. **A. Mizrahi and Y. Fainman, “Negative radiation pressure on gain medium structures,” Opt. Lett. **35**, 3405–3407 (2010). [CrossRef]

**22. **R. Alaee, J. Christensen, and M. Kadic, “Optical pulling and pushing forces in bilayer PT-symmetric structures,” Phys. Rev. Appl. **9**, 014007 (2018). [CrossRef]

**23. **D. Gao, R. Shi, Y. Huang, and L. Gao, “Fano-enhanced pulling and pushing optical force on active plasmonic nanoparticles,” Phys. Rev. A **96**, 043826 (2017). [CrossRef]

**24. **K. J. Webb and Shivanand, “Negative electromagnetic plane-wave force in gain media,” Phys. Rev. E **84**, 057602 (2011). [CrossRef]

**25. **A. S. Shalin, S. V. Sukhov, A. A. Bogdanov, P. A. Belov, and P. Ginzburg, “Optical pulling forces in hyperbolic metamaterials,” Phys. Rev. A **91**, 063830 (2015). [CrossRef]

**26. **A. Salandrino and D. N. Christodoulides, “Reverse optical forces in negative index dielectric waveguide arrays,” Opt. Lett. **36**, 3103–3105 (2011). [CrossRef]

**27. **M. I. Petrov, S. V. Sukhov, A. A. Bogdanov, A. S. Shalin, and A. Dogariu, “Surface plasmon polariton assisted optical pulling force,” Laser Photon. Rev. **10**, 116–122 (2016). [CrossRef]

**28. **V. Kajorndejnukul, W. Ding, S. Sukhov, C. W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics **7**, 787–790 (2013). [CrossRef]

**29. **K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commun. **5**, 3300 (2014). [CrossRef]

**30. **D. Hakobyan and E. Brasselet, “Left-handed optical radiation torque,” Nat. Photonics **8**, 610–614 (2014). [CrossRef]

**31. **D. G. Baranov, A. Krasnok, and A. Alù, “Coherent virtual absorption based on complex zero excitation for ideal light capturing,” Optica **4**, 1457–1461 (2017). [CrossRef]

**32. **Y. Ra’di, A. Krasnok, and A. Alù, “Virtual critical coupling,” ACS Photon. **7**, 1468–1475 (2020). [CrossRef]

**33. **H. Li, A. Mekawy, A. Krasnok, and A. Alù, “Virtual parity-time symmetry,” Phys. Rev. Lett. **124**, 193901 (2020). [CrossRef]

**34. **G. Trainiti, Y. Ra’di, M. Ruzzene, A. Alù, Y. Radi, M. Ruzzene, and A. Alù, “Coherent virtual absorption of elastodynamic waves,” Sci. Adv. **5**, 1–8 (2019). [CrossRef]

**35. **L. Novotny and B. Hecht, *Principles of Nano-Optics*, 2nd ed. (Cambridge University, 2012).

**36. **A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monticone, and A. Alú, “Anomalies in light scattering,” Adv. Opt. Photon. **11**, 892–951 (2019). [CrossRef]

**37. **A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monticone, and A. Alú, “Anomalies in light scattering,” Adv. Opt. Photon. **11**, 892–951 (2019). [CrossRef]

**38. **A. Archambault, M. Besbes, and J. J. Greffet, “Superlens in the time domain,” Phys. Rev. Lett. **109**, 097405 (2012). [CrossRef]

**39. **V. Grigoriev, A. Tahri, S. Varault, B. Rolly, B. Stout, J. Wenger, and N. Bonod, “Optimization of resonant effects in nanostructures via Weierstrass factorization,” Phys. Rev. A **88**, 011803 (2013). [CrossRef]

**40. **C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. **1**, 16048 (2016). [CrossRef]

**41. **K. Koshelev, G. Favraud, A. Bogdanov, Y. Kivshar, and A. Fratalocchi, “Nonradiating photonics with resonant dielectric nanostructures,” Nanophotonics **8**, 725–745 (2019). [CrossRef]

**42. **A. Krasnok, M. Caldarola, N. Bonod, and A. Alú, “Advanced optical materials spectroscopy and biosensing with optically resonant dielectric nanostructures,” Adv. Opt. Mater. **6**, 1701094 (2018). [CrossRef]

**43. **A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, B. Luk’yanchuk, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science **354**, aag2472 (2016). [CrossRef]

**44. **I. Staude and J. Schilling, “Metamaterial-inspired silicon nanophotonics,” Nat. Photonics **11**, 274–284 (2017). [CrossRef]

**45. **D. G. Baranov, D. A. Zuev, S. I. Lepeshov, O. V. Kotov, A. E. Krasnok, A. B. Evlyukhin, and B. N. Chichkov, “All-dielectric nanophotonics: the quest for better materials and fabrication techniques,” Optica **4**, 814–825 (2017). [CrossRef]

**46. **A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express **20**, 20599 (2012). [CrossRef]

**47. **N. O. Länk, P. Johansson, and M. Käll, “Directional scattering and multipolar contributions to optical forces on silicon nanoparticles in focused laser beams,” Opt. Express **26**, 29074 (2018). [CrossRef]

**48. **M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particle,” Opt. Express **18**, 11428 (2010). [CrossRef]

**49. **D. A. Shilkin, E. V. Lyubin, M. R. Shcherbakov, M. Lapine, and A. A. Fedyanin, “Directional optical sorting of silicon nanoparticles,” ACS Photon. **4**, 2312–2319 (2017). [CrossRef]

**50. **V. Valuckas, R. Paniagua-Domínguez, A. Maimaiti, P. P. Patra, S. K. Wong, R. Verre, M. Käll, and A. I. Kuznetsov, “Fabrication of monodisperse colloids of resonant spherical silicon nanoparticles: applications in optical trapping and printing,” ACS Photon. **6**, 2141–2148 (2019). [CrossRef]

**51. **C. F. Bohren and D. R. Huffman, *Absorption and Scattering of Light by Small Particles* (Wiley, 1998).

**52. **C. W. Hsu, B. Zhen, J. Lee, S. L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature **499**, 188–191 (2013). [CrossRef]

**53. **M. G. Silveirinha, “Trapping light in open plasmonic nanostructures,” Phys. Rev. A **89**, 1–10 (2014). [CrossRef]

**54. **F. Monticone and A. Alù, “Embedded photonic eigenvalues in 3D nanostructures,” Phys. Rev. Lett. **112**, 213903 (2014). [CrossRef]

**55. **I. D. Toftul, K. Y. Bliokh, M. I. Petrov, and F. Nori, “Acoustic radiation force and torque on small particles as measures of the canonical momentum and spin densities,” Phys. Rev. Lett. **123**, 183901 (2019). [CrossRef]