## Abstract

Gears are rotating machines, meshing with each other by teeth to transmit torque. Interestingly, the rotating directions of two meshing gears are opposite, clockwise and counterclockwise. Although this opposite handedness motion has been widely investigated in machinery science, the analogue behavior of light remains undiscovered. Here, we present a simple nanophotonic directional coupler structure which can generate two light beams with opposite handedness of polarization states—optical gears. Due to the abrupt phase shift effect and birefringence effect, the angular momentum (AM) states of photons vary with the propagation distance in two adjacent waveguides of the coupler. Thus, by the choice of coupling length, it is able to obtain two light beams with opposite handedness of polarization, confirming the appearance of optical gears. The full control in the handedness of output beams is achieved via tuning the relative phase between two orthogonal modes at the input port. Optical gears thus offer the possibility of exploring light-matter interactions in nanoscale, opening up new avenues in fields of integrated quantum computing and nanoscale bio-sensing of chiral molecules.

© 2017 Optical Society of America

## 1. Introduction

The abrupt phase shift is a fundamental phenomenon in many classical and quantum resonant systems where energy exchange is possible, such as RLC circuits [1], coupled pendulums [2] and quantum dots (QDs) [3], as shown in Fig. 1(a). In optics, this phenomenon has been observed both in bulk and nanophotonic systems, i.e., interface reflections [Fig. 1(b)] [4], metasurfaces [5–7] and directional couplers [8]. The abrupt phase shift has been recently gathering increasing interest, as it plays an important role in many light-matter interactions with exotic effects, such as negative refraction and reflection [9], photonic spin Hall effect [10], spin-orbit coupling [11] and chiral beam distinguishing [12–14].

In particular, this abrupt phase shift occurs when light coupling in a directional coupler consisting of two photonic waveguides (WGs). Already a number of integrated optics devices have been realized based on directional couplers, such as optical filters [15], 3-dB splitters [16], polarization beam splitters [17, 18], PT-symmetric nonlinear couplers [19], entangled photon-pairs sources [20], two-photon quantum interference [21], integrated quantum logical gates [22], and all-optical data processing [23]. However, most of those demonstrations have been focused on a single quasi-linearly polarized mode and mainly discussed the energy exchange between two adjacent waveguides. Innovations concerning multi-polarized modes, i.e. quasi-circularly (or elliptically) polarized modes [24, 25], and the abrupt π phase shift, which is introduced by the coupling process, remain largely unexplored.

In this work, we show that they have much potential for creations of novel devices as well. For example, a directional coupler is a crucial ingredient for the manipulation of angular momentum (AM) of light in nanophotonic waveguides when two orthogonal polarized modes are involved. The AM of photons has great potential in nanophotonics [26]. Therefore, it is of much practical significance to study the AM behavior in a directional coupler. In our findings, by engineering the length of coupling region, it is possible to construct an optical analogue of two meshing gears, where the quasi-elliptically polarized modes have opposite handedness in two adjacent waveguides at the output ports. To our knowledge, it is the first time that this new concept of optical gears is proposed. In addition, the handedness of the output modes can be manipulated via the choice of the relative phase ($\Delta {\varphi}_{0}$) of quasi-TE and -TM modes at the input port [Fig. 1(c)]. Interestingly, our scheme is conceptually different from previous methods for manipulation of AM, such as birefringence effect caused by optical crystals [27] and abrupt phase change introduced by nano-resonators [28]. Instead, we show that, in the coupling region of a directional coupler, the phase lag between the two orthogonally polarized modes is modulated via two factor: the abrupt phase shift and the birefringence effect that happens in the coupling process.

## 2. Results and discussion

The proposed scheme is sketched in Fig. 1(c). The directional coupler consists of two uniform parallel silicon (Si, *n* = 3.476) waveguides. The width (w) and height (h) of each waveguide are identical, w = h = 340 nm, and the gap between them is g = 40 nm. We assume the whole structure is surrounded by silica (SiO_{2}, *n* = 1.444) and the operating wavelength is 1.55 μm.

We first discuss the abrupt phase shift and the birefringence effect in the coupler, and then the opposite handedness of polarization behavior. As light propagates along the coupler, it couples from the first waveguide (WG1) to the second one (WG2) and then couples back to the first one again. By using the coupled mode approach, the light field dynamics of the coupling region is described by

*β*

_{1,2}=

*n*

_{1,2}

*k*

_{0}are the propagation constants, and

*n*

_{1}=

*n*

_{2}the effective indices,

*k*

_{0}= 2π/λ the free space wavenumber, while $\kappa =\pi /(2{z}_{c})$ the coupling coefficient with the coupling length

*z*

_{c}_{.}For our single-mode directional coupler, the light energy can be 100% exchanged between two waveguides, and Eq. (1) can be solved analytically,

_{1}(0) = 1 and a

_{2}(0) = 0. For 0 < z < z

_{c}, Eq. (2) can be simply written as,

The *–j* term in Eq. (3) implies an intrinsic phase lag of π/2 for the light field in the WG2 compared with the one in the WG1. This solution is well described for the energy coupling process, but not sufficient for the description of the phase evolution. To make up this defect, it has to be modified by some mathematical transformations every time when light is totally coupled from one waveguide to another. Thus applying the Euler's formula and some mathematical transformations to Eq. (3), a general description for the energy and phase evolution in the coupler can be written as

_{c}< z < (n + 1)z

_{c}, where n is an integer. Equation (4) is a periodic solution with a period T = 4z

_{c}. In addition, for $z\in \left[2n{z}_{c},(2n+1){z}_{c}\right]$, the phase of light field in WG1 is π/2 in advance compared with the one in WG2, while an opposite situation occurs for $z\in \left[(2n-1){z}_{c},2n{z}_{c}\right]$. Besides, at every point where z = (2n + 1)z

_{c}, a π phase shift happens in WG1 and the abrupt phase shift (π) occurs in WG2 at the points where z = 2nz

_{c}. To visualize this finding, at the bottom of Fig. 2, we plot the evolution of relative phase between WG1 and WG2 in the complex plane for each period.

It should be emphasized that light coupling between waveguides is a resonance phenomenon, which is an analogy to the standing wave of a laser's resonant cavity consisting of two mirrors. There is also a π phase shift between the incident light and the reflected light when the light is reflected by the mirrors, which can be well explained by Fresnel equations [29].

To discuss the birefringence effect in the coupler, we apply the supermode solution to analyse the coupling process. In the coupling region, the coupler can be regarded as a two cores waveguide, and the guided modes can be linearly represented by a symmetric (even, β_{+}) and an anti-symmetric (odd, β_{-}) modes. Usually, the propagation constants of the even and odd modes are not equal (β_{+} ≠ β_{-}) but with small difference. Thus their interference pattern results in a beat in the waveguides, with the beat length z_{b} = 2z_{c} = 2π/(β_{+} - β_{-}), where β_{+,-} = n_{+,-}k_{0} are the propagation constants and n_{+,-} are the effective indices of the even and odd modes. The average propagation constant for the light in each waveguide is$\overline{\beta}=({\beta}_{+}+{\beta}_{-})/2$.

Considering both the propagation effect and abrupt phase change effect, for WG1, the phase of light at different longitudinal positions is thus given by,

^{2}) into 9 areas and plot the polarization state at the central points of them, for a quasi-TM mode [Fig. 3(a)] and a right handed (RH) quasi-circularly polarized mode [Fig. 3(b)]. As expected, the zero intensity of longitudinal component (E

_{z}) at the waveguide central point results in purely transverse light field at that position. At the central point, only the predominant component (E

_{x}for the quasi-TM mode, E

_{y}for the quasi-TE mode, E

_{x}- iE

_{y}for the RH quasi circular mode) exists, reaching its peak while other components equal zero. Thus it is reasonable to use the central point polarization to characterize the predominant polarization of a certain mode in nanophotonic waveguides. Interestingly, for a certain mode, although its polarization is position dependent, its polarization projection in the xy-plane is mainly consistent with its predominant polarization component. For example, for the quasi-TM mode, its polarization projection in xy-plane mainly points alone x-direction while the one for the RH quasi-circular mode is mainly RH elliptically polarized. Note that the projection components in xy-plane are those that can be analysed by conventional wave plates when light emits into free space.

In addition, due to the geometrical symmetry of the waveguide, the centers of between eigenmodes and geometric structure are coincided with each other in both left and right nanophotonic waveguides. Thus, we utilized the polarization states in center points of nanophotonic waveguides to represent the light modes’ polarization states.

Nanophotonic silicon waveguides usually exhibit huge birefringence effect. However, in a rectangle Si waveguide surrounded by silica, where the width and height are equal, the propagation constants of the fundamental (zero order) quasi-TE and -TM modes are equal due to the diagonal symmetry, that is β_{TE} = β_{TM}. However, in the coupler, the even and odd modes for the TE- and TM-like polarized light are dramatically different from each other. Figure 3(c) displays the real parts of the dominant polarization components for two orthogonal polarized modes [Re(E_{x}) for TM and Re(E_{y}) for TE]. Using the Eigenmodes Solver, which is available in finite-different time-domainate (FDTD) Solutions package from Lumerical Inc., the effective indices for the even and odd modes of TE-like polarized light are calculated to be 2.5637 and 2.2581, respectively, while the ones for TM-like polarized light are respectively 2.5311 and 2.1573. Thus, we have${\overline{\beta}}_{TE}=2.4109{k}_{0}$ and ${\overline{\beta}}_{TM}=2.3442{k}_{0}$.

We perform computer FDTD simulation to confirm our theoretical analysis. In our simulation, we use the phase and power of the dominant polarization component (E_{x} for TM and E_{y} for TE) at the waveguide center point to represent the phase and power in each waveguide. Figure 3(d) shows the dependence of the phase on the propagation distance (z) for WG1. According to Eq. (5), the slopes of lines indicate the average propagation constants, ${\overline{\beta}}_{TE/TM}$, which are in good agreement with the simulation results. As for the power (P∝|E|^{2}) of light, the normalized powers of the first and second waveguides have a characteristic given by

_{+}- β

_{-}) / 2 is the coupling coefficient. Interestingly, a π phase shift happens to both TE- and TM-polarized modes in the vicinity where the powers reach their minimum (0), as predicted by our abrupt phase shift theory. As an aid to comprehension, according to Eq. (5), we defined the abrupt phase shift term (

*ϕ*

_{1A}) in WG 1 as

We plot the abrupt phase shift term and power dependence on the propagation distance (z) in Fig. 3(e). Although there are some minor disagreements between the analytical and stimulated results regarding the abrupt phase shift, the abruptness of π phase shift is for sure for both of polarized modes. Thus, Eqs. (5) and (6) are very good approximated methods for the prediction of phase of light in the coupler. Further information about the abrupt shift of π can be found in a former reference [30], which mainly focus on the ingredients that have important influence on the abruptness of this π phase shift.

To investigate the evolution of polarization and angular momentum of light in the coupler, we respectively discuss the power and phase of light. We first assume a quasi-TE and -TM modes simultaneously entering the input port of WG1. These two orthogonal polarized modes will independently undergo different coupling processes in the coupler. As for the relative phase between the quasi-TE and -TM modes in the first and second waveguides, it is given by,

Note that ${e}^{-i\Delta {\varphi}_{1/2}(z)}$ are periodic functions, which means, ${e}^{-i\Delta {\varphi}_{1/2}(z)}={e}^{-i\left[\Delta {\varphi}_{1/2}(z)\pm 2m\pi \right]}$, where m is an integer. We simplify the relative phase by omitting the redundant 2mπ. Thus, Eq. (9) could be written as,

Although at first glance the relative phases $\Delta {\tilde{\varphi}}_{1/2}(z)$ appear to be irregular, we found that the expression $\Delta {\tilde{\varphi}}_{1}(z)-\Delta {\tilde{\varphi}}_{2}(z)$ can only have three discrete values, ± π and 0. This indicates that the handedness of modes’ polarization states in two adjacent waveguides are precisely opposite [$\left|\Delta {\tilde{\varphi}}_{1}(z)-\Delta {\tilde{\varphi}}_{2}(z)\right|=\pi $] or uniform [$\left|\Delta {\tilde{\varphi}}_{1}(z)-\Delta {\tilde{\varphi}}_{2}(z)\right|=0$]. Interestingly, we found that at some discrete positions where the amplitudes of the two polarized modes are equal [${P}_{1\_TE}(z)={P}_{1\_TM}(z)$, or ${P}_{2\_TE}(z)\text{=}{P}_{2\_TM}(z)$], the handedness of light beams’ polarization states in the first and second waveguides are perfectly opposite [Fig. 4(f)]. Equation (7) indicates that the energy coupling [${P}_{1}(z)$ and ${P}_{2}(z)$] is independent of the phase condition [*ϕ*_{1}(z) and *ϕ*_{2}(z)], so that we can analyze the amplitudes and phases independently. At the point (${z}_{e}$) where ${P}_{1\_TE}({z}_{e})\text{=}{P}_{1\_TM}({z}_{e})$, according to Eq. (7), it should satisfy

To help comprehension of this optical meshing gears behavior (the opposite handedness of two light beams’ polarization states in the coupler), in Fig. 5, we plot the polarization states at the center points of two waveguides at these discrete positions. At these positions, usually a right-handed elliptically polarized mode at the left waveguide will accompany with a left-handed elliptically polarized mode at the right waveguide, and vice versa. This theoretical prediction is consistent with the FDTD simulation and the simulated electric field of each cross section is shown in the right side of Fig. 5.

Another interesting finding of the optical meshing gears behavior is that the center points polarizations of output modes can be steered by the choice of initial relative phase ($\Delta {\varphi}_{0}$) between the quasi-TE and -TM modes at the input port. An arbitrary$\Delta {\varphi}_{0}$can be obtained by employing a voltage-controlled polarization controller [31]. Taking the fourth equal amplitude position [z = z_{e}(4) ≈9.12 μm] for example, a change in the initial phase leads to a change of the polarization of elliptical polarized modes in two waveguides at z = z_{e}(4). Indeed, the relative phases ($\Delta {\tilde{\varphi}}_{1/2}[{z}_{e}(4)]$) show a linearly dependence on$\Delta {\varphi}_{0}$, as shown in Fig. 6(a). Also,$\Delta {\varphi}_{0}$is independent of the energy coupling. Therefore, it is able to obtain a right handed (RH) quasi-circular mode in the left waveguide while a left handed (LH) one in the right for$\Delta {\varphi}_{0}=0.29\pi $. In addition, the choice of $\Delta {\varphi}_{0}=-0.71\pi $leads to a LH quasi-circularly polarized mode in the left waveguide while a RH one in the right. Figure 6(b) shows the electric field distributions and center points polarizations of two waveguides at z = z_{e}(4) for different$\Delta {\varphi}_{0}$value (0, −0.71π and 0.29π). In particular, the opposite handedness characteristic of center points polarization at two adjacent waveguides still holds true for various$\Delta {\varphi}_{0}$values, since $\left|\Delta {\tilde{\varphi}}_{1}[{z}_{e}(4)]-\Delta {\tilde{\varphi}}_{2}[{z}_{e}(4)]\right|=\pi $. This finding is important, as it allows for the manipulation of chirality of output modes.

Surprisingly, the electric field distributions (|E|) of the three output modes are spatially different. In relative large waveguide structures, it is believed that the spatial mode distribution is independent of the polarization of light [32]. However, as the dimensions of waveguide scale down, these two quantities will inevitably get connected due to the spin-orbit interactions [33]. In other words, the polarization of light does indeed affect its spatial mode distribution in nanophotonic waveguides, which is our case.

Angular momentum can be decomposed into two components: the spin part (SAM) associated with the polarization and the orbital part (OAM) related to the spatial phase [34]. These two components can get coupled in nanophotonic waveguides. A quasi-elliptically polarized mode is usually accompanied by a longitudinal vortex component (E_{z}) due to the spin to orbital coupling in nanophotonic waveguides [35, 36]. We find that the twisted handedness of the longitudinal component coincides with the center point polarization. For example, the center point of the left waveguide at$z={z}_{e}(4)$witnesses a right-handed elliptical polarization for$\Delta {\varphi}_{0}=0$, spinning clockwise. Accordingly, the longitudinal vortex component of the left waveguide at$z={z}_{e}(4)$ twists clockwise, which is revealed by the spatial phase distribution of E_{z} [Fig. 6(b)].

The gap between two adjacent waveguides is a key parameter in our device. We use Eqs. (9) and (12) to evaluate its influence on the optical gears. Figure 6(c) shows that an increase in the gap leads an exponential increase in${z}_{e}(4)$. Also, our device is wavelength dependent. A linear increase in the wavelength results in an exponential decrease in${z}_{e}(4)$[Fig. 6(d)].

In this paper, the concept of optical gears specifically refers to two light beams with opposite handedness of polarization states at special positions Ze(m) in the coupler. We used the concept of “optical gears” to describe the special polarization states and angular momentum evolution in the coupler mainly based on the following two reasons. Firstly, the opposite polarization rotation behavior of the elliptically polarized modes in the two waveguides system is similar to that of two meshing mechanical gears. Secondly, in nanophotonic waveguides, the z-polarized vortex components are generated due to spin to orbital angular momentum conversion, which often occurs when circularly polarized light is strongly confined in the lateral direction [33]. The z-polarized vortex component in each waveguide, which accounts for around 30% of total energy, has a twisted wavefront [Fig. 6(b)], indicating that it rotates around the mode center. The twisted behavior of the z-polarized components is similar to that of mechanical gears.

We would like to emphasize the importance of this unique characteristic of opposite chirality, which is an optical analogue of two meshing gears transmitting rotational motion. Although some newly discovered optical phenomena, such as photonic wheels [37], polarization of Möbius strips [38] and surface plasmon drumhead modes [39], are often limited by immediately practical applications at the beginning, they may trigger increasing discussions later since they are strongly connected to fundamental physics and a variety of potential applications. It is therefore not unrealistic to expect that the optical gears phenomenon may open up new avenues in various fields, such as the integrated quantum science and on-chip chiral molecules detections, where the handedness of AM of photons is a key requirement. We sincerely hope that the optical gears, a new concept of nano-photonic designs, can be widely recognized and used in a wide range of applications in nanoscale such as light manipulation, quantum computing, biomolecular sensing and other technical aspects where two nano-spot light beams with opposite handedness of polarization states are needed. For example, the light beams could be used to manipulate the nanoparticles spinning on the opposite directions, or transmit two light beams with opposite angular momentum into other quantum circuits simultaneously. In the field of life science, the optical gears may be applied to sensing the chiral structures of the biomolecules in nanoscale through different absorption characteristics (circular dichroism) [40].

## 3. Conclusion

Our results uncover an exotic chirality phenomenon buried under the coupling process in a nanophotonic directional coupler, which has not been previously reported in literature. Namely, we introduce a new idea of optical gears where two light beams with opposite handedness of polarization states can be obtained via a simple coupler structure, and where the chirality of AM is tunable via the choice of initial relative phase between two orthogonal modes at the input port. Also, we find that the polarization of modes vary along with the propagation distance in the coupler when two orthogonal mode involved, because of the abrupt phase shift effect and birefringence effect. The demonstration of a simple coupler capable of processing complex light beams carrying AM may open many possibilities for applications in fields ranging from fundamental physics and devices designing. For example, it could be applied to make an entangled photon source with tunability, for which the handedness of AM state is an available degree of freedom to encode quantum information.

## Funding

National Natural Science Foundation of China (NSFC) (61574064); Project of Discipline and Specialty Constructions of Colleges and Universities in the Education Department of Guangdong Province (2013CXZDA012); Natural Science Foundation of Guangdong Province (2014A030313446); Program for Changjiang Scholars and Innovative Research Team in University (IRT13064); Science and Technology Program of Guangdong Province (2015B090903078); Science and Technology Planning Project of Guangdong Province (2015B010132009).

## Acknowledgments

The authors thank Dr. Ning Zhu (Institute of Opto-Electronic Materials and Technology, South China Normal University) for helpful discussion. Also, we thank the support of the Innovation Project of Graduate School of South China Normal University.

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