1. Introduction

Photonic integrated circuit has established itself as an attractive technology to realize ultrafast on-chip optical interconnects with low power consumption [1], facilitating a wide range of applications such as optical signal processing [2–4], on-chip sensing [5, 6] and scalable quantum photonics [7, 8]. In addition to conventional silicon-on-insulator material platforms, silicon-nitride (SiN) photonics has entered the vision of researchers for its wide transparent range and low loss in near-infrared region [9–11]. Compatible with current CMOS infrastructure, SiN offers moderately high contrast of refractive index and exhibits excellent performance in realizing passive photonic functions [12–14].

As arrays of optical scatterers with subwavelength spacing and judiciously engineered geometry, metasurface shows unprecedented capacity in tailoring electromagnetic waves [15], underpinning a myriad of applications like planner optics [16], high-efficient holograms [17], orbital-angular momentum generation [18, 19] and optical analog computing [20]. However, current researches primarily focused on the light control in free space [21]. Comparatively fewer attentions are devoted to their paramount potential for guided electromagnetic waves. Previous researches on this regard include controlling the propagation of waveguide modes [22–24] and coupling between guided waves and light propagating in free-space [25–28]. As directional coupling is ubiquitous in photonics, it is highly desired to separate light beams into different directions with well-defined modes depending on their polarization states or wavelengths. Optical couplers instigated by metasurface are hence of vital significance [23, 26], for this approach can synergize the flexibility of light control by nanonatennas with the fundamental building block of optical waveguides. However, previous explorations are either limited to double-element antenna pairs [26–28] or unfold restrained control over polarization and phase [29, 30]. The dipole interference model [27–29] is only suitable for designing few-element antennas and a given array can solely impart a certain phase profile to one specific incident polarization. Recently directional coupling using waveguide-integrated phase-gradient antenna array is also reported [30], but the designs are restricted to geometric metasurface and only work for circular polarizations. In terms of the applications in waveguide coupling and guided wave control, a comprehensive design scenario applicable for arbitrary elliptical polarizations accommodating both propagation [17] and geometric phase (or Pancharatnam-Berry phase) metasurface [31] is still elusive.

Here we transfer the Jones matrix model [18, 32–34] to waveguide applications targeting versatile on-chip couplings around telecommunication wavelength of 1.55 μm with Si metasurface-patterned SiN photonic waveguides. Dielectric metasurface is focused here for plasmonic antennas generally exhibit high Ohmic loss. Two operation scenarios are given. (1) Two independent phase profiles can be locally imparted to guided electromagnetic waves for arbitrary pair of orthogonal incident polarizations. As a proof of concept, we numerically demonstrate a set of chip-integrated directional couplers and polarization splitters for several representative orthogonal polarization pairs, by imposing two opposite phase gradients to the light coupled by Si nanoantennas. (2) Nanoscale complete control of both phase retardation and polarization of the guided modes. Directional waveguide couplers with well-defined output fundamental TE or TM mode are reported for this case. Moreover, this metasurface-on-waveguide configuration can be further engineered to cover multiple working wavelengths to realize color routing [27, 32]. Our research further translates the concept of metasurface into the integrated optics, serving as a positive reference on complete and versatile control over guided electromagnetic waves for chip-integrated photonic applications such as directional couplers, polarization sorters, light routers and multi-wavelength optical switches.

2. Fundamentals and design principles

Generally the most fundamental attribute of metasurface is to locally encode a configurable and polarization-dependent phase to incident electromagnetic wave [15]. Previous explorations can be ascribed into two categories leveraging either geometric or propagation phase [33]. Geometric phase metasurface utilizes optical antennas with same geometry but spatially varying orientations to impose equal and opposite phase profiles to two circular polarizations. In contrast, propagation phase design applies elaborately engineered antenna geometry but fixed orientation to independently tailor the phase for two orthogonal linear polarizations [26]. The Jones matrix model [18, 32] that combines the two scenarios can venture photonic integrated circuits into new territories, offering promising opportunities for versatile mode coupling and conversion applications.

First, we consider a periodic array of identical Si antennas resting on an infinitely large SiN substrate. As is illustrated in Fig. 1(a), the square lattice period here is fixed at $\mathrm{\Lambda }=1$ μm, which is smaller than the telecom light wavelength to avoid non-zero diffraction orders [21, 33–36]. The antenna geometry ($l_x\times l_y\times l_z$) and rotation angle θ along z axis can be independently engineered.

 

Fig. 1 (a) Schematic of an antenna unit. (b) and (c) Comparison of $\left|\mathbf{E}\right|$ distribution along the $XOZ$ or $YOZ$ plane respectively for a Si antenna irradiated by linearly x- or y-polarized light. Antenna geometry: $l_x\times l_y\times l_z=0.3\times 0.6\times 1$ μm and θ = 0. (d) and (e) Phase retardation φ_x and transmittance $\left|t_x\right|$ of the x-polarized light respectively under different combinations of l_x and l_y. A linearly x-polarized plane wave normally irradiates the antenna (electric field vector aligned with antenna axis). (f) Corresponding complex amplitude transmission rate $t_x=\left|t_x\right|\exp (i{\phi }_x)$ in the complex plane.

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The scattering characteristics of the dielectric antennas can be comprehended as form birefringence [36, 37] or truncated waveguides with azimuthal asymmetry. Figures 1(b) and 1(c) delineate the electric field norm distributions $\left|\mathbf{E}\right|$ of an unrotated Si antenna under the irradiation of linearly x- and y-polarized light respectively. Light filed inside an antenna with its dominant electric field polarized along the antenna longer axis has larger effective index and hence exhibit bigger phase retardation. Consequently, a periodic antenna array behaving like a birefringent wave plate can be described by a Jones matrix J [33, 37],

The scattering properties of the periodic Si antenna array are analyzed via full-vector Finite-Difference Time-Domain (FDTD) method. Antenna height l_z is fixed as 1 μm, while other structure parameters l_x and l_y are flexibly changed to engineer φ_x and φ_y. Figures 1(d) and 1(e) depict the eigen-phase φ_x and transmission rate $\left|t_x\right|$ of the light transmitted into SiN substrate respectively [21] as a function of antenna geometry at $\lambda =1.55$ μm. Data sets are numerically generated by sweeping the combinations of l_x and l_y from 0.1 to 0.9 μm with a step of 25 nm, followed by a cubic spline interpolation. In Fig. 1(f) we plot the corresponding complex transmission rate t_x of the array under the illumination of x-polarized light, where the red dashed circle indicates an averaged transmission rate of 0.89. Considering the system’s two-fold (C2) rotational symmetry [33, 39], the phase and amplitude response of φ_y and $\left|t_y\right|$ can be easily obtained by exchanging the x and y axis in Figs. 1(d) and 1(e).

Next we will briefly summarize the design principles for two different operation scenarios illustrated in Fig. 2(a). Figure 2(b) manifests the outline of the design methodology used in this paper (detailed explanations are given later). Corresponding examples for versatile on-chip couplings will be demonstrated subsequently.

 

Fig. 2 (a) Summary for the two operation scenarios, where the geometry of each pixel l_x, l_y and θ can be chosen freely. The red and orange color-highlighted phase and polarizations denote configurable parts, while the black orgrey colored phase and polarizations indicate given or not configurable factors. (b) Flow chart introduction to the whole design methodology.

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2.1. Complete phase and polarization control for one arbitrary incident polarization

Considering the generalized Snell’s law of refraction [21, 34], the refraction angle is configurable by introducing a phase gradient $\text{dΦ}/\mathrm{d}x$ along the interface. In terms of guided waves, we substitute the refractive index of the transmitted media n_t from the original equation [21] into the effective index of the guided modes: $n_{\text{eff}}\sin ({\theta }_t)-n_i\sin ({\theta }_i)=\text{dΦ}/\left(\mathrm{d}x\cdot k_0\right)$, where wave vector $k_0=2\pi /\lambda$ [30], θ_i represents the incident angle and refraction angle ${\theta }_t=\pm \pi /2$. In the case of normal incidence ${\theta }_i=0$ on a gradient metasurface, we have

Considering an arbitrary elliptical polarization of incidence $|{\mathbf{\lambda }}^{+}=R\left(\alpha \right)\cdot {\left[\cos \epsilon ,\mathrm{i}\sin \epsilon \right]}^T={\left[{\lambda }_1{ }^{+},{\lambda }_2{ }^{+}\right]}^T$ and transmittance $|{\mathbf{\kappa }}^{+}〉={\left[{\kappa }_1{ }^{+},{\kappa }_2{ }^{+}\right]}^T$, α stands for polarization rotation angle and ε is used to characterize polarization ellipse. Under the assumption of unity transmission $\left|{\mathbf{\lambda }}^{+}|=|{\mathbf{\kappa }}^{+}\right|$, we can always find a symmetric and unitary Jones matrix $J\left(p,q\right)$ that can establish the mapping between any arbitrary input and output polarizations [18].

Applying the symmetric and unitary constraints of J [34], we can obtain the following equation.

If M

is not singular, we can solve the matrix element j_xx and j_xy, while other elements can be also obtained from the unitary condition of J as

The coupling process is assumed to be decomposed into two steps [22, 23]. First, the antenna pixel locally converts the incident light into E_antenna with target polarization ellipse and designed phase retardation. ${\mathbf{E}}_{\text{antenna}}\left(x,y,z\right)$ is the near-fields around the antenna pixel, where the waveguide is parallel to x axis and $YOZ$ plane denotes the waveguide cross-section plane. Second, E_antenna will be selectively coupled into certain waveguide modes [23, 28], which is governed by the conversion efficiency η or the spatial mode overlap between the antenna electric fields and waveguide mode.

Consequently, for the controlled coupling into fundamental TE mode, the output polarization can be selected as $|{\mathbf{\kappa }}^{+}〉=e^{i\phi \left(p,q\right)}|\mathbf{y}〉$. For the excitation of TM mode, we configure $|{\mathbf{\kappa }}^{+}〉=e^{i\phi \left(p,q\right)}|\mathbf{x}〉$ to maximize the selective coupling efficiency, where $|\mathbf{x}〉={\left[1,0\right]}^T$ and $|\mathbf{y}〉={\left[0,1\right]}^T$ are unit polarization vectors along x and y axis respectively.

2.2. Independent phase control of guided waves for two orthogonal polarizations

If M in Eq. (5) is a singular matrix, the output polarization can be written as $|{\mathbf{\kappa }}^{+}〉=|{\left({\mathbf{\lambda }}^{+}\right)}^{*}〉e^{i{\phi }^{+}}$, where ${\phi }^{+}$ is a configurable phase. Similar equation can be established by considering another orthogonal incident polarization $|{\mathbf{\lambda }}^{-}〉=R\left(\alpha \right)\cdot {\left[i\sin \epsilon ,\cos \epsilon \right]}^T$. Consequently, in this scenario the output polarization ellipse is preserved while the handedness is flipped. At each antenna pixel $\left(p,q\right)$ we can find a Jones matrix $J\left(p,q\right)$ that guarantees the following mappings simultaneously [18].

The retrieved Jones matrix unifies the geometric and propagation phases [18, 33] and can be decomposed into eigenvectors $R\left(\theta \right)$ and eigenphases φ_x and φ_y by Eq. (2). Then for given input and output, we only need to find the antenna pixels imposing phase shifts φ_x and φ_y under linear x- or y- incident polarizations and rotate it with angle θ along z axis, as shown in Fig. 2(a). For determining the optimal antenna geometry with designed eigenphases φ_x and φ_y, enumerating-search method is applied traversing all the calculated combinations of $\left(l_x,l_y\right)$ to optimize the following objective function.

Although the electric field components of waveguide modes are actually all non-zero, the polarization of the prominent transverse electric field component for a waveguide eigenmode still can be written as the superposition of two output elliptical polarizations $|{\mathbf{\kappa }}^{+}〉$ and $|{\mathbf{\kappa }}^{-}〉$ along the transverse plane [40]. The dimension of the SiN waveguide is selected as width×height = 1.0×0.8 μm, for it satisfies the fundamental mode transmission condition around 1.55 μm and the effective index of fundamental TE and TM mode are almost degenerate. Hence, any approximate elliptical polarizations combined by the dominant polarization vectors of TE and TM modes in the waveguide share approaching phase velocity to preserve the polarization ellipse.

Next we will demonstrate various kinds of functional devices using metasurface-patterned photonic waveguides. As is shown in the flow chart in Fig. 2(b), we can either impart two independent phase profiles to arbitrary pair of orthogonal polarizations to achieve polarization sorters and wavelength routers, or realize directional coupling with mode-selectivity for one incident polarization. For the design process, we first select the desired phase responses for all the antennas. Then we calculate the Jones matrix J and retrieve the eigenphases (φ_x, φ_y) & rotation angle (θ) for each antenna. Computer optimizations are conducted subsequently searching through the phase map datasets [like Figs. 1(d) or 1(f)] to find the antenna geometry (l_x, l_y) with optimal eigenphase responses. Finally, full-vector FDTD simulations are applied to validate the performance of our design.

 

Fig. 3 Chip-integrated metasurface coupler (a) Device structure sketch. (b) Antenna parameters. (c) Incident polarizations. (d) and (e) Transmission rates to waveguide ports under $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ illumination. (f) Coupling directivity $\mathrm{\gamma }=10\times \lg \left(T_{\text{right}}/T_{\text{left}}\right)$ as a function of light wavelength. (g) $\left|\mathbf{E}\right|$ distribution at the left and right ports under the excitation of $|{\mathbf{\lambda }}^{+}〉$ at λ = 1.55 μm. (h) and (i) Electric field component ${\mathbf{E}}_x$ and ${\mathbf{E}}_y$ along waveguide middle plane respectively with $|{\mathbf{\lambda }}^{+}〉$ illumination.

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3. On-chip directional couplers for arbitrary polarization

In this section, a set of on-chip polarization splitters are proposed, which directionally couple two arbitrarily given orthogonal polarizations into opposite directions. We start from representative elliptical polarizations as they accommodate both linear and circular polarizations. Figure 3(a) sketches the device structure imposing two independent phase gradients to two incident plane waves with elliptical polarizations of $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ accordingly. The antenna geometric parameters are detailed in Fig. 3(b). The amorphous Si antenna-patterned SiN waveguide can be fabricated by a combination of chemical vapor deposition (LPCVD & PECVD) [9, 10], electron-beam lithography (EBL) and reactive-ion etching (ICP-RIE) elaborated in literature [22, 23]. Sidewall passivation technique can be applied to etch Si nanofins with high aspect ratio [41]. As is shown in Fig. 3(c), the incident polarization ellipse is aligned with coordinates: $|{\mathbf{\lambda }}^{+}〉=R\left(0\right){\left[\cos (\pi /3),i\sin (\pi /3)\right]}^T=\frac{1}{2}|\mathbf{x}〉+\frac{\sqrt{3}i}{2}|y〉$ and $|{\mathbf{\lambda }}^{-}〉=R\left(0\right){\left[i\sin (\pi /3),\cos (\pi /3)\right]}^T=\frac{\sqrt{3}i}{2}|\mathbf{x}〉+\frac{1}{2}|\mathbf{y}〉$. From left to right, the antennas numbered from -3 to +3 with a center-to-center distance of $\mathrm{\Lambda }=1$ μm. For simplicity but with representativeness, we assign two opposite constant phase gradients: $\mathrm{\Delta }\phi =\pi /3$ for $|{\mathbf{\lambda }}^{+}〉$ and $-\pi /3$ for $|{\mathbf{\lambda }}^{-}〉$. Consequently, according to Eq. (2), $|{\mathbf{\lambda }}^{+}〉$ will be directionally coupled to the right port while $|{\mathbf{\lambda }}^{-}〉$ is directed to the left port. The phase gradient $\mathrm{\Delta }\phi =\pi /3$ is a representative but not fully optimized value. We did not strictly follow Eq. (2) when selecting $\mathrm{\Delta }\phi$ because a relatively loose value of $\mathrm{\Lambda }=1$ μm is applied to mitigate the coupling between adjacent antennas and release potential fabrication challenge.

Figures 3(d) and 3(e) show the transmission rate to the left (T_left) and right waveguide ports (T_right) under the excitation of $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ respectively, where asymmetric coupling is achieved. In Fig. 3(f) we plot the coupling directivity parameter $\gamma =10\times \lg \left(T_{\text{right}}/T_{\text{left}}\right)$ as a function of light wavelength, which is competitive to previous reports [27, 29]. A relatively coarse step of 25 nm is applied when sweeping antenna geometric parameters ($l_x,l_y$) to generate the phase map datasets. If a smaller step is applied, the datasets will be more accurate and the device directivity will be further enhanced. One may consider that the coupling efficiency here is not very high [26, 27], we emphasis that this value is still comparable with previous literatures [26, 27] and is highly dependent on how tightly the excitation beam is focused [28]. In whole simulations, we use diffracting plane wave source filtered by a $x\times y=8\times 2$ μm rectangular aperture. If Total-Field Scattered Field (TFSF) light source is applied with reduced light spot, the maximum coupling efficiency in Fig. 3(d) can be ameliorated to approach 6%. Furthermore, the coupling efficiency can also be enhanced by putting more antennas on the waveguide or applying a decreased period Λ with smaller light spot. In Fig. 3(g) we plot the electric filed norm $\left|\mathbf{E}\right|$ distribution for the left and right waveguide ports under the illumination of $|{\mathbf{\lambda }}^{+}〉$ at $\lambda =1.55$ μm, while the distribution of electric field components ${\mathbf{E}}_x$ and ${\mathbf{E}}_y$ along the middle waveguide plane are shown in Figs. 3(h) and 3(i) respectively.

One may consider that the transmission curves between Figs. 3(d) and 3(e) have different shapes between orthogonal incident polarizations. It originates from the distinction in selective coupling [22, 28]. For TE mode, the dominant electric field component is ${\mathbf{E}}_y$. The prominent component of E_antenna largely inherited from incidence $|{\mathbf{\lambda }}^{+}〉=\frac{1}{2}|\mathbf{x}〉+\frac{\sqrt{3}i}{2}|\mathbf{y}〉$ is also y-polarized. Therefore, light is primarily coupled to fundamental TE mode. However, for TM mode, the preferable electric field component is ${\mathbf{E}}_z$, while the z-component of E_antenna is barely excited because the lack of momentum in z direction from light source $|{\mathbf{\lambda }}^{-}〉=\frac{\sqrt{3}i}{2}|\mathbf{x}〉+\frac{1}{2}|\mathbf{y}〉$. Consequently, the coupling efficiency to TM mode is comparatively lower. To obviate this problem, we can simply rotate the incident polarization ellipse by $\alpha =\pi /4$, for in this case for arbitrary ellipse ε, the x and y component of the excitation source always have same absolute value: $R\left(\pi /4\right)|{\mathbf{\lambda }}^{+}=R\left(\pi /4\right)\left(\cos \epsilon |\mathbf{x}+i\sin \epsilon |\mathbf{y}\right)=\frac{\sqrt{2}}{2}\le [\left(\cos \epsilon -i\sin \epsilon \right)|\mathbf{x}〉+\left(\cos \epsilon +i\sin \epsilon \right)|\mathbf{y}〉]$ versus $R\left(\pi /4\right)|{\mathbf{\lambda }}^{-}=R\left(\pi /4\right)(i\sin \epsilon |\mathbf{x}〉+\cos \epsilon |\mathbf{y}〉=\frac{\sqrt{2}}{2}\left[\left(i\sin \epsilon -\cos \epsilon \right)|\mathbf{x}〉+\left(\cos \epsilon +i\sin \epsilon \right)|\mathbf{y}〉\right]$.

 

Fig. 4 Metasurface-integrated orthogonal polarization splitters for arbitrary elliptical polarizations. (a), (f) and (k) Device structures for polarization ellipse $\epsilon ={50}^{\circ }$, 60° and 70° accordingly. (b), (g) and (l) Corresponding incident polarizations. (c), (h) and (m) $\left|\mathbf{E}\right|$ distribution at left and right waveguide port for $\mathrm{\lambda }=1.55$ μm at $|{\mathbf{\lambda }}^{+}〉$ illumination. (d), (i) and (n) Waveguide transmission spectrum under $|{\mathbf{\lambda }}^{+}〉$ excitation for (a), (f) and (k) respectively. (e), (j) and (o) Corresponding directivity spectrum.

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Polarization splitter for arbitrary elliptical polarization: Figure 4 shows the performance of three on-chip polarization splitters that couple pairs of exemplary orthogonal incident polarizations $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ into opposite directions. The device performances with polarization ellipse parameter $\epsilon ={50}^{\circ }$, 60° and 70° are shown in Figs. 4(a)–4(e), 4(f)–4(j) and 4(k)–4(o) accordingly, where $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ are all rotated by $\alpha =\pi /4$. The directivity curves share the same shape but interchanged direction between orthogonal polarizations for all the cases, with high directivity values ranging from about 10 to 13 dB. Hence, our general method is applicable for any arbitrary polarizations in designing directional couplers and polarization splitters/demultiplexers. In addition, the chirality-sensitive directional coupling here can be also interpreted from the spin-momentum locking effect in the transverse spin of light [42, 43].

Polarization splitter for classic polarizations: Light sources with linear or circular polarizations are the most common cases in optical applications. Figures 5(a)–5(f) and 5(g)–5(l) illustrate the performance for the metasurface-integrated directional couplers working for linear and circular polarizations shown in Figs. 5(b) and 5(h) respectively. At $\lambda =1.55$ μm, incident polarization channels $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ are directed to the right and left port accordingly.

As is shown in the tables below Figs. 5(a) and 5(g), this design method can cover previous geometric or propagation phase designs for guided waves. If we assign the ellipse parameter as ε = 0

or $\pi /2$, similar antenna deignis obtained as propagation phase metasurface. When ε is configured as $\pm \pi /4$, we get identical antenna design with gradient orientation angles θ featured in geometric metasurface. One may consider the θ in Fig. 5(g) is not directly arithmetic sequence. However, considering the C2 symmetry [i.e. satisfies $R\left(\pi \right)J=J$] of the antenna, interchanging l_x and l_y denotes the same transformation as a 90° rotation by $R\left(\pi /2\right)$. Therefore, if we apply identity transform $R\left(\pi \right)$ to antenna No. 2 and $R\left(\pi /2\right)$ to antenna No. -1, 0, 1 to flip l_x and l_y, we will obtain monotonously increased θ.

 

Fig. 5 Device structure sketch for linear polarizations. (b) Illustration for incident polarizations. (c) Distribution of electric field component ${\mathbf{E}}_y$ and ${\mathbf{E}}_z$ along waveguide middle plane at $\lambda =1.55$ μm. (d) $\left|\mathbf{E}\right|$ distribution at left and right ports for $\lambda =1.55$ μm. (e) Waveguide transmission spectrum. $|{\mathbf{\lambda }}^{+}〉$ incident polarization is applied for (c), (d) and (e). (f) Device directivity spectrum. (g)-(l) Same for the device working at circular polarizations.

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4. Directional couplers with mode-selectivity

In previous examples, only the phase profiles for orthogonal incident polarizations are engineered, leaving actually no deliberate control over the coupled mode polarizations. As is elaborated in section 2.1, complete control over the phase and polarization of coupled electromagnetic waves for only one incident polarization is achievable in the antenna-on-waveguide platform. Following Eqs. (5) and (9), here we introduce a phase gradient to one incident polarization $|{\mathbf{\lambda }}^{+}〉$ and fix the output $|{\mathbf{\kappa }}^{+}〉=|\mathbf{y}〉$ as the same time to achieve directional coupling to fundamental TE mode. Figure 6(a) manifests the device structure working for linear polarization $|{\mathbf{\lambda }}^{+}〉$ shown in Fig. 6(b). As is evidenced by the electric filed norm $\left|\mathbf{E}\right|$ distribution in Fig. 6(c) and the vector plot of E in Fig. 6(d), a good fundamental TE mode is observed at the right port for $\lambda =1.55$ μm. The directional coupling performance is shown in Figs. 6(e) and 6(f), where a directivity of 13 dB is achieved.

Figures 6(g)–6(k) show the performance of the device that directionally couples free-space light into left-propagating TM mode. A directivity approaching 12 dB is observed around 1.55 μm. Same design method to Fig. 6(a) is applied, but an imperfect TM mode is observed at the left port, as shown in Figs. 6(i) and 6(j). To mitigate the aberrance and achieve good TM mode quality, a simple optimization constraint can be added for TM mode-based devices. Generally, the selective conversion efficiency η to TE mode is comparatively higher than TM mode. The devices in Fig. 3 and Fig. 4 also show pronounced fundamental TE mode component at the right port. The antenna nearfield E_antennawill largely preserve the polarization components from the source, thus the z-component of the electromagnetic waves emitted by the antenna is almost zero. Hence the spatial overlap between E_antenna and TM mode is smaller, considering the dominant electric component of TM mode is ${\mathbf{E}}_z$ (${\mathbf{E}}_x$ only poses secondary impact). Consequently, effective coupling to TM mode can only happen when the y-polarized electric component of E_antenna is much weaker than the x-polarized component. For unrotated antennas, the easiest way is to apply x-polarized light source and limit the antenna dimension in y-direction.

Figure 6(m) shows the device structure of the directional waveguide coupler with TM mode-selectivity. In previous examples, we go through all the calculated antenna geometric parameters $\left(l_x,l_y\right)$ to optimize the deviation between designed and simulated phases in Eq. (9). Here, we restrict a maximum y-dimension $l_y\le 300$ nm and deploy linearly x-polarized light shown in Fig. 6(n). Consequently, the antenna pixels barely generate y-polarized near fields to guarantee efficient TM mode coupling to the right port. As is shown Figs. 6(o) and 6(p), an excellent fundamental TM mode is observed at the right port around 1.55 μm. In Figs. 6(q) and 6(r) we plot the waveguide transmission and directivity spectrum respectively, where a directivity about 11 dB is achieved at 1.55 μm.

 

Fig. 6 (a) Device structure sketch. (b) Incident polarization. (c) $\left|\mathbf{E}\right|$ distribution for the left and right port at $\lambda =1.55$ μm. (d) Vector map for E at the right port ($\lambda =1.55$ μm). (e) and (f) Waveguide transmission and coupling directivity spectrum respectively. (g)-(l) Same for the directional coupler for fundamental TM mode (coupling to the left port). (m)-(r), Same for the TM mode coupler (coupling to the right port) with constraint condition $l_y\le 300$ nm applied in optimization.

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5. Multi-wavelength waveguide couplers

Generally the Jones matrices obtained from Eqs. (5) or (8) are applicable for a wide wavelength range when Λ still remains a subwavelength value. In finding the optimal antenna geometry $\left(l_x,l_y\right)$, optimization in Eq. (9) is conducted at a fixed wavelength of $\lambda =1.55$ μm. Next we will show that the objective function can be modified to find antenna pixels working for multiple wavelengths, which enables dual wavelength devices or on-chip color routers.

 

Fig. 7 (a) and (g) Device structure for dual wavelength directional couplers working for both 1.35 and 1.55 μm. Incident polarization states are illustrated as (b) and (h) accordingly.(c) and (i) Waveguide transmission spectrum for devices in (a) and (g) respectively. (d) and (j) Corresponding directivity spectrum. (e) and (k) Distribution of ${\mathbf{E}}_{\mathrm{z}}$ at 1.35 and 1.55 μm. (f) and (l) $\left|\mathbf{E}\right|$ distribution at the left and right ports at dual working wavelengths for the two proposed devices accordingly.

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5.1. Dual-wavelength directional couplers

First, we focus on only one linear incident polarization $|{\mathbf{\lambda }}^{+}〉=|\mathbf{y}〉$ to design directional couplers working at both ${\lambda }_1=1.35$ and ${\lambda }_2=1.55$ μm. For ${\lambda }_2=1.55$ μm, the desired phase profile is set identical as Fig. 3(b). For ${\lambda }_1=1.35$ μm, we assign ${{\phi }^{+}\left(m\right)|}_{\lambda ={\lambda }_1}=m\cdot {\mathrm{\Delta }\phi |}_{\lambda ={\lambda }_1}+{\phi }_0$ for the m^th antenna, where m is antenna number, ${\mathrm{\Delta }\phi |}_{\lambda ={\lambda }_1}=5\pi /18$ is an exemplary but not optimized phase gradient. ${\phi }_0=0.89\pi$ is an arbitrary initial phase determined in later optimization process. For $|y〉$ polarized light source here, we can directly optimize eigenphase ${\phi }^{+}={\phi }_y$ for both wavelengths. A new objective function is applied to trim the scattering attributes of each antenna pixel across multiple wavelengths.

Figure 7(a) illustrates the schematic of the metasurface-integrated directional coupler with dual working wavelength: 1.35 and 1.55 μm. Figure 7(b) plots the incident polarization states, while the coupling performance for $|{\mathbf{\lambda }}^{-}〉$ illumination is not engineered. Device performance is shown in Figs. 7(c)–7(e), where directional coupling to the right port is achieved around both 1.35 and 1.55 μm. Hybrid high-order modes are observed at 1.35 μm in Fig. 7(f), because the SiN waveguide no longer satisfies fundamental mode transmission condition at 1.35 μm. In contrast, at 1.55 μm the $\left|\mathbf{E}\right|$ distribution at the right port is dominated by fundamental TE mode component.

Figures 7(g)–7(l) show the dual-wavelength device performance working for elliptical pol-arization $|{\mathbf{\lambda }}^{+}〉=\frac{1}{2}|\mathbf{x}〉+\frac{\sqrt{3}i}{2}|\mathbf{y}〉$. Directivity values of 13 and 15 dB are achieved around 1.35 and 1.55 μm respectively. Similar optimization method are conducted but the target eigenphases φ_x and φ_y retrieved from Eqs. (1) and (ef Eq.8) need to be optimized together. We substitute the term $\left|{{\phi }_{+}^{\text{design}}|}_{\lambda }-{{\phi }_{+}^{\text{simulation}}|}_{\lambda }\right|$ from Eq. (10) into ${\left({{\phi }_{+}^{\text{design}}|}_{\lambda }-{{\phi }_{+}^{\text{simulation}}|}_{\lambda }\right)}^2+{\left({{\phi }_{-}^{\text{design}}|}_{\lambda }-{{\phi }_{-}^{\text{simulation}}|}_{\lambda }\right)}^2$ as an instance for elliptical polarizations, one can also optimize the gradient ${\mathrm{\Delta }\phi |}_{\lambda ={\lambda }_1}$ to further ameliorate device performance.

 

Fig. 8 (a) and (f) Device structure sketch working for linear polarizations $|{\mathbf{\lambda }}^{+}〉$ shown in (b) and (g) accordingly. (c) and (h) Waveguide transmission spectrum for (a) and (f) respectively. (d) and (i) Corresponding directivity curve. (e) Electric field norm $\left|\mathbf{E}\right|$ distribution at waveguide ports for 1.35 and 1.55 μm of the device shown in (a). (j) Distribution of ${\mathbf{E}}_z$ for (f) at 1.5 and 1.6 μm under $|{\mathbf{\lambda }}^{+}〉$ excitation. (k) and (l) $\left|\mathbf{E}\right|$ distribution at waveguide ports for dual wavelength under the illumination of $|{\mathbf{\lambda }}^{+}〉$ and $|{\mathbf{\lambda }}^{-}〉$ respectively.

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5.2. Color routers and demultiplexers

Compared to dual wavelength directional coupling to the same direction, coupling same incident polarization to different directions at multiple wavelengths is more challenging, for one antenna pixel has to encode opposite phase gradient (which is largely governed by geometry) at different wavelengths. Here only linear x- or y-polarizations are considered. We directly engineer solely φ_x or φ_y with no requirement on the other eigenphase. Figure 8(a) depicts the device structure routing ${\lambda }_1=1.35$ and ${\lambda }_2=1.55$ μm light to different directions for $|\mathbf{y}〉$ polarization shown in Fig. 8(b). The target phase profiles are configured as ${{\phi }_y\left(m\right)|}_{\lambda ={\lambda }_1}=\left(m+1/2\right)\cdot {\mathrm{\Delta }\phi |}_{\lambda ={\lambda }_1}=\left(m+1/2\right)\cdot 5\pi /18$ and ${{\phi }_y\left(m\right)|}_{\lambda ={\lambda }_2}=m\cdot {\mathrm{\Delta }\phi |}_{\lambda ={\lambda }_2}=m\cdot \pi /3$. Figures 8(c)–8(e) shows the color routing performance, where the 1.35 μm signal channel is directionally coupled to the left port while the light signals with ${\lambda }_2=1.55$ μm is directed to the right port.

Generally bidirectional coupling between different light wavelengths can be conveniently designed for only one polarization, for the optimization results deteriorate when more constraints persist. However, by introducing more floating variables one may also find designs working for dual wavelengths and both orthogonal polarizations. Here we also employ the phase profile of the m^th antenna ${{\phi }_x|}_{\lambda =1.5\mu \mathrm{m}}\left(m\right)$ and ${i_y|}_{\lambda =1.5\mu \mathrm{m}}\left(m\right)$ as the same distribution of ${\phi }^{+}$ and ${\phi }^{-}$ shown in Fig. 3(b), while leaving no specific constraints on the phase gradients at 1.6 μm. An optimization process traversing several discrete combinations of pending variables is conducted to determine the phase gradients and initial phase at 1.6 μm with an objective similar to that used in Fig. 7(g).

Figure 8(f) shows the on-chip color router that selectively couples x-polarized 1.5 and 1.6 μm light to the right and left port accordingly, while simultaneously splits these light channels to opposite directions for incident y-polarization depicted in Fig. 8(g). As is shown in Figs. 8(h1)–8(j), directional couplings are observed for both polarizations at dual wavelengths with a maximum directivity exceeding 21 dB. In Figs. 8(k) and 8(l) we plot the $\left|\mathbf{E}\right|$ distribution at the left and right waveguide ports for dual wavelengths under the illumination of $|\mathbf{x}〉$ and $|\mathbf{y}〉$ respectively. TM mode-dominated patterns are observed at $|\mathbf{x}〉$ excitation, while for $|\mathbf{y}〉$ illumination TE mode components are more pronounced.

6. Conclusions

We translate the Jones matrix model to address versatile on-chip coupling by leveraging Si metasurface-integrated SiN photonic waveguides. A comprehensive design process applicable to arbitrary elliptical incident polarizations and different device functions is discussed. Two independent phase profiles can be imparted on arbitrary pair of orthogonal incident polarizations, fulfilling the applications for on-chip polarization sorters and directional couplers with directivity ranging from 10 to 20 dB. Meanwhile, we can also achieve complete control over both phase and polarization of the coupled waves for only one specific incident polarization to design directional couplers with excellent mode-selectivity, where the excited waveguide mode and signal propagation direction can be flexibly configured by adjusting antenna geometry. Furthermore, our design method can be expanded to cover multiple working wavelengths across the whole telecommunication band. Chip-scale color routers or wavelength demultiplexers are proposed, opening up new opportunities for multifunctional chip-scale devices such as wavelength/polarization multiplexers, optical switches/routers, and high-bit rate telecommunication applications [26].

In addition to coupling devices, our method could be also potentially applied to tailor the propagation and conversion of two orthogonal waveguide modes, serving as a positive paradigm for compact mode convertors, polarization rotators and versatile nonlinear wavelength conversions [22, 23]. Moreover, the reported phase gradient, packing density and optimization algorithm for the nanoantenna pixels can be further modified to enhance device performance. By introducing antenna combos [44–46] or active materials [47, 48], we envision the dispersion attributes of the antennas can be further engineered with our optimization method to enable broadband, multi-wavelength and tunable devices for photonic integrated circuits.