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Abstract
Metasurface has achieved fruitful results in tailoring optical fields in free space. However, a systematic investigation on applying meta-optics to completely control waveguide modes is still elusive. Here we present a comprehensive catalog to selectively and exclusively couple free space light into arbitrary high-order waveguide modes of interest, leveraging silicon metasurface-patterned silicon nitride waveguides. By simultaneously engineering the matched phase gradient of the nanoantennas and the vectorial spatial modal overlap between the antenna near-field and target waveguide mode profile, either single or multiple high-order modes are successfully launched with high purity reaching 98%. Moreover, on-chip twisted light generators are theoretically proposed with configurable OAM topological charge ℓ from −3 to +2. This work may serve as a comprehensive framework for guided mode meta-optics and motivates further applications such as versatile integrated couplers, multiplexers, and mode-division multiplexing-based communication systems.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As the electrical bottleneck in conventional electronic circuits is getting increasingly prominent in recent years [1], photonic integrated circuits are hatching as a promising technology to potentially revolutionize conventional integrated circuits [2], by providing broadband optical communication paradigms [3,4], ultrafast chip-scale information processing platforms and efficient optical interconnects with low power consumption [5–10]. However, current integrated photonic devices generally encounter limitations from its fundamental building blocks of optical waveguides, in terms of bulk footprint and restrained functionalities [2]. To realize compact and multifunctional integrated optical devices, subwavelength photonic structures such as metasurface [11–16] can be allied with dielectric waveguides to infuse new degrees of freedom to conventional waveguide-based devices to enhance performance or expand device functionalities [17–29].
In recent years, there has been growing interest in integrating various subwavelength meta-structures with waveguide platforms [19,22–24,30–41], which greatly enriches the overall structural design library of integrated photonic devices, opening new opportunities to either largely enhanced device performance or previously inaccessible sophisticated functionalities [18,19]. Efforts have been contributed to apply plasmonic nanoantennas on the top of dielectric waveguides for off-chip beam manipulations [22,42] and high-speed polarization demultiplexing [30,31,43]. However, these devices inevitably inherit the high Ohmic loss from metal nanoantennas that limit the device efficiency. Dielectric metasurfaces-addressed photonic waveguides are also explored with low material loss, where geometric phase (also called as Pancharatnam-Berry phase) [44] metasurfaces are integrated on silicon [34] and silicon nitride waveguides [35] for compact integrated polarization demultiplexers. Nevertheless, the optical fields that get coupled into these waveguides are uncontrollable hybrid modes [34,35], which will severely influence their performance in high-speed optical communication systems due to the inter-mode dispersions. Complex optical field generations and manipulations via metasurfaces in free space have been exhaustively explored [45,46]. However, a systematic physical catalog exploring the excellent capability of using metasurface to completely control diverse guided modes is still elusive to the best of our knowledge.
Here, we present a comprehensive framework targeting the arbitrary manipulation of waveguide modes using silicon metasurface-patterned silicon nitride nanophotonic waveguides. Arbitrary high-order modes of interest can be selectively and exclusively excited with high mode purity reaching 98$\%$ by following our proposed easy-to-implement methods to simultaneously engineer the phase gradient of the metasurface and the spatial modal overlap between antenna near fields and the target guided mode profile to excite. The mechanism of mode competition and selective mode excitation are analyzed as well. Furthermore, via judiciously mixing specific high-order modes, on-chip vortex beam generators are also theoretically demonstrated and configurable topological charge $\ell$ of orbital angular momentum (OAM) from $-3$ to $+2$. This work may serve as a positive paradigm to migrate meta-optics from free space optics into guided mode optics, for catalyzing further researches in photonic integrated circuits and enabling applications such as versatile on-chip couplers, demultiplexers, and OAM-multiplexing-based communication systems [47–49].
2. Fundamentals and design method
The general device structure of the mode-configurable directional coupler is schematically shown in Fig. 1(a). Normally, the incident light beam undergoes consecutive scattering events in the waveguide via the nanoantennas and picks up an effective unidirectional momentum [32,33] provided by the silicon phase-gradient metasurface atop the silicon nitride waveguide. To directionally and selectively excite a specific waveguide mode of interest, we first need to match the phase gradient of the metasurface $\Delta \varphi / d$ with the effective mode index $n_{\mathrm {eff}}$ of the target mode,
Fig. 1. Schematics and spatial modal overlap engineering illustration.(a) Operation schematic. Normally incident light can be directionally coupled into a specific guided mode by interacting with the gradient metasurface. (b) Antenna near field. Black dashed lines indicate the profiles of waveguide and the antennas. (c) Electric field distributions of the ideal guided modes. (d) Calculated spatial modal overlap $\eta$ between antenna near field $\mathbf {E}_{\textrm {antenna}}$ and target guided mode $\mathbf {E}_{\textrm {mode}}$ of devices exclusively launching $\mathrm {TM}_{40}$, $\mathrm {TM}_{41}$, $\mathrm {TM}_{42}$, $\mathrm {TM}_{43}$ and $\mathrm {TM}_{44}$ modes with same waveguide width of 5.0 µm. (e) Calculated mode purity for the same devices to excite $\mathrm {TM}_{40}$, $\mathrm {TM}_{41}$, $\mathrm {TM}_{42}$, $\mathrm {TM}_{43}$ and $\mathrm {TM}_{44}$ modes, showing excellent agreement with $\eta$ to validate our assumption.
In addition to the matched phase gradient, judicious engineering of the vectorial spatial modal overlap $\eta$ is also vital for exclusively launching certain arbitrary high-order waveguide mode,
It is worth pointing out that $\eta$ is a vectorial integration. Therefore, the polarization state of the incident light source and $\mathbf {E}_{\mathrm {antenna}}$ is crucial to determine whether TE or TM modes is preferably excited. From Fig. 1(c) we can elucidate that the electric field component $\mathbf {E}_{\mathrm {y}}$ is dominating for TE modes [36], while $\mathbf {E}_{\mathrm {z}}$ is the biggest component for TM modes. If the excitation free space light source is linear $y$-polarization, $\mathbf {E}_{\mathrm {antenna}}$ will be also largely $y$-polarized, leading to a much larger $\eta$ for directionally coupling to TE modes. In contrast, if we apply linearly $x$-polarized light source to produce mostly $x$-polarized $\mathbf {E}_{\mathrm {antenna}}$, TM modes will be launched instead, because TM modes has minimal $\mathbf {E}_{\mathrm {y}}$ component but relatively larger $\mathbf {E}_{\mathrm {x}}$ component to produce larger a $\eta$ with selective TM mode in this case.
Given that $\Psi _{m, n}$ mode ($\Psi$ is TE or TM) has $m+1$ lobes along $y$ direction, a total of $m+1$ rows of antenna arrays should be applied. By comparing the corresponding columns in Figs. 1(b) and 1(c), we can see that in this case each antenna array will be assigned as local hotspot to match the maximum location of each lobe along $y$ direction. Meanwhile, as each lobe is $\pi$ out-of-phase to adjacent lobes, neighboring antenna arrays should share a displacement around $\Delta x \approx \lambda /\left (2 n_{\mathrm {eff}}\right )$ along $ {x}$-axis to match the $\pi$ phase difference between neighbouring lobes, as shown in Fig. 1(b).
Therefore, in summary, linearly $ {x}$-polarized light should be adopted to excite TM mode series, while $ {y}$-polarization will launch TE mode series. To selectively excite $\Psi _{m, n}$ mode, $m+1$ rows of identical antenna arrays should be applied with a dislocation $\Delta x$ along $ {x}$ axis between adjacent arrays. The mode order number $n$ is thus controlled by the matched phase gradient $\Delta \varphi / d$ to the effective index $n_{\mathrm {eff}}$ of the target waveguide mode for excitation.
In previous similar publications, mode-management is either based on conversion [45] or restrained to only fundamental modes [36,40,50]. In contrast, the proposed method here can simultaneously achieve light coupling and mode conversion in a fully integrated and compact manner. Compared with previous researches that mostly focus on polarization or wavelength demultiplexing [37], this work presents a complete and systematic framework on deploying meta-optics to tailor guided mode optics, realizing diverse high order waveguide modes with much improved mode purity. Moreover, for the first time, we further extend our approach to launch OAM beams with scalable topological charge to even $|\ell |=3$, which is also an obvious increment (approximate one order of magnitude higher) to other similar archived references that only discuss vortex beams with $\ell = \pm 1$ [40,50,51].
3. Selective excitation of arbitrary waveguide modes of interest
To demonstrate the feasibility of our proposed design method, a set of mode-configurable on-chip directional couplers for arbitrary high-order modes are designed and numerically validated. Figure 2 catalogs diverse meta-waveguide couplers to exclusively excite various TM modes as a proof-of-concept, using silicon metasurface-dressed silicon nitride waveguides around the telecommunication wavelength of $\lambda =1.55$ $\mathrm{\mu}$m. For instance, to selectively launch $\mathrm {TM}_{m, n}=\mathrm {TM}_{5,2}$ modes, $m+1=6$ rows of antennas with matched gradient of $\Delta \varphi / d = 266^{\circ }$ are optimized under the illumination of linear $ {x}$-polarized plane wave.
Fig. 2. Mode configurable meta-waveguide couplers using silicon metasurface-patterned silicon nitride waveguides to selective launch diverse TM modes. The device schematics (top view) are shown as the left panels with different scale bars, where the silicon nanoantennas are marked as red rectangles; the silicon nitride waveguides are colored in blue; grey color indicates silicon dioxide substrate. The output electric field norm distributions $|\mathbf {E}_{z}|$ at the right ports are shown as the right panels with mode purity values manifested above. For detailed device structure parameters please see Supplement 1.
As shown in Fig. 2, most devices exhibit high mode purity over 80$\%$, and the highest has reached 98$\%$. Moreover, the proposed devices have shown broad operation bandwidth. As illustrated in Fig. 4(b), the device to launch $\mathrm {TM}_{5, 2}$ mode has a broad optical bandwidth (full-width half-maximum, FWHM) of 155 nm. The corresponding coupling efficiency spectra under total-field scattering-field (TFSF) light source are shown in Fig. 4(e) accordingly.
Massive TE modes can also be effectively launched in a similar manner as cataloged in Fig. 3 with averaged purity about 90$\%$, by following our design principle of guided-mode meta-optics. The device bandwidth and coupling efficiency spectrum of an exemplary device to excite $\mathrm {TE}_{1, 2}$ mode in Fig. 3 are shown in Figs. 4(c) and 4(f) respectively. Figures 4(g) and 4(h) illustrate the electric field component distribution of $\mathbf {E}_{\mathrm {z}}$ for the device to excite $\mathrm {TM}_{5, 2}$ mode and $\mathbf {E}_{\mathrm {y}}$ for the device to launch $\mathrm {TE}_{1, 2}$ mode accordingly, where the incident $ {x}$- and $ {y}$-polarized light beams are directionally coupled to the waveguide right ports to produce the target modes.
Fig. 3. Chip-integrated waveguide mode convertors for exclusively exciting arbitrary TE modes. Device structures and output $\left |\mathbf {E}_{y}\right |$ distributions at the right ports are shown in each unit accordingly.
Fig. 4. Chip-integrated waveguide mode convertors for exclusively exciting arbitrary waveguide modes: (a) $\mathrm {TM}_{4,0}$, (b) $\mathrm {TM}_{5,2}$, (c) $\mathrm {TE}_{1,2}$. Device coupling efficiency calculated under total-field scattering-field (TFSF) light source are shown in (d)-(f) accordingly. (g) and (h) The electric field component $\mathbf {E}_{\mathrm {z}}$ and $\mathbf {E}_{\mathrm {y}}$ along waveguide middle plane for the devices in (a) and (c) respectively.
A practical tip to achieve high mode purity in designing these meta-devices integrated with phase gradient metasurfaces for given mode $\Psi _{m, p}$ is to judiciously select the waveguide dimension. If two modes $\Psi _{m, p}$ and $\Psi _{m, q}^{\prime }$ share same mode order number $m$ (denoting same rows of antenna arrays are involved), and they by accident have a similar value of effective mode index $n_{\mathrm {eff}}$, crosstalk will take place. In this case, it will be easy to simultaneously excite both modes $\Psi _{m, p}$ and $\Psi _{m, q}^{\prime }$, which will deteriorate the mode purity of the interested mode $\Psi _{m, p}$.
For instance, for a certain device for launching TM$_{5,4}$ mode (see Supplement 1 for details), a pronounced disturbing TM$_{5,3}$ mode exists at the telecommunication wavelength of $\lambda = 1.55$ $\mathrm{\mu}$m, as shown in the mode purity spectra of the guided modes in Fig. 5(a). In Fig. 5(b) we plot the effective mode indices distribution of diverse high-order modes with same order number m = 5 supported in this device. Because the effective mode indices $n_ \textrm {neff}$ of TM$_{5,4}$ mode ($n_ \textrm {neff} = 1.500$) and TM$_{5,3}$ mode ($n_ \textrm {neff}' = 1.597$) are approaching, an unwanted TM$_{5,3}$ mode is also excited, hence dragging down the mode purity of the target TM$_{5,4}$ mode.
Fig. 5. The effect of waveguide size on purity. Upper panels: The mode purity spectra of $\mathrm {TM}_{5,4}$, $\mathrm {TM}_{5,3}$, $\mathrm {TM}_{5,2}$, $\mathrm {TM}_{5,1}$, $\mathrm {TM}_{5,0}$ modes in different waveguide sizes. Lower panels show the relationship between effective refractive index and purity of different modes in different waveguide sizes. The waveguide cross-section dimension for the device in (a) is 4.6 $\mathrm{\mu}$m $\times$ 4 $\mathrm{\mu}$m, the waveguide dimension for (c) is 4.6 $\mathrm{\mu}$m $\times$ 3.8 $\mathrm{\mu}$m.
An easy method to suppress the unwanted $\Psi _{m, q}^{\prime }$ mode is to modify the dimension of the waveguide to discriminate and diverge the effective index $n_{\mathrm {eff}}$ of the two modes. As shown in Figs. 5(c) and 5(d), by altering the dimension of the underlying silicon-nitride waveguide from width $\times$ height $= 4.6 \times 4$ $\mathrm{\mu}$m$^2$ to width $\times$ height $= 4.6 \times 3.8$ $\mathrm{\mu}$m$^2$, the difference between effective mode indices $n_ \textrm {neff}$ of target mode TM$_{5,4}$ and main jamming mode TM$_{5,3}$ is enlarged from 0.097 to 0.107 to suppress the excitation of the disturbing TM$_{5,3}$ mode. Then the mode purity of the target mode $\Psi _{m, p}=$ TM$_{5,4}$ will be effectively enhanced.
4. On-chip OAM generators with a configurable topological charge
Beside eigen-TE or TM mode series, our proposed method for arbitrary mode-configurable meta-waveguide couplers can also facilitate the manipulation of complex intra-waveguide light field generations. Selective excitation of either single or multiple specific waveguide modes (eigen-modes or certain hybrid modes) is available as well. Taking optical vortex beams carrying OAM as an instance [51–54], the complex OAM field with scalable and configurable topological charge $\ell$ can be realized using our design method for meta-waveguide platform.
Orbital angular momentum (OAM) is one of the fundamental attributes of photons. A light beam with rotating polarization possess spin angular momentum (SAM), for instance, a SAM of $\pm \hbar$ per photon for circularly polarized light [50]. In contrast, a rotating wavefront leads to light beams carrying OAM with density written as $\textbf {j}=\textbf {r} \times \textbf {p}$ [48], where $\textbf {r}$ is the spatial vector and $\textbf {p}=\epsilon _0 \textbf {E} \times \textbf {B}$ ($\epsilon _0$ denotes dielectric permittivity [55], E and B are electric and magnetic fields respectively). A simple example of an optical vortex carrying OAM has a helical transverse phase distribution of $\varphi = \exp (\textrm {i} \ell \phi )$, where $\phi$ stands for azimuthal coordinates, $\ell =\frac {1}{2\pi }\times \oint _{C}{\nabla \phi (\textbf {r})\textrm {dr}}$ is the topological charge ($C$ is a closed loop surrounding the phase singularity) [48].
An easy approach for OAM generation is mode mixing method [48], where specific OAM light vortices can be obtained by wisely mixing several high-order modes [55]. Leveraging the above-mentioned catalogs for selective launching distinct high-order modes of interest, we have numerically validated a set of on-chip OAM generators with configurable topological charge $\ell$ from $-3$ to $+2$ via mode mixing, using finite-difference time-domain (FDTD) calculations. Figure 6 shows the device structures and simulated output fields of twisted light for $\mathrm {OAM}_{+1}$, $\mathrm {OAM}_{-1}$, $\mathrm {OAM}_{-2}$ and $\mathrm {OAM}_{-3}$ accordingly, which are realized by properly mixing specific high order modes [48]. More details and results are given in Supplement 1. The top-view of the proposed vortex beam emitters are shown in the left panels. The silicon nitride waveguide has quasi-square dimension and its structure is judiciously chosen to support the desired high-order modes for mode mixing, and simultaneously cut off other higher order modes which are not supposed to be excited. A total of $N+1$ silicon antenna array sets are deployed atop the silicon nitride waveguide to excite OAM$_{\pm N}$ mode.
Fig. 6. On-chip vortex beam generators with configurable OAM topological charges for (a) $\ell =+1$, (b) $\ell =-1$, (c) $\ell =-2$, and (d) $\ell =-3$. The device schematics (top view) are shown as left panels. The output electric field distributions for individual antenna arrays, the overall OAM field inside the meta-waveguide, and the output optical fields after exiting waveguide Right Ports after a propagation length around 20 $\mathrm{\mu}$ m are shown as right panels. For device structure details, please see Supplement 1.
Each antenna array set will launch one certain right-propagating high-order mode under the normal incidence of linearly $y$-polarized light from free space. The distance between different antenna sets along $ {x}$-axis are judiciously optimized to achieve the desired phase differences between different high-order modes. This mechanism is schematically shown as the right panels. Taking the third row for $\mathrm {OAM}_{-2}$ as an instance, the mode distributions in $y$-$z$ plane perpendicular to $x$-axis are shown when only one of the antenna arrays (i.e. namely $\mathrm {TE}_{2,0}$ antennas, $\mathrm {TE}_{0,2}$ antennas or $\mathrm {TE}_{1,1}$ antennas) are present. The phase distributions of the mixed output mode after exiting the right waveguide ports are also shown as the right-most panels of Figs. 6(a)–6(d), indicating designer topological charges $\ell$ from $+1$ to $-3$ respectively.
To characterize the purity of the output vortex beams generated from our proposed devices, the OAM spectrum method [56,57] is applied. Figure 7 shows the relative intensity distribution of the OAM components from the proposed twisted light generators depicted in Figs. 6(a), 6(c), and 6(d) accordingly. The detailed calculation method for the spectrum in Fig. 7 is elaborated in Supplement 1. In short, the purity of the generated vortex beams are retrieved from the relative power percentage of the target OAM component (for excitation) to the total power of the output light field. The purity (relative power in arbitrary units) of the OAM component with topological charge $\ell '$ is computed as $p=\left |c_{\ell ^{\prime }}\right |^{2}$[56]:
Fig. 7. Calculate purity of the output vortex light in free space from the proposed OAM generators in Fig. 6(a), 6(c), and 6(d) accordingly. The purity values are retrieved from OAM spectrum method around $\lambda = 1.55$ $\mathrm{\mu}$m (detailed in Supplement 1).
Here, the $\textbf {E}_{\textrm {mode}}$ are mainly $y$-polarized electric field component retrieved at $10$ $\mathrm{\mu}$m, 20 $\mathrm{\mu}$m, and 30 $\mathrm{\mu}$m after the structures for the left, middle, and right panels in Fig. 7, respectively. The calculated OAM light purity for the devices shown Figs. 6(a) (to excite $\textrm {OAM}_{+1}$ mode), 6(c) (to excite $\textrm {OAM}_{-2}$ mode), and 6(d) (to excite $\textrm {OAM}_{-3}$ mode) are $p=89.7\%$, 87.1$\%$, and 73.0$\%$ accordingly. We note that these numerically verified OAM purity values are on par with or even higher than previous similar publications on nanoscale chip-integrated vortex beam generations [22,40,50,57,58]. In the meantime, the calculated mode purity value $p$ also essentially agree with the spatial overlap integral method used in previous Refs. [22,40,50,58] as $p'={|\iint \mathbf {LG}_{0, \ell ^{\prime }}^{*}(y, z) \cdot \mathbf {E}_{\textrm {mode}}(y, z) \textrm {d}y\textrm {d}z |^{2}}/$ $(\iint |\mathbf {L} \mathbf {G}_{0, \ell ^{\prime }}(y, z)|^{2} \textrm {d}y\textrm {d} z \cdot \iint {|{\mathbf {E}_{\textrm {mode}}}(y, z)|}^{2} \textrm {d}y\textrm {d} z )$. Taking the device to launch $\textrm {OAM}_{+1}$ mode [illustrated in Fig. 6(a)] as an instance, the OAM purity of its generated light field calculated from overlap integral method is $p'=89.3 \%$, which is highly approaching with the purity computed from the OAM spectrum method [56,57].
The output vortex light field distributions at different propagation distance of $\Delta L$ in free space for the proposed devices are shown in Fig. 8 and Fig. S3 in Supplement 1, validating that the generated OAM field can propagate in free space in a stable manner (i.e. the helical phase distribution of $\exp (\textrm {i} \ell \phi )$ is well maintained).
Fig. 8. Distribution of the (a) instantaneous electric field component $\textbf {E}_y$ and (b) its phase profiles for the generated $\textrm {OAM}_{+1}$ light at different propagation distance $\Delta L$.
We further note that this scenario is also potentially scalable for higher OAM modes [40,50] with simpler device structures and material requirements than those exploring Parity-Time symmetry [59,60]. The physical intuition-based design method is also straightforward to comprehend and time-saving in design compared with the inversely designed vortex beam emitters [51], which are featured by long computer optimization time around from hours to even days. The operation bandwidth data of the individual antenna arrays to launch OAM$_{-2}$ and OAM$_{-3}$ modes are shown in Figs. 9(a) and 9(b) respectively, where most meta-structures exhibit very broad optical bandwidth. The calculated phase distributions of the mixed output vortex beams inside the waveguide at different light wavelengths are illustrated in Fig. 10. We also illustrate the electric field component distribution of $\mathbf {E}_{\mathrm {y}}$ and the output optical fields after exiting waveguide Right Ports at Wavelengths at both ends of the bandwidth in right panels. The proposed $\mathrm {OAM}_{+1}$ and $\mathrm {OAM}_{-2}$ device have shown a much larger optical bandwidth than the micro-ring resonators approach that typically have a bandwidth of only several nanometers [61].
Fig. 9. Mode purity spectra when only the single antennas are present accordingly. Upper panels: Mode purity spectra of modes in $\mathrm {OAM}_{-3}$. Lower panels: Mode purity spectra of modes in $\mathrm {OAM}_{-2}$.
Fig. 10. Left Panels: Phase distributions of $\mathrm {OAM}_{+1}$ (a) and $\mathrm {OAM}_{-2}$ (b) inside the waveguide showing broad bandwidth. Right Panels: The output optical fields after exiting waveguide Right Ports around 20 $\mathrm{\mu}$m at different wavelengths are shown in right panels.
5. Conclusions
In summary, we have extended the concept of meta-optics into the realm of integrated optics to achieve almost arbitrary control over waveguide modes. By simultaneously engineering the phase gradient of the silicon metasurface resting on the silicon nitride waveguide and the vectorial spatial modal overlap between antenna near field and target guided mode profile, we can selectively and exclusively excite almost arbitrary high-order modes of interest with high mode purity reaching 98$\%$. By judiciously mixing several high-order modes, structured light like optical vortices carrying OAM can also be excited on chip with the configurable topological charge from $-3$ to $+2$.
The proposed device may be further optimized via inverse design algorithms [18,49,54,62–64] to simultaneously couple multiple modes at the same time. Polarization-multiplexed devices can be envisaged as well [30]. The meta-waveguides can also be allied with two-dimensional materials [6,65], phase-change materials [66] or lithium niobate [67,68] for reconfigurable or dynamically tunable devices. Through further optimizations, we believe OAM beams with higher topological charge are also possible, opening new opportunities for chip-scale structured light generations and potentially boosted mode-division-multiplexing-based communication systems.
Funding
National Natural Science Foundation of China (62075113, 61675114).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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