1. Introduction

Silicon-on-insulator (SOI) waveguide photonics and photonic integrated circuit (PIC) technologies have widely been exploited in optical communications systems, on-chip optical interconnection of computers and other advanced areas [1,2]. Goos-Hänchen (GH) shift is a movement of reflected light beam on the interface of two media without external driving, so attracting intensive attention due to its anomalous optical laws for implementing novel functionalities in modern photonics. Since the beginning of the 21^st Century, research on the physical characteristics of GH shift had started [3–8]. First, in 2002-2004 a complex expression for both positive and negative electromagnetic materials at the reflection side and the frustrated total internal reflection (TIR) was defined by Lai and Qing’s teams, and research on the GH shift with negative refractive medium was done by Berman, the GH shift of reflected light-wave within a gap of photonic crystals was studied by Felbacq and co-workers [3–5]. In 2009, the characteristics of GH shifts in different media were studied by Wan and Oh, and Aiello theoretically demonstrated the dural existence of GH spatial and angular shifts, and derived the coherent model [6–8]. Since 2013 the quantum property theories and applications of the GH effect in optical switches were theoretically studied by Sun, then Lukishova et al. demonstrated the quantum science of GH shift via single-photon experiments and further discussed the quantum engineering applications [9,10]. Meanwhile, the coexistence and coherence of GH shifts and Imbert-Fedorov (IF) shifts were theoretically demonstrated by Ornigotti and co-workers [11,12]. Then from 2016 to 2019, some achievements on optical communication components and optical sensors were reported [13–18]. Meanwhile, the more remarkable achievements on active manipulation of light beam were reported [19,20], some were made to enlarge the negative GH shifts of plane-wave [21–23].

As well known, as a new type of optical/photonic materials, metamaterials can change the classical optical laws, then modify the manipulation capabilities of light-waves. So, in the past decade, as research on micro- and nanoscale photonic technologies is quickly extended, the metamaterials-enhanced light-wave systems are presenting exotic phenomena, and their impacts upon the GH shifts have been stirring much more enthusiasm, and even in the metasurface form, leading to an emerging research area and representing a broad avenue for developing new potential optical/photonic functionalities in modern engineering and applications [24–30]. For instance, in 2011-2012, Yu and co-workers systematically reviewed the generalized laws of reflection and refraction of light-wave propagation under the metasurface effect and Aieta et al. studied the out-of-plane light-wave shift, the IF shift, with the anisotropic antenna metasurface, then 5 years later Komar, Baranov and their co-workers studied the electrical and photonic properties of light-waves under the effect of all-dielectric metasurface, and further in 2018 Ornigotti investigated the co-existence of GH and IF shifts [26–30]. Until to 2019, Kong et al. systematically investigated the GH and IF shifts at the TIR condition when a large metasurface is imposed onto the interface by separately considering two linear polarization states [31]. In 2020, a paramount important research achievement in the simultaneous generation of assembly polarization states were reported by Gao’s team with several states of six circularly polarized and linearly-polarized states generated by a single metasurface [32]. Very recently, the anomalous optical laws of light-wave behavior, typically including the generalized Snell’ and Fresnel’s laws of refraction and reflection, the negative refractive index, tilted light-wave, vortex beam, etc. have been developed out, and more advanced functions of metasurfaces have been investigated for the application of the generalized Snell’ and Fresnel’s laws of refraction and reflection [33,34].

In this work, a waveguide deflecting corner structure with one input and three output waveguides and a metasurface of Si-nanoscale semi-sphere array on SOI waveguide platform is introduced first, then the in-depth understanding to the general GH shifts of both plane wave and guided wave, the equivalent permittivity of guided wave in a waveguide, and the optical phase change of guided wave caused by a vertically inserted metasurface and the general GH shifts are theoretically modelled. Moreover, with the optical phase change caused by the metasurface, the GH shifts are simulated, and then the advantages of guided-mode over the plane-wave are presented. Furthermore, we select an optimal construction of devices and conduct the verifications with both finite difference time domain (FDTD) simulations and experiments with the optimally designed and state-of-the-art fabricated device samples. Finally, the conclusions are given.

2. Modelling the Goos-Hänchen shift of guided waves through a metasurface

2.1 1 × 3 waveguide corner with a vertically inserted metasurface

This work is aimed to investigate the sphere-metasurface effect in guided-wave and then improve the applicable manipulation of guided-mode in SOI-PIC by causing an abrupt phase change (APC). Figure 1(a) shows a 1 × 3 SOI-waveguide corner (WC) structure with an input waveguide and a multimode interference (MMI) structure in conjunction with three single-mode output waveguides, which form a TIR interface with the reflector material. A metasurface is vertically inserted between the TIR interface and reflector material as shown in Fig. 1(b), which has a uniform array of Si-nanoscale semi-spheres with a radius of ${\textrm{R}_{ms}}$. In this regime, the single-mode input waveguide is represented in Fig. 1(c) where θ_g is the acceptable angle that not only determines the maximum value of the input angle θ_in, but also determines the divergent angle γ’ at the TIR interface of WC structure and further the divergent angle γ at the end of MMI structure as shown in Fig. 1(d), where the width of the M-mode waveguide is W_mm.

 

Fig. 1. Principle and device concepts of the interaction between a guided-wave and a vertically inserted metasurface. (a) and (b) the schematics 1 × 3 SOI-WC structure and that vertically inserted metasurface structure of Si-nanoscale semi-sphere array, respectively. (c) and (d) the schematic three-layer waveguide and SOI-WC structure for transferring a singe-mode input waveguide to an MMI multimode structure.

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As an incident light-wave from the input waveguide to the TIR is a guided-wave rather than plane-wave, the pure linear polarization states do not exist, namely, the linear P-polarization is replaced by TM-mode and the linear S-polarization is replaced by TE-mode [9]. As shown in Fig. 1(c), for a three-layer waveguide structure with the refractive indices of core and cladding, n₁ and n₀, when the incident angle is θ_in, its numerical aperture is $NA = {\theta _{max}} \approx {n_1}\sqrt {2(1 - {n_0}/{n_1})} $, where θ_max denotes the maximum acceptance angle of input light [35]. As shown in Fig. 1(d), when a guided-mode in the input single-mode waveguide is imposed onto the TIR interface, the guided wave of reflected-mode is assumed to have a normalized reflectivity R_mr= 1 and an optical phase incremental ${\phi _{mr}}$, the reflective coefficient is $r = exp (i{\phi _{mr}})$. Further, this guided wave of reflected-mode must suffer from the process of two optical phase increases before it is reflected into the MMI structure: (i) the optical phase caused by the reflection process and (ii) the optical phase caused by the inserted metasurface. Finally, it is assumed that the relative permittivity of the reflector material is ɛ_mr=ɛ_r+ iɛ_i in accordance with the Equation ${D_\textrm{m}} = \frac{1}{{{R_m}}} \cdot \frac{{\partial {R_m}}}{{\partial \theta }} + i\frac{{\partial {\phi _m}}}{{\partial \theta }} = D_m^R + iD_m^I$[8], where the term $D_\textrm{m}^I$ is relevant to the optical phases of the reflected light-wave at the polarization state m. By referring to the phase-equal principle of guided-mode shown in Fig. 1(c), we set ɛ_eq as its equivalent real relative permittivity, the incident angle θ_in and the acceptance angle θ_g of the input waveguide meet $\sin {\theta _{in}} = {n_1} \cdot \sin {\theta _g}$, and for the traveling length L_t of guided-wave and the waveguide length L_g, which meet $\sqrt {{\varepsilon _{eq}}} \cdot {L_t}\cos {\theta _g} = {n_1} \cdot {L_t}$[36], we have an equivalent real dielectric constant as ${\varepsilon _{eq}} = n_1^2/{\cos ^2}[{\sin ^{ - 1}}(\sin {\theta _{in}}/{n_1})]$.

2.2 Modeling the optical feature response of semi-sphere type metasurface

Based on Ref. [8], by setting the relative permittivity of waveguide core as ${\varepsilon _1} = n_1^2$, for TE- and TM-mode plane-wave, $D_{TE}^I$ and $D_{TM}^I$ are expressed as $D_{TE}^I = D_\textrm{S}^I ={-} 2\sin {\theta _{in}}/\sqrt {{{\sin }^2}{\theta _{in}} - {\varepsilon _1}} $ and $D_{TM}^I = D_P^I ={-} 2{\varepsilon _1}\sin {\theta _{in}}/[\sqrt {{{\sin }^2}{\theta _{in}} - {\varepsilon _1}} \cdot ({\sin ^2}{\theta _{in}} - {\varepsilon _1}{\cos ^2}{\theta _{in}})]$. Then, for the guided wave, the real part of relative permittivity ${\varepsilon _1}$ is replaced by $N_{eff}^2$, $D_{TE}^I$ and $D_{TM}^I$ can be unitized to a general expression $D_\textrm{m}^I$ as

For the semi-sphere metasurface unit with a radius of ${\textrm{R}_{ms}}$, if ${a_1}$ and ${b_1}$ are the scattering coefficients of electric and magnetic dipole moments, respectively, the electric and magnetic dipole polarizabilities, $\alpha _{eff}^E$ and $\alpha _{eff}^M$, of the semi-sphere metasurface in both the x and y directions are defined as [37,38]

As shown in Fig. 1, the nanoscale semi-sphere metasurface is vertically inserted into the WC structure studied in this work and the guided wave transmits through the metasurface, so, as shown in Fig. 2, the output light-wave of a waveguide is received behind the metasurface. Thus, by deleting the effective polarizability of magnetic field in Eq. (25) of Ref. [37], the transmittance of light-wave is refined to be Eq. (5):

Here ${\alpha ^E}({45^o}) = |{t({{45}^o})} |{\alpha ^E}$ when the wave strikes onto the metasurface at 45° incidence.

 

Fig. 2. Schematic propagation processes in a three-dimensional (3D) model of waveguide. (a) and (b) The polarization changing processes of a guided-mode light-wave traveling in homogeneous medium at TE- and TM-mode, respectively.

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2.3 Comparing optical feature responses between plane wave and guided wave

Equations (2) through (5) are not for the plane wave, while for the guided wave, so the physical parameters and constants are drastically different from plane wave [35,36]. For an input guided-wave with TEM-mode, as shown in Fig. 2(a), when input light is TE-mode, i.e., its electric field is at the y-axis direction, its magnetic field does not maintain in a plane perpendicular to the wavevector ${\vec{k}_m} = {N_{eff}}(0){\vec{k}_0}$, and any beam in the conical equal-phase wave-front mode will not have a transverse magnetic field.

Similarly, as shown in Fig. 2(b), when the input light is TM-mode, i.e., the magnetic field is at the y-axis direction, the electric field does not maintain in a plane perpendicular to the wavevector ${\vec{k}_m} = {N_{eff}}(1){\vec{k}_0}$, and any beam in the conical equal-phase wave-front mode will not have a transverse electric field. Thus, the TE- and the TM-mode can never co-exist in a guided wave. This case breaks the optical resonance condition of dipoles where an electric-magnetic coupling occurs [37,38]. Based on the polarization distribution attributes of a guided wave shown in Fig. 2, the optical resonance condition of electric and magnetic dipoles is broken for any off-axis points A and B of the equal-phase wavefront for TE-mode or C and D of the equal-phase wavefront for TM-mode excepts for the point in the axis vicinity.

According to the traditional optical laws, the reflection and refraction are subject to the Fresnel’s and the Snell’s laws, respectively, and an incident angle can individually induce an optical APC. As a result, the reflection angle is affected by the phase change of the reflected beam, further leading to a shift of the reflection angle. For the guided-mode m, if $N_{eff}^{in}(m )$ and $N_{eff}^{rf}(m )$ are the effective refractive indices of input and reflected guided waves, respectively, and λ is the wavelength of light in vacuum, under the TIR condition after a metasurface is imposed on the TIR interface, an effective optical APC, ϕ_ms, is caused to the reflected guided-mode, and X is the GH spatial shift of the reflected wave, we can have the modified Fresnel’s law of reflection for the guided wave as defined by Eq. (6) [25]:

Thus, when the metasurface caused optical APC is ϕ_ms, then from the modified Fresnel's law for the reflected guided-mode defined by Eq. (6), the critical angle and the reflection angle can be defined by Eqs. (7) and (8), respectively, as

Of remarkable difference from the traditional laws of light-wave reflection is that, due to ϕ_ms, both θ_c and θ_r depend on θ_in and the reflection angle no longer equals to the incident angle. By considering the metasurface caused optical phase ϕ_ms and the reflection caused optical phase ϕ_mr, and based on the traditional GH spatial shift $X$ of reflected beam for a guided wave [25], we can define the GH spatial shift X of the metasurface enhanced waveguide corner regime shown in Fig. 1(a) as

3. Simulations for the optical feature responses of Si-nanoscale semi-sphere metasurface for guided waves

3.1 Simulations for the optical responses of Si-nanoscale semi-sphere metasurface in waveguide corner structure

From guided-mode principle, it is concluded that the equal-phase front of output wave is a conical wavefront, so only the thin beam in the conical axis vicinity has the similar polarizability characteristics with plane wave as discussed in Sec.2. By selecting the same waveguide structure as shown in Fig. 2, where the width of single-mode waveguide is set as 2.0 µm, then the effective indices for TE- and TM-mode are obtained to be $N_{eff}^{in}({TE} )= 3.4060$ and $N_{eff}^{in}({TM} )= 3.3066$, respectively. Therefore, for the case of plane wave at the incident angle of 45°, with an FDTD software tool we obtain the simulation results of the optical phase changes with the radius of semi-spheres as shown in Fig. 3(a). Note that both TE- and TM-mode have frequently abrupt optical phase changes between -1 rad and +1 rad. In contrast, for a waveguide shown in Fig. 2, on the five different selected points: O, A, B, C and D on the output wavefront behind the metasurface, we also obtain the optical phase changes of guided wave with the radius of semi-spheres for TE and TM-mode as shown Fig. 3(b). What is astonishing are that the phase changes of TE- and TM-mode at the same point are completely overlap, namely, it is polarization independent. Illustratively, the two horizontally symmetric points A and B have the similar smooth phase changes in amplitude with opposite signs, while the two vertically symmetric points C and D have the completely same smooth optical phase changes in both amplitude and sign. These phenomena imply that the metasurface caused optical phase change is only dependent on the acceptable angle of guided mode, which is determined by incident angle irrespective of polarization state. In addition, of importance, it can be noticed that only the optical phase change of TM-mode on the axis-point-O is unstable compared with all the other points when the radius of semi-spheres is smaller than 380 nm because the quasi-axis beams have the extremely large acceptable angle and they are almost the same as that of plane wave.

 

Fig. 3. (a) and (b) FDTD simulations for the optical abrupt phase change ϕ_ms of plane and guided waves at the five points of output equal-phase wavefront caused by the metasurface for TE- and TM-mode. (c) and (d) Numerical simulations for the critical and reflection angles with incident angle for the APC values: π/4 and π/2.

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Based on the above discussions for both the changes of critical/reflection angles and the dependences of optical APC caused by metasurface on the radius of semi-spheres, with Eqs. (7) and (8) we select the two values of ϕ_ms as 0.25π and 0.50π, then obtain the critical and reflective angles with incident angle as shown in Figs. 3(c) and 3(d), respectively. Note from Fig. 3(c) that the critical angle decreases with incident angle due to the metasurface-enhanced TIR condition, where the critical angle of TE-mode linearly decreases, but that of TM-mode linearly decreases first and then approaches to a stable value. So, it turns out that the APC value caused by the vertically inserted metasurface of Si-nanoscale semi-spheres directly determines the decreasing rate of critical angle. Note from Fig. 3(d) that the reflective angle does not equal to the incident angle any more, while linearly increases with incident angle for TE-mode and slowly decreases with incident angle first and then quickly increases for TM-mode. Similarly, the APC value also determines the increasing rate of reflective angle with incident angle.

An important finding from Fig. 3(c) is that the critical angle θ_c has the different changing trends with incident angle θ_in between TE- and TM-mode, and then the two curves have the intersection point at 39°, which is determined by the device regime. So does the reflection angle θ_r from Fig. 3(d). It turns out that the incident angle of 39° can make the WC structure studied in this work have no discrepancy between TE- and TM-mode, leading to the same APC values and further the same GH shifts between these two polarization modes.

3.2 Simulations for the Goos-Hänchen spatial shifts of light-wave in waveguide corner structure via an inserted metasurface of Si-nanoscale semi-spheres

By considering the significant dependences of GH spatial shifts on incident angle, we select three incident angle values: 35°, 39° and 45°, and then for the two phases of APC: 0∼0.25π and 0.25∼0.50π, we obtain the numerical simulations of the GH spatial shifts with APC as shown in Figs. 4(a) and 4(b), respectively. First note from Fig. 4(a) that for the APC value of 0.25π, the GH shift discrepancy between TE- and TM-mode is about 0.8 µm at the incident angle of 35°, while this value is close to zero at the incident angle of 39°, and it changes to 0.4 µm at the incident angle of 45°. Similarly, note from Fig. 4(b) that for the APC value of 0.50π, at the incident angle of 35° the GH shift discrepancy between TE- and TM-mode is around 0.4 µm, while at the incident angle of 39° this value is also close to zero, which is in accord with the phenomenon in the simulations shown in Figs. 3(c) and 3(d) as discussed in the Sub-Sec. 3.1 that the critical angles for the TIR vs incident angle at TE- and TM-mode have an intersection at 39° and so are the reflection angles, and at the incident angle of 45° it changes to 0.2 µm. Thus, of importance is that: (i) the GH shift discrepancy between TE- and TM-mode does not depend on incident angle, but it dramatically depends on the APC value and (ii) at the incident angle of 39°, TE- and TM-mode have the equal GH shifts. So, these two differences sustain the developments of both the polarization dependent and independent functionalities of applicable devices.

 

Fig. 4. (a) and (b) the numerical simulation results of the GH spatial shifts with incident angle for three incident angles: 35°, 39° and 45° and two APC values: 0.25π and 0.50π for both TE and TM reflected guided modes; (c) and (d) the perspective WC structures for the TIR operation without and with the semi-sphere metasurface; (e) and (f) the FDTD simulations for the reflected mode at TE-mode for the cases: without and with the metasurface, respectively; (g) and (h) that of the reflected mode at TM-mode for these two cases at the incident angle of around 39°.

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Based on the advantages of guided wave in a waveguide, the metasurface caused APC over the corresponding plane wave is analyzed in Sec. 2. Firstly, the structures of two WC devices: (i) the WC structure without metasurface and (ii) the WC structure with a Si-nanoscale semi-sphere metasurface designed on the TIR interface as shown in Figs. 4(c) and 4(d), respectively, the input single-mode waveguide has both the width and thickness values of 2.0 µm, and the output single-mode waveguides have the width and thickness of 12.0 µm and 2.0 µm, respectively. Then, based on the simulation results shown in Figs. 4(a) and 4(b), we select the thickness of 0.45 µm for the metasurface of Si-nanoscale semi-spheres with a diameter of 0.9 µm and a period of 1.0 µm that can generate $0.50\pi $. As a result, with the software tool – FDTD we simulate the propagation processes of optical beam, then we obtain the optical field distributions of the reflected light-wave at TE-mode at the optimal incident angle of 39° as shown in Figs. 4(e) and 4(f), respectively. Similarly, the optical field distributions of the reflected light-wave for TM-mode are obtained as shown in Figs. 4(g) and 4(h), respectively. Note that with the three output waveguides, for the simulation results at both TE- and TM-mode in Fig. 4, the optical field distribution in the (i) case without metasurface almost maintains a large quasi-circular single-mode to the middle output waveguide after being reflected by TIR interface, but in the (ii) case with the metasurface it is re-distributed by the metasurface assisted TIR interface and the reflected mode with a much smaller mode-size selects the output at the right output waveguide where the centroid of optical power mode is almost at the core of the geometrical shape. Of importance from the comparison of results between the output modes in Figs. 4(e)/(f) and 4(g)/(h) is that the WC structure generates the quasi-zero GH shift discrepancy between TE- and TM-mode irrespective of having or having no metasureface on the TIR interface, which presents the advantage of guided wave over plan-wave in the optical response of metasurface to a light wave from physical mechanism. It turns out that both the numerical and FDTD simulation results of GH shifts at the aforementioned optimal incident angle of 39° shown in Fig. 4 sufficiently sustain the numerical results shown in Figs. 3(c) and 3(d).

It can be concluded from the comparison of results between the (i) and the (ii) cases shown in Fig. 4 that without the metasurface caused optical phase change the GH shift cannot make the mode out of middle output waveguide, while with an $0.25\pi \sim 0.50\pi $ optical APC caused by the Si-nanoscale semi-sphere metasurface the large positive GH spatial shift of some 2.0 µm is achieved to make the reflected mode have a displacement from the middle output waveguide to the right one. So, of more importance is that the output modes for these two cases are at two different output waveguides with the effect of the vertically inserted metasurface.

Additionally, from the above FDTD simulation results, we tabulate the output positions at the TE- and TM-mode into Table 1. It can be found that although at the (i) case without the metasurface, with the appropriate construction optimization of WC structure, TE- and TM-mode have a completely same small GH spatial shift, and at the (ii) case with the vertically inserted metasurface of Si-nanoscale semi-spheres, the GH spatial shift caused displacements of the reflected mode at both TE- and TM-mode are also the same to be ∼2.0 µm with a ∼62% reflectivity, the unshifted mode with a ∼25% reflectivity are still maintained in the middle waveguide. This attribute is owing to the isotropic scattering of the Si-nanoscale semi-sphere of metasurface unit. From the FDTD simulation results shown in Figs. 4(c) through 4(f), it can be found that the wavefront distribution of a reflected beam in the (ii) case with the semi-sphere metasurfaceh has a large discrepancy from what obtained in the (i) case without metasurface. So, apart from the optical APC to generate the anomalous reflection of guided-mode, the other advanced attribute of metasurface lies on the change or re-shape of wavefront. It turns out that the FDTD simulation results depicted in Table 1 no doubt gives the powerful evidence to the above numerical simulation results shown in Figs. 4(a) and 4(b).

Table 1. FDTD simulation results of the GH shift caused displacements and the mode transfer efficiency of SOI WC with a metasurface of Si-microscale semi-sphere array:

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The controllable displacement and re-shaping of the reflected mode caused by the metasurface can create new physical laws of light waves, and the other advantage of the WC structure is that it has the same displacement between TE- and TM-mode no matter it has or has no metasurface when the incident angle is set at 39°, which is agreeable with the numerical simulation results shown in Figs. 4(a) and 4(b) owing to the same physical mechanism.

4. Experimental verification and analysis

According to the above designs of the WC structure, we fabricate a sample of 1 × 3 SOI-WC device with a state-of-the-art technique as shown in Fig. 5(a) where the center-to-center spacing of output ports is set to be 3.0 µm and the input single-mode waveguide is designed to have an incident angle of 39°, then the experimental output modes without the metasurface layer are observed at the three ports as shown in Figs. 5(b) and 5(c) for TE- and TM-mode, respectively. In contrast, when a semi-sphere metasurface with an APC value of ${\sim} 0.5\pi $ is designed for WC structure, the experimental output modes at the three ports as shown in Figs. 5(d) and 5(e) for TE- and TM-mode, respectively. Note that the experimental results are agreeable with the FDTD simulation results shown in Fig. 4, so it turns out that the metasurface caused optical phase change of ${\sim} 0.5\pi $ can lead to the controllable GH shifts of a guided-mode for both TE- and TM-mode, which is ∼2.0 µm and enough for many applicable devices such as the high-performance polarization beam splitters, high-performance optical switches, etc. It is also noticed from Figs. 5(b) and 5(c) that because the output beam is aimed to the middle port without the metasurface at the TIR interface, both TE- and TM-mode almost have no beam shift, then note from Figs. 5(d) and 5(e) that TE-mode has a positive GH shift with about 65% fraction and ∼25% fraction has no shift, in contrast, TM-mode also has a positive GH shift with about 60% fraction and ∼30% fraction has no shift. Thereby, the experimental results are agreeable with the FDTD simulations at the two cases of WC structure: without and with the metasurface at the TIR interface and the results are also agreeable with the above numerical simulation values. The sum of output powers at two ports at both TE- and TM-mode is around 90%, implying a sum of the scattering loss caused by two reflection processes at two TIR interfaces and the scattering loss caused by the sidewall roughness of waveguide during the propagation of guided wave is about 10%, which could be decreased as the fabrication quality is improved. In addition, the output power differences between Figs. 5(b)/(d) and 5(c)/(e) are due to the fabrication errors from the optimal design.

 

Fig. 5. Device sample and experiments. (a) the SEM photo of 1 × 3-WC structure sample where CM_1,2 are corner mirrors, the metasurface is set on CM₂, (b) and (c) the experimental results: the photos of output modes without metasurface for TE- and TM-mode, respectively, while (d) and (e) the output modes with a designed semi-sphere array metasurface that can cause APC value of ∼0.5π for TE- and TM-mode, respectively, where the normalized output of reflected mode with ∼65% and ∼60% selects the right output waveguide and the reflected mode with ∼25% and ∼35% still selects the middle output waveguide, respectively.

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The great agreements among the experimental results, both the FDTD simulations and the numerical simulations can lead to a convincible conclusion that an active manipulation of GH shift can efficiently be controlled by the accurate abrupt optical phase change of such a metasurface of Si-nanoscale semi-sphere array that is vertically inserted into the WC structure. Then, it is known that the functions of a metasurface can be efficiently controlled by the variable material property, unit structure, etc.

5. Conclusions

As a conclusion, this work in theory and experiment has demonstrated the significant advancements of the guided-mode from waveguide over the plane wave in improving the metasurface caused optical phases for the development and applications of new functionalities. For the TIR case of SOI waveguide mode in a WC structure, first behind the TIR interface a metasurface layer of Si-nanoscale semi-sphere array vertically inserted into the WC structure, then the equivalent permittivity of a guided wave is discovered and modelled, furthermore for the optical phase change of reflected mode and the optical phase change of both plane wave and guided wave caused by the metasurface are investigated and analyzed by comparing the discrepancies of electric polarizabilities between plane wave and guided wave. Further, GH shifts of the reflected guided-mode caused by the metasurface are systematically investigated with respect to the WC structure and the metasurface unit structure. As a result, an important advantage of the guided wave over the plane wave is that the manipulation of metasurface-caused optical phase change is more controllable. Consequently, the new theoretical models of the GH shifts are derived, and then the numerical calculation values, the FDTD simulation values and the experimental results are in excellent accord with one another, which is powerfully sustaining the innovation, providing the opportunities for developing the new physical functions and devices such as polarization independent devices, ultrafast micro/nanoscale photonic switching devices, and wavelength selective switches, etc. Illustratively, the GH shifts of TE- and TM-mode are almost same at the incident angle of 39°, which is conducive to developing the polarization independent PIC devices.