Optical gears in a nanophotonic directional coupler

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 photons remains undiscovered. Here, we present a simple nanophotonic directional coupler structure which can generate two meshing gears of angular momentum (AM) of light, optical gears. Due to the abrupt phase shift effect and birefringence effect, the 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 AM, 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 ports. 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.

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 multipolarized modes, i.e. quasi-circularly (or elliptically) polarized modes, 23 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. By engineering the length of coupling region, it is possible to construct an optical analogue of two meshing gears, where the quasielliptically 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 ( 0   ) of quasi-TE and -TM modes at the input port ( Figure 1c). Interestingly, our scheme is conceptually different from previous methods for manipulation of AM, such as birefringence effect caused by optical crystals 24 and abrupt phase change introduced by nano-resonators. 25 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.

RESULTS AND DISCUSSION
The proposed scheme is sketched in Figure 1c. The directional coupler consists of two uniform parallel silicon (Si) 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 (SiO2) 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 AM 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 1 1 0 1 2 , where β is the propagation constant. Correspondingly, the initial conditions are a1(0) = 1 and a2(0) = 0. For 0 < z < zc, 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 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. 26 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 where floor(x) is the floor function such that floor(x) is the largest integer not greater than x, and 1 (0)  is the initial phase at the position z=0. The first term suggests that the light propagate along the +z direction while the second term indicates the abrupt phase shift (π) introduced by light coupling. Accordingly, the phase distribution of light for WG2 is, 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 βTE = βTM.
However, in the coupler, the even and odd modes for the T=4z c waveguides have a characteristic given by where κ = (β+ -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 (φ1A) in WG 1 as We plot the abrupt phase shift term and power dependence on the propagation distance (z) in Figure 3c.
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. Also, this abruptness of π phase shift is independent of the coupling length zc ( Figure S1, Supporting Information). Thus, eq 5 and 6 are very good approximated methods for the prediction of phase of light in the coupler.
To investigate the evolution of 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 1/2 () iz e   are periodic functions, which means, , where m is an integer. We simplify the relative phase by omitting the redundant 2mπ. Thus, eq 9 could be written as, The solution to eq 11 is, where m is an integer. In our case, ze(m) ≈ m*2.28 μm or ze(m) ≈ m*22.74 μm. The latter case is relative large. In

Supporting Information
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.XXXXXXX.
Author Contributions # F. Zhang and Y. Liang contributed equally to this work.

Notes
The authors declare no competing financial interest.