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Abstract
In photonic systems, bilayer or multilayer systems exhibit numerous exciting phenomena induced by twisting. Thus, it is highly desired to explore the twisting effect by engineering the light-matter interactions. Optical torque, an important means in optical micromanipulation, can rotate micro-objects in various ways, enabling a wide range of promising applications. In this study, we present an interesting phenomenon called “pure optical twist” (POT), which emerges when a bilayer structure with specific symmetry is illuminated by counter-propagating lights with opposite spin and/or orbital angular momentum. Remarkably, this leads to zero net optical torque but yet possesses an interesting mechanical effect of bilayer system twisting. The crucial determinant of this phenomenon is the rotational symmetries of each layer, which govern the allowed azimuthal channels of the scattered wave. When the rotational symmetries do not allow these channels to overlap, no resultant torque is observed. Our work will encourage further exploration of the twisting effect through engineered light-matter interactions. This opens up the possibility of creating twisted bilayer systems using optical means, and constructing a stable bilayer optical motor that maintains identical rotation frequencies for both layers.
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1. Introduction
Objects, in general, experience optical torque when exposed to an external optical field, as has been extensively studied [1–3]. As an important means in optical micromanipulation [4–11], optical torque [12,13] can rotate micro-objects in different ways [14–16]. It is capable of doing mechanical work and can provide various promising applications, such as optical spanners [17,18], optical motors [19–22], optical matter machines [2,23–25], among others [26–29]. However, zero optical torque typically requires a force that acts through the center of mass, or the particle possessing pertinent symmetries, or just happens by chance, such as when the torque is switching signs. Here we identified a new mechanism that enforces zero net optical torque but yet produces an interesting mechanical effect of bilayer system twisting. This could assist in constructing a stable bilayer optical motor that keeps the rotation frequencies identical for both layers. Moreover, in photonic systems, bilayer or multilayer material systems exhibit numerous exciting phenomena induced by twisting [30,31], such as broadband strongly bianisotropic responses [32], photonic topological transition [33–35], and so on [36,37]. Thus, it is highly desired to explore the twisting effect by engineering the light-matter interactions [38,39].
In this work, we consider an arbitrary bilayer structure, where the upper and lower layers are identical and have the degree of rotational symmetry $m_s$. The bilayer structure laying on the $xy$-plane is illuminated by two incident lights propagating along $\pm z$ direction with opposite spin and/or orbital angular momentum [see Fig. 1(a) as an example]. In certain bilayer structures with specific $m_s$, the optical torque acting on each layer can be equal and opposite during rotation, irrespective of their relative orientation or the overall symmetry of the system. This optical twist with zero net optical torque is referred as pure optical twist (POT), which is an analogy definition with the pure spin current (a flow of spins without any net charge current) in condensed matter physics. The POT relies on the rotational symmetry of each layer rather than other structural details. Whereas certain rotational symmetries of bilayer systems presenting robust POT are closely related to the azimuthal channels of each incident light, which is determined by the angular analogue of Floquet’s theorem [40–42]. We will first numerically demonstrate the existence of POT by full-wave simulations and then analytically formulate its underlying principle.
Fig. 1. (a) Schematic of a bilayer structure illuminated by two counter-propagating circularly polarized plane waves with opposite helicities. (b)-(e) Optical torques versus the twist angle $\varphi$ for bilayer structures with different degrees of rotational symmetry $m_s = 1,2,3,$ and $4$. The structures are depicted in the insets. The permittivity of dielectric spheres is $\varepsilon = 12.25$ [43,44] with the radius $R = 100$ nm. The nearest sphere distance within the same layer and the interlayer distance are arbitrarily set to $D = 2R+300$ nm (center of sphere to center of sphere). The wavelength and intensity of each incident light are $\lambda = 532$ nm and 1 mW/$\mu$m$^2$, respectively.
2. Pure optical twist in bilayer systems
To enhance clarity and comprehension, we begin by considering an incident light composed of two counter-propagating plane waves with opposite helicities ($\sigma _1 = 1$, $\sigma _2 = -1$) [41,45], as illustrated in Fig. 1(a). The electric fields of the two plane waves are expressed as
For the ease and accuracy of computation and interpretation, we start by considering a bilayer structure consisting of a collection of identical dielectric or metallic spheres. Here, the particle structures are chosen to satisfy different rotational symmetries, disregarding the formation of these structures. The distance between particles is chosen arbitrarily as it doesn’t affect the occurrence of the observed phenomena (see Fig. S1 in the Supplement 1). We stress that our conclusion remains valid for a bilayer structure comprising continuous pieces of material, as evidenced later. The two identical and parallel layers of spheres stack along the $z$ direction, with the lower and upper layers laying symmetrically at $z<0$ and $z>0$, respectively. The time-averaged optical torques exerted individually on the upper and lower layers of the bilayer structure are calculated by [44]
Figures 1(b)–1(e) show the optical torques $\Gamma _{z}^{U, L, T}$ versus the twist angle $\varphi$ between the two layers of bilayer systems with four different rotational symmetries. It is immediately seen that the net optical torques $\Gamma _{z}^{T}$ are in general nonzero, i.e., $\Gamma _z^U\neq -\Gamma _z^L$, for the systems with rotational symmetry $m_s = 1,2$. In contrast, zero net optical torque is generated for systems with $m_s = 3,4$, demonstrating POT. Usually, the optical torque can be affected by the phase because altering the phases $\phi _1$ and $\phi _2$ in Eq. (1) leads to a change in the polarization angle of the optical field and hence the optical torque. Figures 1(b)–1(c) ($m_s = 1,2$) indeed demonstrate a clear dependence of the optical torque on the phase. However, we observe a strikingly different behavior in Figs. 1(d)–1(e) ($m_s = 3,4$), where the POT appears to be completely unaffected by any changes in the phase. The phase shift-insensitive POT greatly facilitates the experimental implementation. Moreover, POT demonstrates robustness against weak symmetry breaking caused by the structure. For instance, for the bilayer structure used in Fig. 1(d), POT remains robust even if the two layers lack strict three-fold symmetry or exhibit slight differences (see Fig. S2 in the Supplement 1 for more details). This robustness is helpful to experimental observation of the POT phenomenon where imperfect settings always exist.
One can image that, in cases where there is no net angular momentum inflow from the total incident light, the scattered light in a system with certain symmetry can also possess zero net angular momentum, resulting in the system acquiring zero torque. However, the scattering process is complex. In order to rigorously demonstrate the reason behind the absence of net torque (i.e. POT), we resort to the equivalent expression of the optical torque by the T-matrix method [40–42,48],
The terms $\Gamma _{1,z}$ and $\Gamma _{2,z}$ represent the optical torques generated by the first and second plane waves, respectively. Since the helicities of these two plane waves are opposite, $\Gamma _{1,z}$ and $\Gamma _{2,z}$ cancel each other. Consequently, the net optical torque is solely determined by $\Gamma _{3,z}$, termed the interference contribution. The expression for $\Gamma _{3,z}$ indicates that only partial waves involving the same azimuthal channel $m$ contribute to the $z$ component of the net optical torque.
According to the angular analogue of the Floquet’s theorem [40,41] (see also, Ref. [42] for a simple proof), the allowed azimuthal channels $m$ of the scattered photons from a scatter with the degree of rotational symmetry $m_s$ are
where $m_i$ is the azimuthal channel of the $i$-th incident wave, with $i = 1,2$ and $n = 0, \pm 1, \pm 2$,…. An important consequence of Eq. (9) is that the interference term $\Gamma _{3,z}$ in Eq. (8) does not vanish only when the degree of the system’s rotational symmetry $m_s$ satisfies where $N$ is an integer. When using the incident fields described by Eq. (1) with the incident azimuthal channel $m_1 = 1$ and $m_2 = -1$ [24,42,49], Fig. 2 illustrates the allowed azimuthal channels of scattered photons when the incident light is scattered by a bilayer structure with $m_s$ degrees of rotational symmetry. The first row of circles represents the azimuthal channels when the incident light with azimuthal channel $m_1 = 1$ is scattered by the bilayer structure. Similarly, the second row of circles represents the azimuthal channels when the counter-propagating incident light with azimuthal channel $m_2 = -1$ is scattered by the same structure. For bilayer structures with a degree of rotational symmetry of $m_s = 1$ or $2$, circles with the same value can be found from the upper and lower rows. For example, when $m_s = 1$, the value of circle $1 - 3 \times m_s$ in the first row equals $-1 - 1 \times m_s$ in the second row. These circles are connected by a red line. Similarly, for $m_s = 2$, the value of circle $1 - 3 \times m_s$ in the first row equals $-1 - 2 \times m_s$ in the second row, and they are connected by a blue line. However, for bilayer structures with rotational symmetry of $m_s = 3$ or larger, circles with the same value cannot be found in these two rows. Instead, they are represented by green cross lines. Therefore, only systems with the degree of rotational symmetry $m_s = 1$ and $2$ are capable of scattering both incident waves into identical azimuthal channels, thus contributing to the value of $\Gamma _{3,z}$. While for $m_s\ge 3$, the angular analogue of the Floquet’s theorem [Eq. (9)] and its resultant condition [Eq. (10)] prohibit the two incident waves from being scattered into the same azimuthal channel $m$.Fig. 2. The circles represent the allowed azimuthal channels of scattered photons when incident lights, described by Eq. (1), are scattered by a bilayer structure. The upper/lower row of circles corresponds to the azimuthal channels when incident light with azimuthal channel $m_1 = 1$/$m_2 = -1$ is scattered by a bilayer structure with a rotational symmetry of $m_s$. For bilayer structures with rotational symmetry of $m_s = 1$ or $2$, circles with the same value can be found in both rows, connected by red or blue lines, indicating that they correspond to the same azimuthal channel. However, for bilayer structures with rotational symmetry of $m_s = 3$ or larger, circles with the same value cannot be found in these two rows, represented by green cross lines, indicating that the same azimuthal channel of scattered photons cannot be generated.
Consequently, this leads to the vanishing of the interference torque $\Gamma _{3,z}$ and, in turn, the vanishing of the POT. The vanishing of $\Gamma _{3,z}$ for $m_s\ge 3$ can be understood as the scattered fields from both layers are “orthogonal” in terms of angular momentum calculation. As a result, POT is immune to the phase shift in both beams, as illustrated in Figs. 1(d) and 1(e). As its underlying physics originates fundamentally from the rotational symmetry of the structure, the POT is not limited to a system composed of spherical particles, as will be shown latter. Moreover, Eqs. (7)–(8) can be applied to structures consisting of any number of layers. As long as the rotational symmetry of the structure does not satisfy Eq. (10), the total optical torque can be ensured to be zero, that is, the occurrence of POT (see Fig. S3 in the Supplement 1 for more details).
Before moving on to more generalized cases, we discuss the possibility of the POT for the structure with $m_s<3$, which is subjected to counter-propagating plane waves illumination. Are there any special cases for structures with $m_s < 3$ that can generate POT? In other words, how can POT be achieved for a bilayer structure whose symmetry satisfies Eq. (10)? In Figs. 3(a) and 3(b), we present optical torques as functions of the phase difference ($\Delta \phi = \phi _2-\phi _1$, $\phi _1$ is, for example, fixed to $\pi /3$) for systems with the degree of rotational symmetry $m_s = 3$ and $m_s = 2$, respectively, while arbitrarily choosing the twist angle $\varphi$ as 130$^{\circ }$. Figure 3(a) confirms that for $m_s = 3$, $\Gamma _z^L = -\Gamma _z^U$ holds true regardless of the phase difference $\Delta \phi$. However, for the system with $m_s = 2$, Fig. 3(b) shows that $\Gamma _z^L$ and $-\Gamma _z^U$ are generally different due to the presence of the non-zero interference term $\Gamma _{3,z}$. Interestingly, at specific points such as $\Delta \phi = \pi /3, 4\pi /3$, the net optical torque can also be zero with $\Gamma _z^L = -\Gamma _z^U$. Further analysis of individual partial wave azimuthal channels in $\Gamma _{3,z}$ reveals that contributions from channels $\pm m$ cancel each other at these particular phase differences. Figure 3(c) illustrates the contributions from dominant azimuthal channels in $\Gamma _{3,z}$ when $\Delta \phi = \pi /3$, corresponding to the red star in Fig. 3(b). The contributions from $m = 1,3$ precisely opposite those from $m = -1,-3$, resulting in a zero net optical torque. It is worth noting that both the total incident field of Eq. (1) situated on the bilayer structure planes [$z = \pm D/2,$ represented by the red arrows in the inset of Fig. 3(c)] and the bilayer structure itself [represented by the black lines in the inset of Fig. 3(c)] exhibit mirror symmetry with respect to the $y$ direction when projected on the $xy$-plane in this situation. Other special points with zero net optical torque correspond to “mirror” symmetries along either the $x$ or $y$ directions. Furthermore, we explicitly present the results for $\Delta \phi = 2\pi /3$ [blue stars in Fig. 3(b)] in Fig. 3(d). Here, the contributions from $m$ and $-m$ no longer cancel each other since the total incident field lose “mirror” symmetry along the $x$ and $y$ directions, as shown in the inset of Fig. 3(d). This finding reveals the crucial role of “mirror” symmetry [40] in achieving the POT when the resultant condition of the angular Floquet’s theorem [Eq. (10)] is satisfied. It represents a typical scenario where the POT occurs “accidentally”. In the following, we will exclude this “accidental” case induced by the particular phase difference and focus on the POT that is due to the symmetry and immune to the phase.
Fig. 3. Optical torques exerted on the lower and upper (opposite value of $\Gamma _z^U$ is taken to facilitate comparison) layers of a bilayer system (twist angle $\varphi$ = 130$^{\circ }$) versus the phase difference $\Delta \phi$ are shown in (a) $m_s$ = 3 and (b) $m_s$ = 2. The wavelength and intensity of the two incident counter-propagating plane waves ($\sigma _1$ = 1, $\sigma _2$ = −1) is $\lambda$ = 532 nm and 1 mW/$\mu$m$^2$, respectively. For $m_s = 2$, the contributions to the torque $\Gamma _{3,z}$ by individual dominant partial wave azimuthal channels are depicted versus $kR$ in panels (c) and (d) for $\Delta \phi$ = $\pi$/3 and $\Delta \phi$ = 2$\pi$/3, respectively. In the insets of (c) and (d), the black solid lines and dashed lines represent the upper and lower layers of the bilayer structure with $m_s = 2$ projected onto the $xy$-plane, respectively. These lines can be visualized as the axis passing through the three spheres in each layer. The red/blue solid line arrow or dashed arrow in the insets of (c)/(d) indicates the direction of the total incident field projected onto the $xy$-plane at $z = D/2$ or $z = -D/2$. Other parameters of the system are $\varepsilon = 12.25$, $R = 100$ nm, and $D = 500$ nm.
Next, we investigate POT on a bilayer structure in the illumination of beams carrying both orbital angular momentum (OAM) and spin angular momentum (SAM). The Laguerre-Gaussian (LG) beams [40,41] serve best for this purpose. When two linearly polarized counter-propagating LG beams carry OAM characterized by $l_1 = 3$, $l_2 = -3$ [40], the corresponding incident azimuthal channels are $m_1 = l_1\pm 1 = 2,4$ and $m_2 = l_2\pm 1 =-2,-4$. Here, the $\pm 1$ values correspond to linear polarization (see Sec. 2 in the Supplement 1), which can be considered as a superposition of circular polarizations with $\sigma = 1$ and $-1$. Thus $m_1 - m_2$ take values of $4,6,8$. According to Eq. (10), there will not be POT if $m_s$ is a divisor of $m_1 - m_2$. Therefore, POT will occur for $m_s = 5, 7, 9,$ and so on, as illustrated in Fig. 4(a) (depicted as blue dots), obtained through numerical calculations based on the multi-spheres scattering theory [46,47]. On the other hand, when the two LG beams are circularly polarized with $\sigma _1 = 1$ and $\sigma _2 = -1$, the incident azimuthal channels are $m_1 = l_1+\sigma _1 = 4$ and $m_2 = l_2+\sigma _2 = -4$. Following Eq. (10), the POT will show up when $m_s$ is not a divisor of $m_1-m_2 = 8$. In other words, the POT occurs for $m_s = 3,5,6,7,9,$ and so forth, as corroborated by numerical results (denoted by red squares) in Fig. 4(a). Furthermore, in the case where $l_1 = 1$, $l_2 = -1$, $\sigma _1 = -1$, and $\sigma _2 = 1$, we find that $m_1 = m_2 = 0$, resulting in any $m_s$ being a divisor of $m_1-m_2 = 0$. Consequently, the production of the POT becomes impossible for any structures under these conditions, as illustrated in Fig. 4(b). Table 1 showcases additional instances of counter-propagating waves carrying spin and/or orbital angular momentum.
Fig. 4. The net optical torque exerted on a bilayer structure composed of identical silver [44] spheres ($R$ = 100 nm, $D$ = 2$R$+200 nm, twist angle $\varphi$ = 30$^{\circ }$) illuminated by counter-propagating LG beams (numerical aperture NA $=$ 0.6, beam power 0.1 W, $\lambda =$532 nm, and phase shift $\phi _1$ = $\pi$/3, $\phi _2$ = $\pi$/4) versus $m_s$ are shown in (a) $l_1$ = 3, $l_2$ = −3 and (b) $l_1 =$ 1, $l_2 =$−1, $\sigma _1$ = −1, $\sigma _2$ = 1. The structures of $m_s = 1,2,3,4$ use the same structures as those shown in Figs. 1(b)-1(e), and structures of $m_s = 5,6,7,8,9$ are rings composed of $m_s$ spheres, see also Fig. S4 in the Supplement 1. The blue dots and red squares in (a) correspond to LG beams with $x$-polarized linear ($\sigma _1$ = $\sigma _2$ = 0) and circular ($\sigma _1$ = 1, $\sigma _2$ = −1) polarizations, respectively. (c) Optical torques calculated via two different approaches exerted on the upper and lower plates of a bilayer structure with $m_s$ = $4$ versus the twist angle $\varphi$. The schematic plot of the whole optical system is shown in the inset. The bilayer structure consists of two identical $30$ nm thick square-shaped gold [20] plates separated by a gap of $86$ nm, with an area of 255 $\times$ 255 nm$^2$. The incident light constitutes of two counter-propagating circularly polarized Gaussian beams with opposite helicities (NA = 0.6, $\lambda =$ 1064 nm, beam power 0.1 W, and phase shift $\phi _1$ = $\pi$/3, $\phi _2$ = $\pi$/4). (d) Optical torques exerted on the upper and lower plates of the bilayer structure contributed by the SAM and OAM.
Table 1. The degree of rotational symmetry $m_s$ for systems producing robust POT in linearly polarized (LP) or circularly polarized (CP) counter-propagating Gaussian or Laguerre-Gaussian (LG) beams. Here, $N$ is an integer.
Our analysis reveals that the emergence of the POT is solely dependent on the appropriate symmetry of the system, irrespective of other system details. To further validate this finding, we examine a bilayer structure comprising two absorptive gold square plates (instead of sphere aggregate), illuminated by two circularly polarized counter-propagating Gaussian beams with opposite helicities [see the inset of Fig. 4(c)]. In this case, one has $m_1 = 1$ and $m_2 = -1$, Eq. (10) indicates the presence of POT for the $m_s = 4$ system. Figure 4(c) indeed exhibits the POT phenomenon, confirming that POT is only determined by the symmetry governed by the angular analogue of Floquet’s theorem and is not influenced by other system particulars. Here, the total electromagnetic fields are obtained by using a commercial finite-element-method package COMSOL Multiphysics (http://www.comsol.com/). The torques are obtained through direct calculations from Maxwell’s theory ($\Gamma _{\text {MST}}$) [42,52], as well as by integrating the SAM and OAM current densities (denoted as $\Gamma _{\text {SAM}}$ and $\Gamma _{\text {OAM}}$, respectively, see Sec. 4 in Supplement 1 for the expressions of SAM and OAM current densities [53]). The results obtained through these two methods agree well. In addition, by comparing Fig. 1(e) and Fig. 4(c), it can be seen that for structures with rotational symmetry $m_s = 4$, the number of peaks for the optical torque exerted on each layer differs within one period (the twist angle $\varphi$ runs from 0$^{\circ }$ to 90$^{\circ }$ ). Strong interlayer coupling enhances the torque exerted on each layer. The enhanced coupling is related to both the symmetry and the specific details of the structure. Specifically, the number of maximum values for optical torque on each layer are determined by the structure’s symmetry, while the number of minor peaks is also associated with the structure’s details (see Fig. S5 and Fig. S6 in the Supplement 1 for more details).
Finally, we explore the conversion between SAM and OAM when the POT occurs. The conversion of SAM to OAM can occur when light is scattered by particles [54,55]. Figures 4(c) and 4(d) use nearly paraxial (numerical aperture NA = 0.6) circularly polarized Gaussian beams. In this context, the incident lights can be regarded as carrying only SAM (refer to Fig. S7 in Supplement 1). However, as depicted in Fig. 4(d), both SAM and OAM contribute to the optical torque in each layer. This indicates that, for each layer, a conversion from spin angular momentum to orbital angular momentum occurs, and the resulting difference contributes to the torque experienced by that layer. Specifically, the conversion between SAM and OAM occurs at individual layer but not on the entire system. As depicted in Figs. 4(c) and 4(d), the lower plate acquires negative SAM from the field while radiating OAM (acquiring positive OAM), manifesting itself as a partial conversion from SAM to OAM, with the difference contributing to $\Gamma _z^L$. Interestingly, the upper plate does the opposite and completely compensates for the consequence caused by the lower plate, thus resulting in the individual conservation of SAM and OAM of the illuminating field while leaving us with a POT in the system.
3. Conclusion
In conclusion, we identified a new mechanism that enforces zero net optical torque but yet possesses an interesting mechanical effect of bilayer system twisting. This intriguing behavior, which we refer to as POT, manifests as the optical torque exerted on each layer of a bilayer structure being exactly equal in magnitude but opposite in direction during rotation. The underlying mechanism can be traced to the angular analogue of Floquet’s theorem, which dictates the possible azimuthal channels of the scattered waves from both layers. When these scattered waves from both layers are “orthogonal”, meaning they possess different azimuthal numbers and cannot interfere, the POT is generated. As the POT stems from the system’s inherent symmetry rather than its specific details, it is believed to be universal and fundamentally different from the accidental vanishing of the optical torque. Our work enhances the understanding of light-matter interactions and opens up the new possibility for developing twisted bilayer structure systems with optical means, a stable bilayer optical motor for DNA unfolding, etc.
Funding
National Natural Science Foundation of China (12174231, 12074084, 12074169, 12074230); Fundamental Research Program of Shanxi Province (202103021222001); Fund for Shanxi 1331 Project; Research Project Supported by Shanxi Scholarship Council of China; Stable Support Plan Program of Natural Science Foundation of Shenzhen Municipality (20200925152152003); Program of Education and Teaching Reform in Shanxi Province (J20230003).
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.
Supplemental document
See Supplement 1 for supporting content.
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