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Abstract
Exceptional points (EPs)—non-Hermitian degeneracies at which eigenvalues and eigenvectors coalesce—can give rise to many intriguing phenomena in optical systems. Here, we report a study of the optical forces on chiral particles in a non-Hermitian system at EPs. The EPs are achieved by employing the unidirectional coupling of the chiral particles sitting on a dielectric waveguide under the excitation of a linearly polarized plane wave. Using full-wave numerical simulations, we demonstrate that the structure can give rise to enhanced optical forces at the EPs. Higher order EPs in general can induce stronger optical forces. In addition, the optical forces exhibit an intriguing “skin effect”: the force approaches the maximum for the chiral particle at one end of the lattice. The results contribute to the understanding of optical forces in non-Hermitian systems and can find applications in designing novel optical tweezers for on-chip manipulations of chiral particles.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Light can induce forces on particles due to momentum conservation. Optical forces play an essential role in small particle manipulations with broad applications in biology, chemistry, and physics [1–6]. In recent years, optical forces induced by near fields have drawn much attention owning to their sizable magnitudes and the associated novel phenomena derived from the rich properties of the near fields [7–16]. Interestingly, near-field interaction can induce a counterintuitive lateral optical force with potential application in sorting chiral particles [9,10,16–20]. In addition, there is growing interest in optical forces in non-Hermitian systems [21–23]. Non-Hermitian systems have a peculiar type of degeneracy called exceptional points (EPs), at which the eigenvalues and corresponding eigenvectors coalesce [24–27]. Many interesting phenomena can happen when non-Hermitian systems approach the EPs, which have important applications in optical isolation [28,29], optical switches [30–32], and unconventional laser photonics [33–35], etc. Study of optical force from the perspective of non-Hermitian physics has uncovered the inherent instability of large clusters in optical trapping [23]. The unusual properties of EPs can significantly enrich optical force phenomena and bring new functionalities for optical manipulations. The research in this direction is still in its infancy. In particular, the effect of non-Hermitian EPs on near-field optical forces remains unknown.
A straightforward approach to realize arbitrary order EPs is by exploiting the unidirectional coupling between multiple resonators. For a two-level system described by a non-Hermitian Hamiltonian $H = \left[ {\begin{array}{{cc}} a&c\\ d&b \end{array}} \right]$, an EP emerges at the condition ${({a - b} )^2} + 4cd = 0.$ This can happen when $a = b$ and either $c = 0$ or $d = 0,$ corresponding to two identical states with unidirectional coupling [36,37]. Unidirectional coupling can be realized via adjusting the interferences of scatterers [36,38] or by utilizing the spin-momentum locking associated with the evanescent wave of waveguide modes [8,9,39,40]. The latter is a type of photonic spin-orbit interaction and has been applied to realize arbitrary-order EPs in coupled dipole spheres [37] and to achieve light funneling with coupled helix particles [41].
In this paper, we investigate the optical forces on chiral particles that unidirectionally couple with each other via a dielectric waveguide and explore the effect of EPs on the optical forces. The chiral particles support a chiral dipole mode that excites the guided wave propagating unidirectionally in the waveguide owning to the spin-orbit interaction [9,41,42]. At the frequency of the chiral dipole mode, the coupling between the chiral particles is unidirectional, giving rise to an EP of arbitrary order determined only by the number of the chiral particles. By applying full-wave numerical simulations and coupled-mode theory (CMT), we show that the EP can induce different optical force spectra for different chiral particles. In particular, it can strongly enhance the chirality-dependent lateral force.
The paper is organized as follows. In Section 2, we introduce the CMT for understanding the mechanism of arbitrary order EPs. In Section 3, we present and discuss the numerical results about the optical forces on the chiral particles at the EPs of different orders. We draw the conclusion in Section 4.
2. Coupled mode theory for unidirectionally coupled resonators
The mechanism of realizing arbitrary order EPs via unidirectional coupling is summarized as follows (a detailed discussion can be found in Ref. [37]). We first consider the coupling of two identical optical resonators, which can be described by the following rate equations according to the CMT [37,43,44]:
For the above effective Hamiltonian, an EP will appear if ${\kappa _{12}} = 0$ and ${\kappa _{21}} \ne 0.$ Assuming the time-harmonic mode field ${a_i} = {A_i}\textrm{exp}({ - \textrm{i}\omega t} )$ and incident field ${a_{\textrm{in}}} = {A_{\textrm{in}}}\textrm{exp}({ - \textrm{i}\omega t} ),$ the mode amplitudes of the resonators can be determined as [37]:
The above results can be straightforwardly generalized to N unidirectionally coupled resonators, where the effective Hamiltonian can be expressed as
Here, only the coupling parameters ${\kappa _{ij}}$ with i > j exist due to the unidirectional coupling. The eigenvalues of this effective Hamiltonian are degenerate at $\omega = {\omega _0} - i\Gamma /2,$ corresponding to a Nth order EP. A special case that is particularly interesting is when the coupling parameter ${\kappa _{ij}}$ take real values, corresponding to the constructive or destructive interferences among the resonators. In this case, the mode amplitudes of the resonators exhibit a monotonic dependence on a spatial dimension similar to the non-Hermitian skin effect [45].
3. Optical forces on chiral particles at exceptional points
3.1 Unidirectional coupling and optical force of a single chiral particle
To achieve unidirectional coupling, we consider the configuration shown in Fig. 1(a), where a single helix sits above a strip waveguide made of silicon (relative permittivity 12). The helix is made of silver whose relative permittivity can be described by the Drude model ${\varepsilon _{\textrm{Ag}}} = 1 - \omega _p^2/({{\omega^2} + i\omega \gamma } )$, where ${\omega _p} = 1.37 \times {10^{16}}\textrm{rad}/\textrm{s}$ and $\gamma = 2.73 \times {10^{13}}\textrm{rad}/\textrm{s}$ [40]. The helix has outer radius R = 92 nm and inner radius r = 11 nm, and it has four turns with a pitch of P = 50 nm. The waveguide has dimensions w = 310 nm and h = 640 nm. The distance between the center of the chiral particle and the upper surface of the waveguide is g = 126 nm. The system is excited by a linearly polarized plane wave with the electric field amplitude E0 = 1 V/m. The wavevector and the polarization direction are both in the yz-plane. The incident angle $\theta $ is defined as the angle between the wavevector k and -y axis. For simplicity, we assume the plane wave only incidents on the particle and neglect the background scattering by the waveguide.
Fig. 1. (a) Chiral particle on a waveguide under the incidence of a linearly polarized plane wave. The helix has four turns with pitch P = 50 nm, major radius R = 92 nm, and minor radius r = 11 nm. The plane wave is linearly polarized in the zy-plane, and the wavevector k forms an angle of θ with the -y direction. (b) Unidirectionality as a function of rotation angle β and incident angle θ. (c) Relative amplitude and phase of the dipole components px and pz. (d) Unidirectional guided wave excited by the chiral particle.
We conducted full-wave simulations of the system by using COMSOL Multiphysics, assuming absorbing boundary condition at both ends of the waveguide to suppress reflections. Under the excitation of the linearly polarized plane wave, the scattering field of the chiral particle will couple to the waveguide. We define the unidirectionality of the coupling as $\alpha = S(+ )/S(- )$, where $S(+ )$ and $S(- )$ represent the time-averaged power of the guided wave propagating in + x and −x directions, respectively. Since the helix is anisotropic, the unidirectionality $\alpha $ varies with the incident angle θ and the rotation angle β. The rotation angle β is defined in the bottom inset of Fig. 1(a), which characterizes the rotation of the helix with respect to its center axis. Figure 1(b) shows the value of $\alpha $ at the frequency f = 118.4 THz for different incident angles and rotation angles. At about θ = 75 degrees and β = 0 degree, $\alpha $ has a local maximum larger than 50, corresponding to near-perfect unidirectionality. To understand the mechanism underlying the near-perfect unidirectionality, we plot the phase difference and relative amplitude of the electric dipole components px and pz induced in the helix, as shown in Fig. 1(c). As seen, px and pz have approximately equal amplitude and a phase difference of 90 degrees at f = 118.4 THz, corresponding to an approximately circularly polarized electric dipole. The spin of the electric dipole directly determines the excitation of the guided wave. Since the evanescent tail of the guided wave contains both longitudinal and transverse field components with a phase difference of $\pi /2$, it carries an intrinsic spin in the transverse direction. Importantly, the guided waves propagating in + x and −x directions carry spin with opposite directions, a property called spin-momentum locking [39]. The spin of the + x-propagating guided wave matches the spin of the circularly polarized electric dipole. Consequently, the excited guided wave propagates in + x direction only, as shown in Fig. 1(d).
The optical force acting on the chiral particle can be numerically evaluated by integrating the Maxwell stress tensor on a closed surface enclosing the particles: ${\mathbf F} = \mathrm{\oint }\left\langle {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathbf T}} }} \right\rangle \cdot \hat{\mathbf n} dS$, where $\left\langle {{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathbf T}} }} \right\rangle = 1/2\textrm{Re}\left[ {{\varepsilon_0}{{\mathbf E}^\mathrm{\ast }}{\mathbf E} + {\mu_0}{{\mathbf H}^\mathrm{\ast }}{\mathbf H} - 1/2({{\varepsilon_0}|{{\mathbf E}{|^2} + {\mu_0}} |{\mathbf H}{|^2}} ){\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathbf I}} }} \right]$ is the time-averaged Maxwell stress tensor; $\hat{\mathbf n} $ represents the outward unit normal vector of the closed surface; E and H correspond to the total fields; and ${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over {\mathbf I}} }$ is the unit tensor. For comparison with the optical force at EPs in Sec. 3.2, we calculate the force on the single particle in Fig. 1(a) for θ = 75 degrees and β = 0 degree. The cartesian components of the force as a function of frequency are shown in Fig. 2(a-c). The Fx component corresponds to the lateral force on the chiral particle, which is induced by the coupling between the particle and the waveguide. The guided wave excited by the chiral particle propagates in + x direction, contributing to a recoil force on the particle in the -x direction. As shown in Fig. 2(a), the amplitude of the lateral force has a maximum value at f = 118.4 THz, where the electric dipole is approximately circularly polarized and the unidirectionality $\alpha $ approaches the local maximum. Figure 2(b) shows the Fy component along the propagation direction of the incident light (corresponding to radiation pressure), which has a peak amplitude at f = 118 THz. Figure 2(c) shows the Fz component of the optical force, which is dominated by the gradient force induced by the strong evanescent near fields. The amplitude of Fz also reaches the maximum at f = 118 THz due to the strong coupling between the resonating chiral particle and the waveguide [12]. To understand the effect of anisotropy of the particle on the lateral force, we calculate the lateral force Fx as a function of the rotation angle β, and the result is shown in Fig. 2(d). As seen, Fx is negative for $\beta \in [{ - 90,\; 90} ]$ degrees and is positive for other rotation angles, indicating the lateral force can change sign when the helix rotates. The lateral motion of the particle is mainly determined by the negative lateral force since its peak magnitude is much larger than that of the positive lateral force.
Fig. 2. Optical force acting on the chiral particle as a function of frequency: (a) x component, (b) y component, and (c) z component. (d) The x component of the optical force as a function of the rotation angle β (degree) at the resonance frequency f = 118.4 THz.
3.2 Optical forces at the second order exceptional point
We now consider two identical helices separated by the distance d couple with each other through the waveguide, as shown in Fig. 3(a). We set the rotation angle of the helices to be β = 0 to obtain a high unidirectionality of the helix-waveguide coupling, which is the key to the realization of the EPs. The excitation is the same as in Fig. 2(a). In this case, the guided waves excited by the two chiral particles both propagate in + x direction, which gives rise to the unidirectional coupling from particle 1 to particle 2, and a second order EP is achieved. Figure 3(b) shows the unidirectionality $\alpha $ as a function of d/λ and frequency for this case. We see that $\alpha $ reaches the maximum near the resonance frequency of the particles, and it varies with d/λ due to the interference between the two guided waves. At d/λ = 2.5, the guided waves interfere destructively, leading to almost vanished $S(+ )$ and thus small $\alpha $. On the other hand, the unidirectionality is enhanced by the constructive interference that happens at integer values of d/λ. Figure 3(c) and 3(d) show the dipole amplitudes p1 and p2, respectively, as a function of frequency and d/λ. Evidently, p1 is insensitive to the variation of d/λ, while p2 exhibits an interference-induced profile changing with d/λ. This agrees with Eq. (4) of the analytical CMT. At the resonance frequency $\omega = {\omega _0}$, the p2 in Eq. (4) is reduced to $\left|{4\sqrt {{\gamma_c}} {A_{\textrm{in}}}({\Gamma /2 - i{\kappa_{21}}} )/{\Gamma ^2}} \right|,$ where the complex coupling parameter ${\kappa _{21}}$ depends on the separation distance d Thus, the value of p2 will change with d. We notice that the maximum (minimum) value of p2 does not appear at integer (half-integer) values of d/λ, which is attributed to the phase of the coupling between the particles and the waveguide.
Fig. 3. (a) Two chiral particles couple via a silicon waveguide under the incidence of a linearly polarized plane wave. (b) Unidirectionality as a function of the frequency and coupling distance d/λ. The amplitude of the electric dipole moment induced in (c) the first and (d) the second particle as a function of frequency and the coupling distance d/λ.
We numerically evaluate the optical forces on the two chiral particles. In what follows, we use Fix, Fiy, and Fiz to denote the cartesian components of the force on the ith particle. Figures 4(a) and 4(b) show the results of the lateral forces F1x and F2x, respectively. We see that F1x is negative and remains approximately unchanged as d/λ varies, while F2x undergoes large variation. The value of F2x is negative (positive) when d/λ take integers (half-integers). This can be understood as follows. When d/λ is an integer, the guided wave propagating in + x direction has a larger power due to the constructive interference, leading to an enhancement of the recoil lateral force F2x. When d/λ is a half integer, the guided wave propagating in + x direction is suppressed due to destructive interference, and the small reflection of the guided wave at the second particle gives rise to a small positive F2x. Figures 4(c) and 4(d) show the forward optical forces F1y and F2y acting on the two chiral particles. It is seen that F1y remains approximately unchanged when d/λ varies, while F2y has large variation due to the interference. The y-component optical force is attributed to the scattering of the chiral particles, which is enhanced in the case of constructive interference of the guided waves and is reduced in the case of destructive interference. Figures 4(e) and 4(f) show the forces F1z and F2z, which are dominated by the gradient force on the chiral particles. The gradient force can be understood as resulting from the interaction between the induced dipoles and their image dipoles, which is proportional to the amplitude of the electric dipole moment. Therefore, F1z is nearly independent of d/λ, while the magnitude of F2z is amplified (reduced) when p2 is enhanced (suppressed) due to the interference. This agrees with the results in Figs. 3(c) and 3(d).
Fig. 4. The x component of the optical force acting on (a) the first and (b) the second chiral particle as a function of frequency and coupling distance d/λ. The y component of optical force acting on (c) the first and (d) the second chiral particle as a function of frequency and coupling distance d/λ. The z component of optical force acting on (e) the first and (f) the second chiral particle as a function of frequency and coupling distance d/λ.
3.3 Optical forces at higher order exceptional points
To illustrate the effect of higher order EPs on the optical forces, we consider four identical chiral particles separated with a distance d, as shown in Fig. 5(a). The structural parameters and the excitation are the same as in Fig. 3(a) to maintain a high unidirectionality of the coupling. The system can give rise to a fourth order EP according to the theory in Sec. 2. Figure 5(b) shows the numerically computed unidirectionality $\alpha $ as a function of the frequency and separation d/λ, where four peaks appear due to coupling of the four particles. The value of $\alpha $ is much larger than that of the second order EP in Fig. 4(b). Figure 5(c-f) shows the electric dipole amplitudes of the four particles. Generally, the n-th particle has n−1 local maxima in the electric dipole moment, which is attributed to the couplings from the n−1 particles sitting on its left side. The dipole moment of the particle 1 (as labelled in Fig. 5(a)) remains unchanged when d/λ varies, while the other particles undergo large variations due to the interferences of the unidirectionally propagating guided waves. In addition, the dipole moment increases from particle 1 to particle 4 generally.
Fig. 5. (a) Four chiral particles couple via a silicon waveguide under the incidence of a linearly polarized plane wave. The particles are labelled as 1-4. (b) Unidirectionality as a function of the frequency and coupling distance d/λ. The dipole moment of (c) the first and (d) the second and (e) the third and (f) the fourth chiral particle as a function of frequency and the coupling distance d/λ.
We evaluated the optical force on each chiral particle, and the results are shown in Fig. 6. Figure 6(a-d) shows the x-component optical force (i.e., lateral force) on each particle as a function of the frequency and the separation d/λ. As seen, the forces have similar features as the electric dipole moments in Fig. 5. From particle 1 to particle 4, the number of local maxima and minima increases due to the increased number of guided waves in the interference. The lateral force F1x on particle 1 is negative at the resonance frequency and is independent of d/λ. In contrast, the lateral forces on the other particles can become positive when d/λ changes, due to the interference of the guided waves. Interestingly, at the condition of constructive interference (i.e., d/λ takes integer values), the lateral forces are significantly enhanced, with the strongest lateral force appears for the particle 4. Figure 6(e-h) shows the y-component optical forces on the four particles. The forces remain negative when d/λ changes, and their magnitudes increase when constructive interference happens at integer values of d/λ. In addition, the constructive-interference-enhanced force monotonically increases from the particle 1 to the particle 4. Figure 6(i-l) shows the z-component optical forces of the particles. Akin to the second order EP system, the force patterns are consistent with the patterns of the dipole moments in Fig. 5. The forces are enhanced when the electric dipole moments are enlarged due to the interference of the guided waves. Under a typical laser power with electric field amplitude of ∼107 V/m [46], the lateral forces Fx and forward scattering forces Fy in this case are on the order of ∼0.1 pN, while the gradient forces Fz are on the order of ∼10 pN.
Fig. 6. The x component of the optical force acting on (a) the first and (b) the second and (c) the third and (d) the fourth particle as a function of frequency and coupling distance d/λ. The y component of optical force acting on (e) the first and (f) the second and (g) the third and (h) the fourth particle as a function of frequency and coupling distance d/λ. The z component of optical force acting on (i) the first and (j) the second and (k) the third and (l) the fourth particle as a function of frequency and coupling distance d/λ.
4. Conclusion
In conclusion, we investigate the optical forces exerted on coupled chiral helix particles at non-Hermitian EPs of different orders. We show that the optical forces can be strongly enhanced by the EPs, and higher order EPs generally give rise to larger enhancement of the forces. In addition, the optical forces have large variations as the particle separation changes, and different particles exhibit different force patterns depending on their relative positions in the system. The phenomena are attributed to the interference of the unidirectional guided waves excited by the chiral particles. The results contribute to the understanding of chiral light-matter interactions in non-Hermitian systems and may find applications in optical force sorting of chiral particles and chiral sensing.
Funding
National Natural Science Foundation of China (11904306); Research Grants Council of the Hong Kong Special Administrative Region, China (AoE/P-502/20, CityU 11301820).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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