Optical meta-waveguides for integrated photonics and beyond

The growing maturity of nanofabrication has ushered massive sophisticated optical structures available on a photonic chip. The integration of subwavelength-structured metasurfaces and metamaterials on the canonical building block of optical waveguides is gradually reshaping the landscape of photonic integrated circuits, giving rise to numerous meta-waveguides with unprecedented strength in controlling guided electromagnetic waves. Here, we review recent advances in meta-structured waveguides that synergize various functional subwavelength photonic architectures with diverse waveguide platforms, such as dielectric or plasmonic waveguides and optical fibers. Foundational results and representative applications are comprehensively summarized. Brief physical models with explicit design tutorials, either physical intuition-based design methods or computer algorithms-based inverse designs, are cataloged as well. We highlight how meta-optics can infuse new degrees of freedom to waveguide-based devices and systems, by enhancing light-matter interaction strength to drastically boost device performance, or offering a versatile designer media for manipulating light in nanoscale to enable novel functionalities. We further discuss current challenges and outline emerging opportunities of this vibrant field for various applications in photonic integrated circuits, biomedical sensing, artificial intelligence and beyond. Recent years have witnessed substantial potential in allying meta-optics with diverse waveguide platforms to enable exotic manipulation of guided light signals. This review cataloged recent advances on meta-waveguides for photonic integration.


Introduction
Efficient manipulation of guided electromagnetic waves is of vital significance in numerous applications in nanophotonics and integrated optics. Distinctive from its electronic counterparts, photonic integrated circuits deploy guided light waves instead of electrical signals to carry information 1,2 . As the bandwidth and power density limit of data transport in electrical wires are increasingly manifesting with higher integration density 3 , integrated optical scenarios have shown promising inroads towards ultrafast and broadband information processing with low power consumption that may potentially circumvent current electrical bottleneck [4][5][6] . Various applications have been reported in this field such as chip-scale optical signal processing, communications, and analog computing [7][8][9] , as well as emerging technologies in quantum, biomedicine, and sensing 10,11 .
However, recent advancements of photonic integrated circuits are hindered by limitations in its most fundamental building block of traditional optical waveguides, in terms of restrained accessible functionalities, compromised efficiency and bulk footprint [12][13][14] . The very limited design library of conventional waveguide structures substantially constraints their function-alities 12,15 to mostly mere waveguiding. For instance, dielectric waveguides are ubiquitously applied to control on-chip propagating modes, while plasmonic waveguides play an essential role in guiding surface waves. Optical fibers are ideal for long-distance information communications and optical connections between chips and boards. However, the continuous trend towards miniaturized and versatile photonic systems demands more complicated device functions to be realized in a compact, multifunctional, configurable and CMOS-compatible way 1 .
Fulfillment of these tasks will entail novel waveguide structures. Fortunately, recent advent of metasurfaces and metamaterials opens a new pathway towards powerful light manipulation by engineering photonic structures in subwavelength scale 14,15 . Transferring the concept of meta-optics into guided waves can help overcome the abovementioned challenges 12 , by infusing new degrees of freedom into waveguide landscapes to dramatically boost device performance and enable novel functionalities [12][13][14][15][16] .
Optical metasurfaces and metamaterials are generally composed of judiciously designed artificial structures with feature size much smaller than light wavelength 16,17 . Metamaterials constructed by three-dimensional bulk subwavelength architectures can realize spatially changing exotic optical parameters (including the permittivity and permeability), giving rise to the transformation optics for invisibility cloaks and slow light phenomena [18][19][20] . Metasurfaces, in contrast, applying two-dimensional arrays of scatterers [21][22][23] , also exhibit unprecedented flexibility in controlling the fundamental attributes of electromagnetic waves, such as the amplitude, phase, polarization, wavefront, and so on. Fruitful applications are also reported such as metalens [24][25][26] , efficient holograms 27,28 , functional coatings 29 , color display 30,31 , LiDAR 32 and nonlinear optics 33,34 . Previous research attentions are mainly devoted to free-space applications. However, recent years have seen a tremendous interest in synergizing meta-optics with various optical waveguides to largely empower conventional photonic devices [12][13][14] . The advancement of subwavelength meta-structured waveguides can not only extend meta-optics physics to the realm of guided electromagnetic waves, but also promise to reshape the landscapes of photonic integrated circuits and massive emergent applications such as lab-on-chip technologies and neuromorphic photonics 8,12,13,59,60 .

Meta-waveguide fundamentals and properties
As is illustrated in Fig. 1a, b, meta-waveguides can be classified via either design methods or underpinning waveguide platforms 12,13 : • On the one hand, meta-waveguides can be conceived by physical intuition-based approaches (namely forward design henceforth) by leveraging the toolbox of metasurfaces and metamaterials with waveguide optics (discussed in Sections "Dielectric waveguide-integrated meta-structures", "Optical meta-fibers", and "Plasmonic meta-devices for controlling surface waves").
In the following, we start by briefly introducing the fundamentals and unique features of subwavelength meta-structured waveguides. General design process and device operation scenarios are concisely covered. Next, we comprehensively review latest researches derived from three different types of forward-designed meta-waveguides in separate sections, according to three distinctive underpinning waveguide platforms: dielectric waveguides, optical fibers and plasmonic waveguides (shown as Fig. 1b). Then we catalog inverse-designed metamaterial waveguides, focusing on waveguide-based design tutorials and algorithms, key applications, and comparisons of different design approaches for tailoring guided wave. Finally, we discuss current challenges and outline exciting opportunities of this vibrant field for integrated photonics and beyond.

Definitions, features and properties
Meta-waveguides here refer to a set of physical structures with engineered subwavelength features that guide electromagnetic waves. A simple case is the waveguiding media itself is man-made metamaterials [129][130][131][132][133] consisting of intricate subwavelength building blocks. Leveraging the mathematical technique called transformation optics 20 , researchers can design waveguides with artificial refractive index distribution to realize exotic waveguiding phenomena such as theoretical arbitrary waveguide bend and light trapping [129][130][131][132][133][134][135] . However, in addition to the loss issues, these scenarios encounter experimental challenges in fabricating sophisticated bulk structures at high optical frequencies 17,20 , prohibiting its widespread implementation.
waveguiding mechanism. As is illustrated in Fig. 2b 12 , structures of subwavelength dimension are crucial for meta-waveguides, because in this case the guided electromagnetic waves are experiencing an effective media with designed optical responses 18,19 . This mechanism is physically distinctive from photonic crystals (PhC) that rely on photonic bandgap 140 . Waveguiding in PhC is realized by introducing a line defect supporting guided modes against the photonic bandgap formed by the Bragg-like diffractions from periodic refractive index variations 140 . Taking subwavelength grating waveguides consisting of periodic silicon segments as an instance [12][13][14] , effective medium theory should be applied. Floquet-Bloch modes are supported without propagation loss 12 if the period Λ is much smaller than light wavelength λ. As Λ gets bigger, light starts to decay as photons are forbidden from propagating within the photonic bandgap. Then the structure will behave as a diffraction grating when the structure period steps into radiation regime 12 .

General design directions
Ramping up from scattered explorations by applying subwavelength structures 141,142 and nanoparticles on waveguides 143,144 , meta-waveguides can now be systematically investigated by either forward or inverse-design methods.
As shown in the lower panel of Fig. 2c function is firstly specified in order to determine objective functions. Then computer optimizations are conducted to retrieve device structure under given constraints, such as gradient-based iterations with adjoint methods 15,147 and deep learning algorithms 148 . Comprehensive design tutorials are detailed in later Section "Inverse-designed metamaterial waveguides". Forward-designed meta-waveguides have explicit physical pictures and are relatively straightforward to design with excellent performance. In contrast, inverse-designed metamaterial waveguides are computationally heavy in the implicit optimization process but can realize some sophisticated functionalities that may be hardly assessable by forward design 57,60 .

General application scenarios
Considering all linear optical devices are mode converters 149 , meta-device functionalities can be ascribed to two general scenarios: linear mode conversion and nonlinear hybrid applications.
In conventional waveguides devoid of structural perturbations, conversion between two arbitrary electromagnetic modes is generally inaccessible due to wavevector mismatch 83,149 . However, in meta-waveguides, the meta-structures can provide an effective momentum k eff to enable desired mode conversion 51,82 . The schematic is sketched as Fig. 2d, where Mode A and B can be free-space light beams, propagating waveguide modes, or surface waves. Lightwave propagating in a meta-waveguide undergoes consecutive and subwavelength-scale delicate field modulations (induced by resonance or form birefringence 35,[150][151][152] ) from the engineered structures, leading to an overall wavevector change of the electromagnetic modes 51 . Besides the collective scattering events from the structural perturbations, the mode conversion can be also interpreted as the contribution of multi-path interference from these subwavelength-dimensioned features 153 .
For nonlinear applications, phase-matching is pervasively required owning to energy and momentum conservations. Pronounced nonlinear phenomena usually demand rigorous phase-matched conditions. However, this requirement can be cir-cumvented by integrating meta-structures to waveguides of nonlinear materials to break mode conversion symmetry 52,94,154,155 . Considering the pumping optical mode as TE 00 (ω 1 ) as an example, the generated nonlinear signal TE 00 (ω 2 ) are then converted to other modes with same frequency ω 2 but different mode orders: TE mn (ω 2 ) and TE pq (ω 2 ), which will be preserved and accumulated with propagation 52,154 . Back coupling from the retained nonlinear modes to TE 00 (ω 2 ) is prohibited due to phase mismatch 52 and minimal field overlap, as the effective wavevector k eff provided by the meta-structures is unidirectional 51,82 . In addition, the nonlinear process can also be enhanced by metastructures with optimized nonlinear overlap 95 .

Dielectric waveguide-integrated meta-structures
Enabled by the ever-increasing resolution offered by nanofabrication technologies, dielectric meta-waveguides open exciting venues towards versatile chip-integrated applications. This section mainly includes dielectric waveguides with subwavelength surface structures and forward-designed metamaterial waveguides, focusing physical model, design method and applications.

Tuning scattering properties of nanoantennas on a waveguide
Analogy to well-established microwave and radiofrequency antennas, optical antennas are essential building blocks for manipulating light radiation at subwavelength scale [21][22][23]156 . Before delving into the details of metasurface-patterned dielec-tric waveguides, we begin with easy interference model of plasmonic optical antennas 40,[48][49][50]94 and discuss their rational implementations on waveguides for directional coupling applications and so on.
Starting from the most fundamental model, the response of an optical antenna with dimension much smaller than light wavelength can be approximated by an electric dipole P a ¼ A exp i k Á r þ α a ð Þ ½ under external electromagnetic stimulus 37,40,48,49 , where A denotes radiation amplitude, k j j ¼ 2π=λ is wavevector, λ stands for light wavelength and r = (x, y, z) is a spatial vector unit 157 . Acting as a resonator and scatterer, the optical antenna will have a characteristic phase α a depending on its shape, material, environment index, and light wavelength 21,156 . As is illustrated in Fig. 3a, by applying another antenna dipole , directional emission can be achieved 48 by properly controlling the antennas displacement and initial phase responses α a and α b . For instance, if we have destructive interference (α a + α p ) − α b = π in the right side (α p is the propagation phase determined by antenna displacement) 40 , directional emission to the left side is realized.
Leveraging this intuitive mechanism, antenna combos can be designed on top of dielectric waveguides to realize directional coupling 40,94 . Alternatively, some antenna elements can also appear as aperture, which can be investigated similarly using Babinet's principle 48 . For example in Fig. 3b, the combination of a gold nanodisk and air nanoslit forms a Fano antenna 49 , which is featured by the asymmetric Fano line-shape originating from the overlap of a broad and narrow resonance from the nanodisk and nanoslit respectively. By saddling this Fano antenna (with opposite bidirectional scattering attributes on two different wavelengths) on rectangular 48 or planar waveguides 49 , wavelength-selective demultiplexers are realized. In addition to the coherent interference from double-element antennas, directional emission can also take place in a single antenna, when the radiation from different internal plasmonic modes interferes destructively in the other side (Fig. 3c) for directional couplers 158 .
A more general antenna array configuration for high directivity is Yagi-Uda antennas, which are inspired by its radiofrequency counterparts consisting of metal rods acting as feed, reflector and directors 156,159,160 . Endeavors are pursuit for implementing Yagi-Uda antennas to waveguides 37,40,50,[160][161][162] . Waveguide-hybridized gold Yagi-Uda antennas can not only directionally in-couple free-space light source to guided waves, but also outcouple waveguide modes to free space and localized plasmons 37 , with further applications in on-chip data communications, directional lasers, and metrology [160][161][162][163] .
Moreover, the scattering attributes of the antenna elements are also polarization-controllable. For instance, the two gold nanorods horizontally resting on a silicon waveguide along the x axis (left panel of Fig. 3d) 50 directionally couple linear x-polarization E x to rightpropagating TM mode. However, this array barely interacts with y-polarized light E y , as the electrical current in the slender nanorod antenna cannot be efficiently excited when the incident electric field vector is perpendicular to its orientation 21 . In contrast, the structure shown in the middle panel of Fig. 3d can couple incident E y into left propagating TE mode with high directionality. By combining the two antenna arrays with opposite coupling direction and distinctive response to incident polarizations, chip-integrated polarization demultiplexers for high-speed optical communications are realized 50 .
Plasmonic chains are also incorporated to various dielectric waveguides to explore photon-plasmon interactions, coherent perfect absorptions 164 , coupling 165 , slow light phenomena 166 , all-optical switches 167 and lab-on-achip applications [168][169][170] , by engineering the interplay x   between the localized surface plasmon modes and propagating waveguide modes 144,165,166,171,172 . Furthermore, this waveguide-hybrid platform is particularly promising for largely miniaturized spectroscopic and sensing applications [168][169][170] . Taking the device in Fig. 3e as an instance 168 , a subtle change in surrounding index can be captured by the highly enhanced field in the narrow gap between the gold antennas, resulting in a shift in plasmonic resonance harvested by the waveguide. Compared with conventional bulky free-space optical setups, plasmon resonance can be conveniently measured in this antennas-loaded dielectric waveguides for surfaceenhanced infrared absorption spectroscopy, with high coupling efficiency over 70% and compact footprint 168 around 2 µm 2 .

Phase-gradient nanoantennas array on waveguide
Besides elementary interference model, next we discuss how to apply metasurface toolbox to design dielectric meta-waveguides for more sophisticated device functions with larger antenna arrays.

Guided modes to free-space light applications
Metasurface-decorated dielectric waveguides can efficiently out-couple waveguide signals to complex freespace light fields 79,173 . A guided wave-driven metasurface is schematically shown as the upper panel of Fig. 3f. The total phase shift of the extracted wave from waveguide-fed metasurfaces consists of two parts: (i) the abrupt and spatial-varying phase shift Δϕ(x) provided by each metaunit 21 at coordinate x, and (ii) phase βx accumulated from the propagation of guided waves, where β is the propagation constant 35 . Therefore, phase profile of the extracted wave along x direction can be formulated as below.
By judiciously designing the meta-units to tailor Δϕ(x), diverse applications can be realized, such as LiDAR 32,174 , optical communications and display [79][80][81] . For instance, using the metal-dielectric-metal sandwiched antennas (for approximate 2π phase shift range) to from a phasegradient metasurface atop of silicon waveguide, off-chip beam deflection and focusing can be achieved 35 . By arranging this metasurfaces on an active InGaAsP/InP microring waveguide to break the degeneracy of clockwise-and counterclockwise-propagating whispering gallery modes (as lower panels in Fig. 3f), a photonic integrated orbital angular momentum (OAM) laser is experimentally demonstrated 35 . In addition to waveguide top surfaces, subwavelength antennas or gratings-like structures can be implemented on waveguide sidewalls as well for structured light and OAM generations 46,175,176 .

Guided-mode conversions
As already discussed in the 'General application scenarios' section, the meta-structures resting on waveguide can bridge the wavevector mismatch between different modes to enable integrated mode convertors 51 . As is illustrated in Fig. 3g, when guided wave propagates against the unidirectional phase gradient k eff ¼ dΦ=dx offered by the collective scattering effect from gradient metasurface, the total internal reflection angle decreases, which corresponds to conversion from low-order to highorder waveguide modes. In contrast, when light propagates along k eff , its wavevector k mode ¼ n eff Á 2π=λ picks up k eff , leading to coupling from high-order to low-order modes 51 (n eff denotes effective mode index and λ is vacuum light wavelength).
If fundamental modes propagating along k eff are injected, it will be coupled into surface waves and get absorbed by the plasmonic metasurfaces. However, light wave with opposite propagating direction can get passed with much lower loss while converting to high-order modes. This asymmetric power transfer facilitates broadband reciprocal optical diodes 82 . Optical neural networks can be also explored utilizing tunable gradient metasurface-based waveguide mode convertors. As shown in Fig. 3h, using Ge 2 Sb 2 Te 5 (GST) phase-change materials as programmable metasurface waveguide mode convertor, the conversion of two waveguide spatial modes (TE 0 and TE 1 modes) can be precisely controlled with 64 distinguishable levels to encode the weight parameters in matrix-vector multiplication computation 177 . A prototypical optical convolutional neural network with 2 × 2 array of the GST convertors is experimentally demonstrated to perform image processing and recognition between handwriting digits "1" and "2". The experimental recognition results are shown in the right panel of Fig. 3h 177 . Besides, invisibility cloaks and chip-integrated spectrometers can be also envisaged in similar platform 67,93 .
Aperture antennas: The waveguide-integrated plasmonic metasurfaces discussed above inherit Ohmic loss from metals 150 . Next, we discuss meta-waveguides with dielectric subwavelength architectures, which have lower optical loss and better CMOS-compatibility 12,18 . Specifically, the antennas can appear as air apertures defined on waveguides. Either fully-or partially etched aperture antennas can be designed on dielectric waveguides to facilitate mode conversion [83][84][85][86][87][88][89] and efficient coupling applications 38,39 using effective medium or coupled mode theory 83 . An exemplary device structure is shown in the upper panel of Fig. 3i, where subwavelength features are partially etched atop of a rectangular silicon waveguide 84 to create periodic index variations Δε(x, y, z) along the propagation direction and a graded index profile along the transverse direction. The periodic subwavelength structures along the propagation direction provide an additional momentum k eff to enable phase-matched coupling to the desired output mode 83 , while the aperiodic transverse graded index profile enhances coupling strength by optimizing spatial modal overlap 83 . The lower panel of Fig. 3i illustrates the desired refractive index profile offered by the meta-waveguide for converting input TE 00 mode to TE 10 mode with high mode purity around 95% and high transmission of 88% 84 .
Besides shallowly etched structures [83][84][85]88,89 , dielectric waveguides with fully etched subwavelength apertures 86,87,178 can also enable efficient mode conversions with simplified fabrication process. Figure 3j shows the SEM image of a fully etched meta-waveguide capable of converting TE 00 mode to TE 10 mode within an ultrashort length about 2.42 µm around λ = 1.55 µm 87 . Integrated multifunctional mode convertors can be also conceived by applying complex surface meta-structures that are also aperiodic in waveguide longitudinal direction 89,178,179 . For instance, the multimode convertor shown in Fig. 3k for simultaneously converting 3 different modes with low insertion loss and acceptable crosstalk 89 . Furthermore, polarization convertors 85 and high-efficiency optical couplers ( Fig. 3l) 38,39 can be also designed using spatially penetrated waveguides.

Nonlinear mode conversions
Metasurfaces can also interface nonlinear mode conversions 180,181 when incorporating on nonlinear waveguides, as already discussed in Section "General application scenarios". Figure 3m shows the integrated lithium niobate (LN) waveguide patterned with dielectric gradient metasurface for phase-matching-free second harmonic (SH) generations 52 . As is shown in Fig. 3m, the generated SH signals TE 00 (2ω) from the pump TE 00 (ω) is coupled to high-order SH modes TE mn (2ω) and TM mn (2ω). They are then preserved and strengthened during propagation, as the inverse conversion from high-order SH modes to TE 00 (2ω) is prohibited due to phase mismatch 52,154 .
Simultaneous SH generation and radiation are also reported using grating metasurface-patterned LN slab waveguides 155 . By encoding the desired phase and amplitude information to the grating metasurfaces under phase-matching condition (Fig. 3n), the wavefront of the generated SH signals can be efficiently controlled for nonlinear beam-shaping functions like dual focusing and Airy beam generation 155 . Other nonlinear metawaveguides 94,95,182 also judiciously optimize nonlinear spatial overlap to significantly enhance nonlinear coupling efficiency 95,183 .

Jones matrix model for waveguide-integrated metasurfaces
Next, we discuss geometric metasurface-patterned dielectric waveguides and then introduce a more general Jones matrix model for waveguides-integrated metasurfaces.

Geometric phase metasurface-on-waveguide
Geometric metasurface or Pancharatnam-Berry phase (PB) metasurface 152,184 utilizes optical antennas with identical geometry but spatially varying rotation angle θ to tailor wavefront of circular polarizations. Meta-units with angular rotation distribution θ(x, y) can locally encode a dispersionless phase profile φ − (x, y) to one certain circular polarization |σ − 〉 and flip its handedness.
The PB phase shift φ − can be traced from the paths of polarization change on the Poincaré sphere (polarization state space) 185 . A PB phase metasurface acting like a halfwave plate will convert incident left-handed circular polarizations (LCP) σ À j i to right-handed circular polarization (RCP) σ þ j i with designer phase modulations 152 : σ À j i ! e i2θ x;y ð Þ σ þ j i. In the meantime, the same structure will impart a conjugate phase profile φ + (x, y) = −2θ(x, y) to the orthogonal circular polarization σ þ j i with polarization conversion: σ þ j i ! e Ài2θ x;y ð Þ σ À j i. This attribute has been exploited to develop free-space applications such as dual-polarity metalens and holograms 186 .
To transfer geometric metasurface to integrated optics, gold and silicon antennas are patterned on silicon waveguides to realize integrated polarization sorters 41 . The gradually increased antenna rotation angle (with 30°step) creates opposite phase gradient between LCP and RCP light, leading to directional coupling of orthogonal circular polarizations to opposite directions 41 . Figure 3o shows the spin-and wavelength-selective demultiplexers 42 , with simulated coupling efficiency above 50% using silicon antennas-patterned silicon nitride waveguide.

Free-space Jones matrix model for metasurface
Despite the simple relationship between imparted phase φ and antenna rotation angle θ for convenient design, meta-waveguides discussed above inherit limitations from geometric metasurface. First, the design methods are only applicable for circular polarizations. Second, the optical fields that coupled into the waveguide are hybrid modes 41,42 , which impedes high-speed optical communications applications due to inter-mode dispersion. Another important category of metasurface exploring propagation phase, where antenna orientations θ are fixed but the width l x and height l y of each antenna cell are individually engineered to tailor the eigen-phases φ x and φ y encoded to two orthogonal linear polarizations 152 : x j i ! e iφ x x j i and y j i ! e iφ y y j i. Propagation phase stems from form birefringence 43,151 , as the dielectric antennas acting as truncated small waveguides have different effective mode indices under different incident polarizations. This leads to different accumulated phase retardations when light passing through the antennas with different geometry. Jones matrix metasurface model combines both geometric phase and propagation phase 151,152 . A periodic metasurface resembling a birefringent waveplate can be described by a Jones matrix J (connecting two input and output polarization vectors as J λ j i ¼ κ j i with two operation scenarios, as illustrated in Fig. 3p: (i) Impart two independent and arbitrary phase profiles φ + (m, n) and φ − (m, n) to an arbitrary pair of orthogonal ellipticalpolarizations λ þ and λ À j i respectively 151,152 , where (m, n) denotes different antenna pixels 43 . The unitary and symmetric matrix nature of J guarantees the following two mappings simultaneously 152 , where superscript * represents complex conjugate. Here, the two independent phase profiles φ + (m, n) and φ − (m, n) can be arbitrarily assigned, but the output polarizations κ þ j i and κ À j i are fixed (as left panels of Fig. 3p): κ þ j i ¼ λ þ À Á Ã and κ À j i ¼ λ À ð Þ Ã j i, denoting preserved polarization ellipse but flipped handedness. The desired Jones matrix J(m, n) can thus be solved to determine antenna structure 151 .
(ii) Complete control over both the phase φ + (m, n) and output polarization κ þ j i of one certain incident polarization λ þ , while the encoded phase φ − (m, n) for its orthogonal polarization λ À j i is not configurable 151 (right panels of Fig. 3p).

Implementation on waveguide
Integrating the Jones matrix metasurface model on dielectric waveguides can propel integrated optics into new heights by offering a more general design method for versatile multiplexers for arbitrary polarizations and complete mode control over coupled lights. By assigning opposite phase gradients to two orthogonal incident polarizations λ þ and λ À j i under operation scenario (i), integrated polarization demultiplexers can be devised with high directivity over 20 dB for arbitrary incident elliptical polarizations 44 (lower panel of Fig. 3p). Furthermore, by optimizing antenna geometries at multiple wavelengths 43,44 , dispersion-managed wavelength demultiplexers or ultrabroadband directional couplers can be also conceived.
Notably, when transferring dielectric meta-units designed by Jones matrix method 151,152 to waveguides, spatial modal overlap between antenna scattering nearfield and target waveguide mode for excitation requires further attention 51 . For instance, to selectively excite TE m, n mode, m + 1 rows of antennas are required 44,187 . The mode index n is then controlled by properly engineered phase-gradient Δφ/d of the metasurface by combining generalized Snell's law with phase-matching condition 21 , where Δφ and d are the phase difference and distance between adjacent antennas respectively, n eff represents the effective index of the target waveguide mode to excite, k 0 is wavevector. As depicted in Fig. 3q, specific highorder mode of interest can be exclusively launched by the mode-configurable coupler with high purity 187 over 90% under operation scenario (ii). On-chip OAM generators with configurable topological charge is also proposed using mode mixing method 44,187 .

Holographic gratings and hybrid subwavelength surface structures
For chip-scale structured light generations, subwavelength holographic gratings are applied on top of silicon waveguides 45,77,78,188,189 to generate surfaceemitting vortex beams carrying OAM with specific topological charge 77 '. To design such device, the holographic grating can be retrieved by interfering the target OAM mode where A and B are amplitudes, θ is azimuthal angle and k mode denotes the propagation constant of the waveguide mode propagating along x direction. Figure 3r shows the proposed broadband OAM generator using two superposed subwavelength holographic forks atop of a silicon waveguide with a compact footprint 45 of 3.6 × 3.6 µm 2 , where G fork is converted to binary phase hologram considering current fabrication conditions. Similar subwavelength holographic surface architectures are leveraged to realize waveguide-integrated holography as well ( Fig. 3s) 80,81 . Besides, inspired from the integrated trench metalens and one-dimensional transmit-array 190 , on-chip wavefront shaping and mathematical optical signal transformations are demonstrated using standard silicon-oninsulator (SOI) platform with subwavelength metalens apertures 53,191,192 . Metamaterials are also saddled on waveguides for various integrated photonics applications such as optical modulation, coupling and sensing 193,194 .

Subwavelength grating waveguides
The meta-waveguides discussed in previous subsections are featured by subwavelength patterns on waveguide top surfaces, as summarized by Fig. 3. In the following two subsections, we focus forward-designed metamaterial waveguides (see Fig. 4), which are devised mainly by effective medium theory or transformation optics [16][17][18][19] .
Among them, subwavelength grating (SWG) waveguides exhibit promising potentiality to revolutionize conventional photonic integrated circuits 12 . Given that this topic is already reviewed in previous literatures [12][13][14] , here SWG waveguides as a specific member of dielectric meta-waveguides family are only very briefly mentioned to keep this review intact.
A typical structure for SWG waveguide is illustrated as Fig. 4a, where periodic silicon segments with subwavelength spacing Λ forms the waveguide core 12 . When is Λ much smaller than the half-wavelength of the guided light, localized Floquet-Bloch modes are supported without scattering loss in propagation 14 . Distinctive from Bragg and conventional diffraction gratings, the SWG structure instead behaving as a homogeneous uniaxial crystal 195 controlled by grating structures and duty cycle. 'Bridge' elements are commonly applied for the low-loss transition between SWG and homogeneous waveguide parts.

Other forward-designed metamaterial waveguides
Integrated photonics applications    56 . c Polarization mode convertor 196 . d Broadband polarization beam splitter 198 . e SWG ring resonator biosensor 90 . f Integrated fiber-to-chip coupling interfaces. Inset: details of the SWG waveguide coupler 203 . g Ultrabroadband chip-integrated all-silicon polarizer 205 . h, i Silicon metamaterial waveguides for exceptional coupling 55 and evanescent wave suppression 54 . j Left-handed metamaterial waveguiding heterostructures to slow light 131 . k Schematic of waveguide bend transformation 135 . l Integrated zero-index waveguide 216 . m Terahertz band-stop filer using a metamaterial slab waveguide 219 . n Metamaterial waveguide for qubits. Insets: SEM images for the capacitive qubit coupler (lower panel) and coupled microwave resonators respectively (upper panel) 225 198 ; ACS (l) 216 . telecommunication band 205 , using a 180°-bend silicon waveguide coupled with SWG metamaterial claddings. Moreover, SWG 'claddings' can be exploited to realize exception coupling and suppress evanescent wave as well (see Fig. 4h, i) 54,55 , for higher integration density and crosstalk mitigation. Optical beams focusing and transformations for guided waves signals are also reported 206,207 .

Exotic optical physics
Metamaterial waveguides are ideal platforms for exploring exotic physical phenomena to theoretically enable light slowing and trapping 131 , such as the axially varying waveguiding heterostructure with a metamaterial core in negative refractive index (Fig. 4j). Although the pace of photons may not be stopped in realistic metamaterial waveguides with appreciable bandwidth 132 , abundant optical physics for light-matter interactions can still be explored, with potential applications in chip-integrated optical signal processing and communications 133,208 .
Leveraging transformation optics (TO) 19,209 , integrated metamaterial waveguides can be conceived 129,135,[210][211][212][213] . Figure 4k illustrates the schematic for designing a lowcrosstalk multimode waveguide bend with feasible fabrication constraints using TO 135 , where darker color indicates higher index. Compared with conventional bending waveguides, significant inter-mode coupling mitigation (over 14 dB) in this TO waveguide is experimentally verified 135 . Waveguide tapers, connectors and beam expanders are also designed under TO theory 129,210,211 .
Moreover, metamaterial waveguides with exotic refractive indices are also an active field of research 214 . For instance, theoretical perfect waveguide bending can be realized in zero-index metamaterials 134,215 . Figure 4l illustrates the on-chip metamaterial waveguide based on standard SOI platform with a refractive index approaching zero 216 , where phase-matching free light propagation is directly observed. Other exotic optical phenomena are also explored at optical and lower frequencies, such as strong field enhancements in epsilon-near-zero materials 217,218 , magnified light absorption ( Fig. 4m) 219,220 , enhanced spin Hall effect and broadband mode conversions 130,[221][222][223] , to name a few.

Quantum applications
Recently, quantum supremacy is demonstrated using programmable superconducting processors for qubits 224 , where the computation time can be dramatically accelerated from over millenniums to minutes compared with classical processors under certain calculation scenarios. In the meantime, computational advantages using quantum photonics are also reported 6 . In contrast to conventional structures, metamaterial waveguides can enable new degrees of freedom in controlling qubits [225][226][227] . Figure 4n shows a superconducting metamaterial waveguides to tailor the transition lifetimes of qubits 225 , where metastable qubit states with ultralong lifetime (about 24-fold enhancement) and short-lived states are observed and selectively tuned.

Optical meta-fibers
Optical fibers are another well-established platform to guide propagating electromagnetic modes. Integrating meta-structures with fibers gives rise to optical metafibers as an important member of meta-waveguide family. Distinctive from dielectric nanophotonic waveguides for various on-chip applications, optical fibers are widely used for massive long-haul optical communications and versatile interconnects that interface chips to exterior systems 2,3 . Moreover, their flexibility, biocompatibility, and mechanical robustness have made optical fibers with small cross-section and extreme aspect ratio as excellent candidates for remote, in situ and in vivo applications 228,229 beyond photonic integrated circuits, such as biochemical sensors and endoscopic optical imaging 230 .
However, conventional optical fibers are limited by silica material properties (in terms of transparency window, cut-off frequency and nonlinearity) and the cylindrical waveguide geometry pervasive in fiber drawling process 229 . Compared with microfibers devoid of subwavelength features 230 , optical fibers employing subwavelength meta-structures allow for much powerful control over light attributes, including phase, polarization, amplitude, dispersion, and optical impedance 229 . A novel technological roadmap (lab-on-fiber) was thus launched to develop a novel class of all fiber devices and components, by judiciously integrating functional structures and materials onto optical fiber substrates at the micro-and nano-scale 105,[231][232][233] . This vison is expected to provide the foundational basis to enlarge the functionalities pertaining to optical fiber technology towards 'plug-and-play' platforms to be exploited in many strategic applications, ranging from optical processing and computing to environmental monitoring, life science, safety and security 75,234 .
In this section we discuss optical fiber 'meta-tips' (fiber facet-integrated meta-structures) and 'metafibers' (intra-fiber subwavelength features), showcasing representative applications and corresponding fiber fabrication technologies.

Optical fiber meta-tips
As highlighted in past review papers 22,229 , the judicious integration of metasurfaces to optical fibers could be a game-changing direction for next-generation fiber-optic devices and components to disruptively expand their conventional functionalities. As is illustrated in Fig. 5a, metasurfaces would, indeed, open unprecedented paths to manipulate light using optical fibers to significantly accelerate relevant developments such as active beam profilers, spatial light modulators and fiber-optic tweezers, just to name a few 231 . Moreover, translating flat optics onto unconventional substrates, as the case of optical fibers, would allow the creation of lab-on-fiber assisted platforms with extraordinary capabilities in biomedical imaging, scanning near-field optical microscopy, and single-molecule detection 103,[106][107][108][109][110][248][249][250][251][252][253] .
Following this intriguing suggestion, one of the important milestones as 'optical fiber meta-tip' was first demonstrated by Principe et al. 75 , with extraordinary light manipulation capability. The proof-of-concept was realized via focused-ion-beam (FIB) milling of rectangular nanoantennas arrays on the gold-coated termination of a single-mode optical fiber. The metasurface was designed to display a linear-phase profile to split an impinging light beam into an ordinary polarized component, and an anomalous extraordinary component (with different polarization state) for beam steering, as shown in Fig. 5b. By judiciously designing phase-gradient metasurface on fiber tips, the same group later reported the efficient excitation of plasmonic surface waves with unprecedented localization levels for developing all fiber optrodes with unparalleled sensing capability for life science applications ( Fig. 5c) 76 . A phase-gradient plasmonic metasurfacebased device was demonstrated to detect biomolecular interactions, with a sensitivity more than two orders of magnitude higher than that of a gradient-free counterpart.
The alluring potential of integrating meta-structures on optical fibers are soon recognized by researchers, followed by massive emerging optical fiber meta-components and meta-devices for various applications. As shown in c Upper panels: Metasurface-enhanced lab-on-fiber bio sensors. Lower panels: Gradient-free benchmark 76 . d SEM images of the fiber facet-integrated metamaterial dispersive gratings 107 . e Multiparameter nanoprobe using a plasmonic crystal cavity on fiber end-face 254 . f Metalens on photonic crystal (PhC) fiber facet: schematics and SEM images of the meta-structures 100 . g Photograph of the nano-optic endoscope using metalens 74 Fig. 5d, a high-quality metamaterial dispersive gratings is integrated on fiber facet with the quality factor exceeding 300 to efficiently tailor light dispersion 107 . Lab-on-fiber nanoprobes using in plane integrated distributed Bragg reflectors on the end-face of a single-mode optical fiber is also reported, enabling the excitation of two spatially separated high-Q resonance modes (Fig. 5e) 254 . Figure 5f shows the first focusing optical meta-tip made by directly patterning a gold meta-lens on the facet of a photonic crystal fiber 100 . Superfocusing is achieved at 1550 nm wavelength with maximum enhanced optical intensities reaching 234%. This concept could also have an enormous impact on next-generation fiber-optic imaging systems, especially in life science applications 231 , with exceptional spatial resolutions and miniaturization levels 255 . The reliable integration of metalenses could considerably overcome traditional difficulties associated with optical aberrations, alleviating also the well-known trade-off between the transverse spatial resolution and depth of focus, which significantly limits the scope of optical imaging in precision medicine. Optical fiber meta-tips coupled with optical coherence tomography (OCT) tools may provide the key asset for advanced nano-endoscopes, for in vivo imaging with spatial resolutions that are hardly possible with conventional platforms. Figure 5g shows a novel matalens-assisted endoscopic OCT platform that can be easily integrated in needles and catheters to achieve neardiffraction-limited imaging through negating nonchromatic aberrations 74 . The proposed tool was validated in case of endoscopic imaging of human lung specimens and sheep airways (shown in Fig. 5g inset), demonstrating superiority as compared to commercial OCT endoscopes featuring to improve and enlarge the clinical utility of OCT platforms.
The extraordinary capability of metasurfaces to control light at nanoscale would also open potential disruptive developments in optical signal processing applications. As illustrated in Fig. 5h, coherently controlled absorption in a fully fiberized and packaged switching meta-device was recently reported 96 . An optical fiber meta-tip platform was conceived demonstrating logical functions (XOR, NOT and AND) at wavelengths between 1530 and 1565 nm. The proposed metadevice has been successfully tested at up to 40 gigabits per second, opening new opportunities for 100 THz bandwidth single-photon operation with potential impact in quantum information networks 97,98 . Meanwhile, fiber facet-integrated metastructures are also exploited for structured light generations and beam shaping [256][257][258][259] . Figure 5i sketches the SEM images of the meta-facet fiber for generating OAM beam with topological charge ' ¼ þ1 257 .
These valuable examples highlight future potentials of lab-on-fiber technology, optical communications, and related optical fiber meta-tips to drive a new technological revolution in optical fiber technology, for many realistic applications with unrivaled advantages in terms of functionalities, miniaturization, power consumption and overall performances. Next development in this domain may also take advantage from the advances in integrating functional materials onto optical fibers with a full spatial control at nanoscale. It is now possible to conceive and create any arbitrary 2D and 3D micro and nanostructure on the facet of optical fibers, as demonstrated by the smallest micro-house never realized on the optical fiber termination using robotics nano-factory and origami techniques 260 (see Fig. 5j).

Metamaterial fibers
Innovations can be also made beyond fiber facets. Compared with optical meta-tips, metamaterial fibers provide a broader stage to deploy meta-structures to tailor the propagation and dispersion of fiber modes, yet they also demand more stringent quality control over the desired subwavelength structures along the entire fiber 229 . For meta-fibers with complex sidewall structures, Fig. 6a shows the SEM image of a microfiber Bragg gratings fabricated by FIB milling working around 1.55 µm communication wavelengths, with 576 nm pitch and 100 nmdepth grooves 236 . This fiber device with subwavelength features exhibit high transmission dip around 15 dB 236 and high sensitivity for refractive index and temperature 237,[261][262][263] . Simple nanostructures like a single gold sphere can be also applied to airlclad nanofibers to explore spin-orbit interaction of light and controlled directional coupling 238 (Fig. 6b). Intra-fiber metasurfaces are proposed for mode conversion 239 . Other sophisticated fiber sidewall textures may be envisaged by adapting emerging fiber fabrication technologies to optical fibers [264][265][266] .
Subwavelength structures that are uniform along fiber length direction can be manufactured by preform-based fiber drawing methods (detailed in next subsection) 267 276 and multiple cores 246,247,276,277 are reported. Figure 6c shows the cross-section of an anisotropic metamaterial fiber with radially distributed internal nanopores around the aluminum core supporting exotic modes 274 to strongly modify the waveguiding fiber attributes. For instance, hollow-core fibers with hyperbolic metamaterial claddings can guide light beyond cut-off frequency for enhanced light coupling 278 and low latency communications. Metamaterial cladding with various material combinations enables powerful dispersion control from ultraviolet 240 , infrared 241,242 to terahertz (THz) 243,278 and microwave frequencies 279 to circumvent the structure and material property limits of conventional silica optical fibers 280 . The meta-fiber shown in Fig. 6d exploits surface plasmon and classical fiber waveguiding 240 . Figure 6e sketches a single-mode, single-polarization hollow-core fiber with metal-dielectric hybrid metamaterial cladding for high-density integration of THz systems 243 .
Epsilon-near-zero (ENZ) materials such as indium tin oxide (ITO)-embedded hollow step index fiber is also proposed as Fig. 6f 245 , with strong field enhancement in the subwavelength ITO shell and potential applications in sensing, nonlinear optics and enhanced quantum emissions 281 . Figure 6g depicts the cross-section of a THz metamaterial fiber with hexagonally arranged subwavelength indium wires for sub-diffraction imaging 246 produced by fiber drawing of the preform. This metamaterial fiber inspired by hyperlenses with indefinite permittivity tensor can collect high spatial frequencies over optically long distances to enhance imaging resolution from THz 277 to infrared frequencies 282 .

Fabrication technologies
The waveguiding flexibility, high aspect ratio and microscopic cross-section of optical meta-fibers provide unique advantages, but simultaneously challenge conventional nanofabrication technologies developed for large planner substrates 228 . Nevertheless, the maturity of diverse fiber fabrication techniques has made optical fiber metatips and metamaterial fibers one step closer from lab demonstrations to potential market products. Figure 7a summarizes currently available approaches to fabricate meta-structured optical fibers 76,97,104,228,229,[283][284][285][286][287][288][289][290][291][292] . For optical meta-tips, ordinary photo-and electron beam lithography generally confront challenges in sample mounting and uniform resist coating for the small fiber facet 229,287 . Instead, nanotransfer 103,110,229,288,289 and FIB milling 75,76,96 can be applied to define exquisite subwavelength meta-patterns at fiber facets. Laser writing 290,293 and nanoimprinting 283,284,291,294 generally have comparatively lower resolution but higher yield. For metamaterial fibers, preform-based fiber drawing methods are most promising for mass-production. Other approaches such as FIB milling and nanoimprinting can produce subwavelength architectures at fiber sidewalls 236,237,266 . Different fabrication technologies for meta-structured fibers are briefly compared as the following.

Nano-transfer
Nanotransfer is a straightforward method that transports as-fabricated meta-structures to fiber facets to circumvent the difficulty of direct nanofabrication on small fiber tips, as shown in Fig. 7b. The transferred meta-structures inherit the high resolution from conventional nanofabrication methods. However, most transfer processes have relatively low throughput and defects can be induced during the transfer. Figure 7b1 illustrates a wet-transferred meta-tip working as a flexible passband filter 288 using sacrificial layer and water bath scooping. The metastructures can be also fabricated by 'nanoskiving' techniques in an inexpensive manner 229,295 for wet-transferred to fiber facets. For dry transfer, a 'decal transfer' technique using sacrificial films (Fig. 7b2) 296,297 , and a cost-effective template stripping method using ultraviolet (UV) lightcurable hybrid polymer (Fig. 7b3) 289 are proposed. Another 'stamping-like' method is sketched as Fig. 7b4 103 , where the plasmonic metasurface is mounted at fiber facet via epoxy 298 . Direct mechanical transfer is also reported 299 (Fig. 7b5). Figure 7b6 depicts a template transfer approach 300 to produce high quality and robust plasmonic structures for real-time biosensing 110,301 .   236 . b Air-clad silica nanofiber with a gold nanoparticle resting on its surface 238 . c SEM image of the metamaterial fiber with aluminum core and radially oriented nanopores 274 . d Cross-section schematic of a rolled-up Al 2 O 3 /Ag bilayers hollow fiber 240 . e Microscope image of the fiber facet with subwavelength indium wires uniformly embedded in Zeonex host around the hollow-core 243 . f Hollow-core ENZ fiber embedded with ITO nano-shell 245 . g Metamaterial fiber for subwavelength-resolved THz focusing and imaging 246 . Panels adapted from: OSA (a 236 , c 274 , e 243 ); AAAS (b) 238 ; ACS (d) 240 ; Springer Nature (f 245 , g 246 ).

Direct laser writing
Leveraging femtosecond laser pulses and two-photon polymerization 302   Nevertheless, fiber tips-integrated subwavelength metastructures can still be envisaged with the steady trend of resolution improvement. Fig. 7c1 shows the SEM image of the laser-written micro-spiral zone plate at fiber facet for OAM generation 290 .
Although currently available material library for twophoton polymerization is slightly restrained, this approach, more importantly, can fabricate threedimensional (3D) complex structures that are hardly accessible by conventional top-down lithography 292,305 . Polymerization only takes place in the vicinity of the laser focal spot and the location of the solidified voxels can thus be manipulated much freely 302 . Beside fiber facets, intrawaveguide sophisticated structures may be also defined by this technology 308,309 . Optomechanical 307 and bionic structures 303 can be fabricated as well. Figure 7c2 shows the colored SEM image of a compact triplet lens objective attached to an optical fiber 305 for miniaturized endoscopes and imaging applications.

Focused-ion beam milling
FIB milling is another technology to produce intricate 3D nanostructures with high fabrication resolution. Precise subwavelength features can be defined at fiber facets 75,76,96 and fiber sidewall 236,237,[261][262][263]266 . Figure 7d1 shows the SEM image of an optical fiber meta-tip 76 , where a thin gold layer is first deposited on fiber facet and the metasurfaces are then written by FIB milling. Besides, elegant fiber sidewall architectures are also demonstrated (see Fig. 7d2, d3) 261,262 . Albeit to the advantages in fabrication accuracy and 3D nanostructuring, FIB has a slow throughput that precludes mass production. The optical response of the engraved nanostructures may be influenced by the gallium ions doping during the process 229 .
Nanoimprinting Nanoimprint lithography is a high-throughput and costeffective technology that utilizes molds to replicate predefined stamp topographies 138 . This method can produce various nanopatterns on fiber facets 284,291,294,[310][311][312] and fiber sidewall in parallel 266 . Figure 7e1 illustrates a nanoimprint process to transfer subwavelength period metal grating from the mold to fiber facet 310 with a sub-15 nm feature using PDMS and SU-8 for chip-scale probing and testing, where U-grooves are optimized to settle the alignment issue 291 . However, for complex metasurface-on-fiber structures, the imprinting and alignment accuracy may still need amelioration. 3D nanostructures are also accessible 284,311 , such as the 3D beam splitter shown in Fig. 7e3 311 . Meanwhile, when combined with fiber drawling, nanoimprinting may also produce surface structures at fiber sidewall using roller molds (Fig. 7e4) 266 . Figure 7e5 depicts the fabricated patterns for resolution test. For its potential applications to optical fibers, the imprinted nanostructures have to stay close to the fiber core in order to evanescently interfere with the optical modes.

Electron beam and photolithography
Electron-beam lithography (EBL) is a time-consuming process but can produce high-quality subwavelength structures with high resolution. To apply EBL to define patterns on the tiny facets of optical fibers, modifications on apparatus are quired 313 , as conventional spin coating and exposure platforms are designed for big wafers. Figure  7f illustrate the exemplary procedures to pattern on facets using EBL (left panels), and the SEM image of the fabricated structures after ion-beam etching (right panel) 101 . 'Dip and vibration' coating technique 314 is proposed for better resist coating uniformity. Customized rotating chuck and exposure holders are also explored to obtain better lithographic patterns on fiber facets 313 .
Photolithography has higher yield when working for large substrates, yet it also encounters challenges on resist coating, sample mounting and alignment of the optical fiber 315 . Figure 7g shows an 'align-and-shine' photolithography process that can transfer microstructures from the mask fiber to target fibers 286 . Interference lithography is also applied to produce periodic nanopillars array on fiber facets 285 (see Fig. 7g1). This approach has good resolution with simple experimental setup, but it is not applicable for metasurface with aperiodic or arbitrarily designed antennas. Furthermore, photolithography on fiber sidewall is also explored (as Fig. 7g2) 316,317 .

Preform-based fiber drawing
Fiber drawing of specific preforms can produce various metamaterial fibers with designed cross-sectional patterns extending over long lengths [267][268][269]277 , showing a promising inroad to high volume production. As is illustrated in Fig. 7h, a macroscopic preform assembled by stacking indium wires is applied to hot furnace and thermally drawn to produce microscopic features in the thin fiber 246 . During the process, the feature dimensions from the original preform can be reduced by order of magnitude 268 . The preform can be fabricated by assembly, drilling and 3D printing 264,318,319 . Currently demonstrated metamaterial fibers made by single drawling are mainly applied for THz and mid-infrared frequencies due to fabrication resolution 246,247,282 . Figure 7i shows metamaterial fiber preform with tin/soda-lime wire arrays and an exemplary transition region after drawling 282 .
To further produce structured fibers with subwavelength features, iterative drawling can be applied 320,321 (see Figs. 7j, k1). After successive drawling, the diameter of the intra-fiber nanowires can be reduced from hundreds of microns to about 15 nm with good radial and axial uniformity 320 . As shown in Fig. 7k2, the drawn fibers can be further assembled into different metamaterial preforms with various spatial structure distributions for further thermal size reduction 321 . Despite these demonstrations are non-optical fibers, this technology may be further adapted to fabricate multifunctional optical metamaterial fibers by improving optical loss and structural quality.
Plasmonic meta-devices for controlling surface waves Distinctive from previous chapters discussing propagating modes in meta-structured dielectric waveguides and optical fibers, in this section we will elaborate on the excitation and manipulation of surface wave (SW) modes using plasmonic meta-structures.
As a comparison, the optical modes in conventional fibers and waveguides are electromagnetic (EM) guided modes based on total internal reflection. The size of these optical elements is thus still wavelength scale constrained by diffraction limit, which severely hinders the high demand on device miniaturization. In contrast, EM SWs, including surface plasmon polaritons (SPPs) and their low-frequency counterpart spoof SPPs, are also eigen EM modes but instead highly confined at the material interface, which can find numerous applications such as superresolution imaging 209 , enhanced light-matter interactions 322 , high-integration optical circuits [323][324][325] , bio-326 and chemical-sensing 327 , thanks to their deepsubwavelength and local-field enhancement characteristics. While natural SPPs only exist in optical frequency, the so-called spoof SPP modes are created in terahertz and microwave frequencies via patterning the highly conducting metal surfaces with subwavelength structures 328,329 . Recently, various meta-devices, including meta-coupler, meta-waveguide and circuits, are proposed to control SWs, providing us versatile possibilities for future on-chip optoelectronic applications 150 . Next, we will introduce the recent advances in SPP excitation and wavefront manipulations using these meta-devices.

Excitations of surface waves using plasmonic metawaveguide couplers
To utilize the novel properties of SW modes, the first step is to excite them efficiently. Unfortunately, because of momentum mismatch, the impinging free-space light cannot be directly coupled to SWs in general. To this end, various conventional optical devices were proposed, including prism coupler and grating coupler 330,331 . However, these elements are either too bulky and/or lowefficiency, which greatly hampers their further applications in integrated photonics 332 .
In 2012, Sun et al. proposed a new strategy to couple free-space propagating waves (PWs) to localized SWs based on a gradient metasurface in metal-insulator-metal (MIM) configuration 70 . As the meta-atoms are illuminated by EM waves, anti-parallel currents will be induced inside two metallic layers, creating a magnetic resonance. Via carefully tuning their local geometric parameters, such metasurface can provide a linearly changed reflection phase with its gradient denoted as ξ ¼ dφ=dx. While the reflection phase changes slowly (i.e., ξ < k 0 , and here k 0 denote the total wavevector of EM waves in vacuum), the metasurface will deflect the impinging wave to nonspecular direction. Nevertheless, as ξ is large than k 0 , the metasurface cannot construct a far-field EM wave with an equal-phase plane, thus achieving PW-SW conversion, as shown in Fig. 8a. Both of the anomalous reflection and SW conversion effects can be described by the following equation: where k r x represents the parallel wavevector of reflection beam and θ i is the incident angle of impinging waves. It implies that the gradient metasurface will introduce an additional wavevector to input EM waves, addressing the momentum mismatch issue between PWs and SWs. Considering that the generated SWs on such inhomogeneous metasurface are not eigen EM modes, the authors further constructed a SPP meta-coupler via connecting a mushroom structure with the metasurface that can guide out the "driven" SWs to eigen spoof SPPs. Such generic idea was soon realized based on other meta-structures working at various frequency regimes 115,116,333 . For instance, a polarization-dependent directional SPP metacoupler working at telecom wavelengths was experimentally demonstrated 115 . The building block consists of a gold nanopatch and a thick gold film that are separated by a 50 nm-thick glass spacer. Via carefully designing both widths of atop nanopatches, they constructed twodimensional gradient-phase metasurface that can couple input xor ypolarized PWs to eigen SPPs propagating along two orthogonal directions, as depicted in Fig. 8b.
Apart from the resonance-based metasurfaces, Pancharatnam-Berry (PB) phase metasurfaces are also exploited to achieve helicity-dependent directional SPP excitations. As shown in Fig. 8c, a transmissive metasurface is composed by the elongated aperture array with constantly varied orientation angles 113 . Except for Bragg reciprocal wavevector, such metasurface can also provide an additional geometric phase based wavevector for impinging light, achieving the helicity-dependent directional SPP excitations at near-infrared wavelengths. While the handedness of input CP light is flipped, the excitation direction of SPP will be switched to opposite direction. In a following work, a MIM typed PB meta-coupler is further demonstrated to couple SPPs with their amplitude and phase independently modulated by the polarization state and polarization angle of incoming light 114 . However, the early PB meta-couplers usually suffer from low-efficiency issue. To address such challenge, a new scheme is reported to solve two fundamental issues widely existing in previous studies 334 . First, via carefully optimizing the polarization conversion efficiency of the building block, the PB meta-coupler is able to convert impinging EM waves to driven SWs with nearly 100% efficiency. Second, the guided-out plasmonic metal should be carefully designed to support both TM and TE spoof SPP mode, solving the polarization mismatch issue between input CP wave and guided-out SPPs. A realistic meta-coupler working in the microwave regime is designed and fabricated, exhibiting a high spoof SPP excitation efficiency of over 80%.
While PB phase metasurfaces possess the advantages such as dispersionless phase response and easy design/ fabrication, they still encounter a strong restriction of spin-correlated functionalities. To overcome this limitation, Mueller et al. proposed an approach to design a single metasurface carrying two arbitrary and independent phase distributions for the input lights of two orthogonal polarization states (including linear, circular and elliptical ones). The combination of propagation phase and PB geometric phase enables us to delink two functionalities of the metadevices for two orthogonal polarizations 152 . Various multifunctional meta-structures leveraging Jones matrix toolbox are demonstrated for achieving either SPP excitations or non-specular beam deflections modulated by the spin states of incoming CP lights 118,335 . A new design scheme was soon proposed to simultaneously achieve high efficiency surface plasmon excitations and wavefront controls based on a single meta-coupler 119,120 . Except for designing a linear-phase profile along x direction, which is mainly responsible for surface plasmon excitations, they also added a specific phase profile along y direction to simultaneously modulate the wavefronts of generated SPPs. Via combining both resonance phase and PB phase together, such metacoupler can achieve either SPP focusing, or deflection effect shined by different spin-polarized EM waves (see Fig. 8d).
Except for the meta-couplers based on wavevector compensation scheme, narrow apertures are also widely utilized to launch SPP modes depending on diffraction effect 336 . In 2013, an aperture-based SPP meta-coupler was proposed 112 to realize polarization-modulated directional SPP coupling. For a single aperture, it approximately behaves like an in-plane dipole source that can emit SPP waves along both sides but with a phase difference of π. As illustrated in Fig. 8e, such meta-coupler can enable constructive or destructive interference of SPP on two opposite sides via carefully designing the individual orientation angles and spatial separation of the aperture pairs. Such aperture-based metasurfaces can not only launch directional SPPs, but also tailor their wavefront simultaneously 122,337,338 . For instance, Fig. 8f depicts a plasmonic meta-slit that can achieve SPP focusing with its direction and focal length flexibly switched by the helicity of input CP light 122 . The meta-devices are composed of single or double arrays of subwavelength nano-slits with their orientation angles satisfying the parabolic distribution. To construct more complicated near-field patterns, the double-lined metasurface was adopted to modulate both the local amplitude and phase of generated SPP via controlling the tilted angles of nanoslit pair. Benefiting from these two degrees of freedoms, the plasmonic SPP caustic curves, Airy beam, and complex holography (Fig. 8g) are successfully demonstrated 337 . a Physical principle of PW-SW conversion by phase-gradient metasurface 70 . b Polarization-dependent directional SPP excitation at telecom wavelengths 115 . c Helicity-dependent directional SPP excitations via Pancharatnam-Berry (PB) phase metasurfaces 113 . d Schematic of terahertz Metasurfaces for spin-delink spoof SPPs excitation combining the resonant and PB phase gradient. Inset: Partial picture of the fabricated meta-structure 119 . e Schematic of aperture-based SPP meta-coupler for polarization-modulated directional SPP coupling. Inset: image of multiple parallel column aperture pairs spaced λ SPP apart 112 . f A plasmonic meta-slit that can achieve flexible SPP focal length switching by the helicity of input circular polarized (CP) light 122 . g Ring-shape array of nano-slits for SPP hologram 337 . Panels adapted from: Springer Nature (a 70 , b 115 , c 113 ); Wiley (d 119 , g 337 ); AAAS (e) 112 ; OSA (f) 122 .

Manipulations of surface waves using plasmonic metawaveguides
In the previous section, we have discussed how to employ meta-couplers to efficiently convert free-space light to near-field SPPs. For the future integration photonic applications, versatile on-chip manipulation the SPP beams is highly desirable, including SPP guidance, wavefront tailoring, and far-field emission.

Surface wave waveguides and circuits
Massive plasmonic meta-structures can be applied to construct meta-waveguides for SW transports and controls in deep-subwavelength scale, such as V-shaped grooves, hybrid nanowires, and plasmonic tapers [339][340][341] . In low-frequency counterpart, spoof SPP waveguides were also developed based on various structured surfaces 328,342,343 . In particular, an ultrathin and flexible comb-shaped plasmonic surfaces is reported to transport the conformal surface plasmons (CSPs) at the microwave regime 344 . As shown in Fig. 9a, such CSP modes can be bent, folded and even twisted with very low propagation and absorption losses on such surfaces with various curved configurations.
Considerable efforts have also been devoted to SPP based integrated optical circuits. In 2011, a simple doublenanowire network was proposed to perform compact logic operations, via illuminating the two nanowire terminals by excitation lasers of specific polarizations and relative phases. Based on the plasmonic interference effect, such plasmonic network can direct guided SPP waves to corresponding output, exhibiting various functionalities of beam splitting, routing and switching 345 . Soon, a binary NOR gate was experimentally verified in a plasmonic network of four-terminal nanowires via cascading OR and NOT gates 346 Fig. 9 Manipulations of surface waves via meta-structures. a Spoof SPP transmission lines constructed by subwavelength comb-shaped corrugated metal strips 344 : Fabricated samples (first row) and simulated electric field distributions (second row). b All-optical nanoscale SPP logic gates: Schematics (left panel) 347 and representative SEM images (right panels). c SPP silver nanowire waveguide integrated with a single layer of MoS 2 for second harmonic generation. Inset: Fourier imaging microscopy collecting signals 348 . d Inhomogeneous nanohole array fabricated on the silver film for SPP Airy beam generation by non-perfectly matched diffraction 354 . e Hyperbolic metasurface for SPP PSHE in visible regime 124 . f Gradientphase meta-wall for spoof SPP anomalous reflection. Inset: simulated abnormal spoof SPP reflection results in x-y plane 73 . g Schematic of the transmissive metawall router for the transformation of 2D SPP to 1D SPP in waveguides. Inset: Simulated current density in plasmonic router 71 . h SPP hologram meta-grating for vortex beam generation 362 . i SPP decoupling via phase-gradient metasurface 127 . Panels adapted from: PNAS (a) 344  (including OR, NOT, XOR, and XNOR ones) based on plasmonic nanoslot waveguides, exhibiting high intensity contrast and lateral confinement (see Fig. 9b) 347 . The local-field enhancement effect in plasmonic devices sheds light on many on-chip photonic applications, such as tunable sources, nonlinear microscopy, alloptical switching, etc. [348][349][350][351] . For instance, Li et al. 348 experimentally demonstrated the second harmonic generation (SHG) in a plasmonic nanowire waveguide (Fig. 9c). The sample consists of a crystallized silver nanowire and a single layer of MoS 2 with large second order susceptibility. Shined by the excitation laser, the excited SPP mode will propagate and then be reflected back inside the nanowire. For these two counterpropagating SPPs, the momentum matching condition is automatically addressed, thus facilitating the nonlinear effect inside such system.

Wavefront control of surface waves
While some basic optical elements (e.g., dielectric triangular prism and cylindric lens) are initially adopted to control the propagation of SPP, they usually possess bulky size and limited functionalities 352 . Relying on Bragg scattering effect, the concept of plasmonic crystals was proposed to realize SPP beam reflection, splitting, interference and demultiplexing 347,353 . To construct more complicated near-field wavefronts, the non-perfectly matched diffraction effect is employed to generate SPP Airy beam and experimentally demonstrate its non-diffracting and selfhealing properties, as shown in Fig. 9d 354 . An inhomogeneous nanohole array is fabricated on the silver film to provide a nonlinear phase modulation for input SPP. In addition, various proposals such as transformation based devices 355,356 and holographic plates 357,358 , also showcase diverse opportunities for on-chip SPP manipulations.
The rapid development of plasmonic metawaveguides has also opened up a new direction for near-field manipulations. In 2013, Liu and Zhang proposed theoretically a plasmonic metasurface constituted by metallic subwavelength grating that can support SPP modes with non-trivial (including flat and hyperbolic) dispersion relations. Several alluring phenomena such as anomalous-diffraction, non-diffraction and negative refraction of SPPs were numerically demonstrated 123 . As illustrated in Fig. 9e, such concept of hyperbolic metasurfaces for surface plasmon controls was soon experimentally verified in visible regime 124 . Other than these bulky meta-deivces, Dong et al. proposed a new concept to design ultrathin gradient metawalls (i.e., near-field version of metasurfaces) to achieve the arbitrary wavefront tailoring of SPP, such as SPP anomalous reflection (Fig. 9f), nearfield focusing and Bessel beam generations 73,359 . Such meta-wall is constituted by ultrathin gradient-index microstructures and a metallic plate that can totally reflect incoming SPPs with specific reflection phases profile, leading to the versatile wavefront reshaping of SPP. This idea was soon further developed to achieve one-dimensional SPP line wave excitations with a transmissive metawall (see Fig. 9g) 71 . This find may be applied in modulating on-chip photonic networks.

Out-of-plane emission of surface waves
For numerous photonic applications, we also need to decouple near-field SWs to far-field PWs in a desired manner. While encountering some discontinuous interface (e.g., bump, hole, groove), SW can be partially scattered to free space but with the radiation intensity and direction hardly controlled 360 . Periodic apertures are utilized for achieving directional out-of-plane emission of SPP 361 . As shown in Fig. 9h, holographic images can be created by incorporating the computer-generated hologram technology with bragg meta-grating to decouple SPP to free-space 362 . Other complex wavefronts, like accelerating Airy beam, focused beam, vortex beam are also suitable for this method [363][364][365] . In addition, Fig. 9i shows H-shaped gradient-phase metasurfaces to guide the spoof SPP modes on metallic strips to directional radiative waves 127 . According to the generalized Snell's law, the emission direction can be flexibly modulated by the phase gradient of the metasurface. Further utilizing the dispersive nature of spoof SPP mode, a broadband frequency scanning antenna is created based on such gradient metasurface, exhibiting a continuous scanning range from 4.8°to 37.2°operating at 8.8-10.7 GHz 128 .

Inverse-designed metamaterial waveguides
Forward-designed metamaterial discussed in previous sections generally starts from a well-established library of structure templates, followed by customized tuning with a handful of specific parameters based on applications. Despite the clear physical guidelines offered by these intuition-based strategies, the device structures are usually not fully optimal and do not consider the massive design space of irregular structures. Furthermore, some sophisticated functionalities such as ultracompact and multifunctional devices are hard to design using forward design methods 15 .
To settle these challenges, various numerical algorithms have been developed that allow a computer to design and optimize freeform metamaterial waveguide structures with dramatically boosted degrees of design freedom to maximize device performance. These computer algorithm-based inverse-design strategies offer a potentially effective and automated method to 'design-by-specification' 15,147 , in a manner following computer aided design techniques that are well established in the integrated circuit design community.
Inverse-designed metamaterial waveguides can be classified into two categories: analog and digital 62,67-69 . To design a device, the design region is first specified and then discretized into M × N unit shapes, called "pixels". The "pattern" is thus considered as a matrix of the pixels with permittivity distribution of [ε] M×N , where each pixel (i, j) has a permittivity of ε i,j accordingly. Then, algorithms are employed to find an optimized permittivity distribution to fulfill figure-of-merit (FOM). As exemplified in the upper panel in Fig. 10a, analog metamaterial waveguides generally have relatively small pixels [366][367][368] (λ=10 ð Þ to meet the perturbation theory 369 . Thus, the permittivity ε i,j or its shape information can vary continuously to calculate the gradients information of the FOM (discretization is conducted later to produce binary devices) 15 . In contrast, pixel dimensions of a digital metamaterial waveguide are much larger (~λ/10) (lower panel in Fig. 10a) 68,370 , and the permittivity ε i,j is restricted to binary distribution, e.g. corresponding to the logical "1" or "0" state.
In this section, we present explicit design tutorials and highlight representative applications using inverse-deigned metamaterial waveguide platforms. Diverse design algorithms are outlined and compared in terms of its versatility and applicability, including gradient-based, non-gradientbased and deep learning-inspired optimization methods.

Analog metamaterial waveguides
Analog metamaterial waveguides using sufficiently small pixels may have tremendous degrees of design freedom. To manage and explore this high dimensional design space in a computationally tractable manner, two classes of gradientbased optimization methods are typically considered: density topology optimization (TO) and level-set method, as shown in upper panels of Fig. 11. These methods, based on the adjoint method, are computationally scalable to systems comprising thousands and even millions of pixels because the number of simulations required to calculate gradients is decoupled from the total number of pixels 147,367 , which vastly broadens the generality and computational efficiency of inverse design. Gradient-based adjoint methods were introduced into nanophotonics in the early 2000's by researchers from the mechanical engineering 371 and control theory 372 communities, where the adjoint method was established. Foundational results Initial studies in the density topology optimization of nanophotonic devices are most linear devices, where the electromagnetic field satisfies the Hermitian condition. These demonstrations include, e.g. photonic crystal (PhC) waveguide Z-bend 373 (Fig. 11a) and 90°-bend 374 (Fig. 11b), where the optimized structure shows a significantly lower loss for a wider range of frequencies  395 . v Fabry-Perot microresonator 396 . w Broadband chip-fiber grating coupler 394 .

Density topology optimization
x Non-reciprocal pulse router 397 . y Laser-driven particle accelerator 398,540 . z Vertical coupler 399  compared to forward designs. Density topology optimization has since been exploited to inversely design various waveguide devices with free-formed subwavelength features (Fig. 11c-f), including, for instance, high-performance 90°w aveguide bend 375 , polarization beam splitter 376 , threechannel mode multiplexer 377 , and nanolens with structural integrity 378 . More recently, density topology optimization was used to tailor guided-mode phenomena in freeform metasurfaces, leading to high efficiency [379][380][381] and multifunctional devices 382,383 . New diffraction phenomena are enabled as well by analyzing these complex mode scattering dynamics 384 .
In contrast to typical TO schemes, an "objective first method" was proposed to allow both electromagnetic filed and permittivity values to optimize individually 385 , which allows for non-zero physics residuals. Alternating directions method of multipliers (ADMM) algorithm were employed in the optimization process without using the adjoint method. A wider range of functional devices were investigated using the method 58,386,387 , including mode converter (Fig. 11g), hub (Fig. 11h), fiber coupler, optical diode, and polarization beam splitter, as well as multi-function device handling simultaneous focusing and wavelength demultiplexing (Fig. 11i). For nonlinear devices, the electromagnetic field no longer satisfies the Hermitian condition and adjustments to the adjoint method are required. A 2D microcavity was designed for nonlinear frequency conversion by solving a linear adjoint equation with time-dependent material 388 (Fig. 11j). Simplified nonlinear optimization was then explored based on a linear adjoint equation with timeinvariant material 366 , followed by devising a compact photonic switch in Kerr nonlinear material (Fig. 11k).

Level-set method
Design procedures (i) The boundary of the material subjected to φ(r) = 0 is allowed to vary continuously. During the optimization process, the boundary is time dependent and can be expressed as the zero solution of the level-set function .
The negative solution and positive solution of ψ(r, t) correspond to the regions filled with permittivity of two boundary values respectively. (ii) Then, the evolution of the boundary can be considered as a change of the levelset function with respect to t via gradients, as shown in the bottom right panel in Fig. 10b.
Foundational developments A shape derivative method was proposed based upon the adjoint variable method and perturbation theory in the early 2000's 389 , which enables the gradient descent algorithm based on shape derivative method. Level-set method is mathematically distinct from the shape derivative method, but essentially equivalent. Initial studies in the level-set method involved bandgap engineering of PhC devices 390 , as shown in Fig. 11l. Later research validated the feasibility of using this method to devise metamaterial waveguides. By parameterizing the boundary of the material with the level-set method, a high-performance silicon Y-junction splitter was designed in an ultra-compact area 367 (Fig. 11m). A general level-set method was proposed for subwavelength featured waveguides that directly incorporates fabrication constraints, especially for curvature constraints on device boundaries 391 . A 1 × 3 splitter, mode demultiplexer, directional coupler, wavelength demultiplexer, and cavity-waveguide coupler 391,392 were later reported (Fig. 11n-r). Besides, level-set method was used to design nonlinear waveguides, such as PhC fibers with large nonlinear frequencyconversion efficiencies (Fig. 11s) 393 .

Density topology optimization combined level-set method
Design procedures (i) First, an optimized analog structure with gray permittivity is obtained using TO method. (ii) Then, the structure is binarized employed level-set method. The level-set method could theoretically prevent intermediate gray structures. Figure 11t shows a wavelength demultiplexer with wide bandwidths designed by simultaneous TO and level-set method 368 . The method was also used to design general 1D grating couplers without any human input from start to finish (including a choice of initial condition) 394 , as depicted in Fig. 11w. Various functional integrated devices had also been demonstrated as Fig. 11u-z [395][396][397][398][399] , such as mode multiplexer and mode splitter, Fabry-Perot micro-resonator, non-reciprocal pulse router for chip-based LiDAR, laserdriven waveguide-integrated particle accelerator, and diamond photonic waveguides.

Fabrication constraints for gradient-based algorithms
Despite the benefits discussed above, such analog devices nevertheless encounter challenges in terms of fabrication constraints. The TO method results in intermediate "gray" structures, but only binary permittivity distribution can be efficiently implemented in experiments for most cases. A wide variety of proposals are reported to mitigate this issue of "gray" areas in structure, such as density filters 400 , sensitivity filters 401 , penalty functions 402 , artificial damping 403 , and temperature parameter trick 404 .
In level-set method, small features, such as narrow gaps or bridges, are usually inevitable. A typical approach is to exploit shape parameterizations that automatically satisfy the desired minimum feature constraint, known as a geometry projection method 405 .
Another approach is to evaluate a dilated and eroded version of the device during the optimization process with improved fabrication robustness and length scale constraint 406 . Curvature limit techniques are also useful for the removal of narrow gaps 391 . To enforce hard design constraints, methods based on reparameterization have been developed in which geometric parameters are initially defined in a continuous latent space and then mapped onto a physical device 407 . This mathematical mapping is specified such that devices that violate a given physical constraint, such as minimum feature size, are not even considered. This method has the advantage not only of enforcing hard constraints, but also of simplifying the design space and improving the reliability of the optimization process through elimination of non-compliant device patterns.

Digital metamaterial waveguides
Analog metamaterial waveguides utilize curvilinear device structures with high degrees of freedom but entail high fabrication resolution requirements. In contrast, digital patterns that utilize regular arrays of spatially coarse pixels can be designed and fabricated in an easier fashion. To date, most popular optimization methods for designing digital metamaterial waveguides fall into three classes: brute-force searching, heuristic, and adjoint methods, as shown in lower panels of Fig. 11.

Brute-force searching method
Design procedures Take the direct-binary search (DBS) algorithm as an instance. (i) Fist is reversing the logical state of a randomly chosen or specifically designed pixel and evaluating the FOM. (ii) If FOM is improved, the new pixel state is kept and the next pixel is reversed; otherwise, the original state is recalled (Fig. 10c). (iii) One iteration ends when all the pixel states are inspected, and the final pattern is set as the initial one of the next iteration.
Foundational developments DBS algorithm is a representative brute-force searching method introduced from digital hologram communities 408 Fig. 12 Artificial neural networks for nanophotonic inverse design. a-c Inverse design with discriminative ANNs. a A convolutional neural network (in conjunction with DBS algorithm) for designing all-dielectric metasurface 439 . b Tandem network (an inverse network connected to a pretrained forward modeling network) for designing multiple thin film stacks 440 . c Meta-waveguide inversely designed by residual deep neural network (ResNet) 487 . d-h Generative ANNs. d A conditional VAE network for designing analog power splitters with various splitting ratios 61 , where etch hole positions are the design space. e A GAN network for metasurface that supports tailored transmission spectral responses 448 , with three networks: the generator, the simulator, and the critic. f A conditional GAN network for designing meta-grating 449 . g A modified GAN network consisting of three principal features: progressive network growth, the self-attention mechanism, and progressive growth of the training set 450 . h A global topology optimization network (GLOnet) 452 . Optimization is done by coupling a generative network with direct electromagnetic solver. i Inverse design of metasurface hologram with reinforcement learning 457 . Initial structure used as the input double deep Q-learning network (DDQN) at each step. Panels adapted from: OSA (a) 439  metamaterial waveguides were invigorated using DBS algorithm, such as free-space-to-waveguide coupler 67 (Fig. 11a1), polarization beamsplitter 68 (Fig. 11b1), waveguide cloak 64 (Fig. 11c1), power divider 409 (Fig.  11d1), nanolens 410 (Fig. 11e1), and mode converter 411 (Fig. 11f1), optical diode 66 , 180°waveguide bend 65 , polarization rotator 63 , active optical switch 412 , polarization splitter-rotator 413 .
In Fig. 11g1, a photonic-crystal (PhC)-like metamaterial structure was proposed for 3 dB power divider that exploits partial-filling hole pixels to suppress the etch depth fluctuations caused by the reactive ion etching (RIE) lag 414 and thus helps to improve fabrication tolerances 370 . A 5 dB performance increment was achieved compared with a device of similar size optimized based on full-filling rectangle pixels. Various PhC-like waveguides were also reported, including mode multiplexer 415 (Fig. 11h1), on-chip routing 69 (Fig. 11i1), star crossing 416 , wavelength demultiplexer 417 , polarization rotator 418 , multimode bend 419 , dual-mode crossing 420 , and power splitter 421 . Furthermore, a rotatable DBS algorithm that adds a calculation of rotational dimension was proposed as Fig. 11j1 for simultaneous mode converting and power splitting 422 .

Heuristic method
Design procedures Take the genetic algorithm (GA) as an instance. (i) Generating initial populations as the first generation. Each population is a matrix of pixel distribution. (ii) Evaluating each population's FOM. (iii) Producing new populations with sequential application of selection, crossover, and mutation. Multiple strategies can be selected, e.g., roulette wheel or tournament selection. Each of the selected population has a specified or random probability to have a crossover with another population, where the crossover point is randomly picked (the left panel in Fig. 10d). After crossover, each population in the new generation has a probability of mutation, where a random number of pixels are selected to reverse (right panel of Fig. 10d). Iterations are thus formed by going back to Step (ii) to evaluate the new generation and population reproduction in Step (iii).
Foundational developments As DBS algorithm mostly converges to a local optimum, it is more suitable for optimizations in a small parameter space. With a large parameter space, heuristic method, such as genetic algorithm (GA) and particle swarming optimization (PSO), are more competitive. Fig. 11k1 sketched a GA-optimized SiO 2 /SiON telecom-fiber to ridge-waveguide coupler 423 , with a 2 dB enhancement in efficiency compared with direct coupling. As shown in Fig. 11l1-p1), GA was then extended to a wider range of technologically relevant applications, such as silicon planar resonator 424 , reflector 425 , long-pass filter 426 , wavelength router 427 , and light modulator 428 . More recently, GA method was combined with simulated-annealing algorithm to enable accurate phase control in freeform meta-waveguides, contributing to high efficiency devices for complicated mode conversion from guided modes to free space (e.g., orbital angular momentum emitter 47 shown in Fig. 11q1). On the other hand, the particle-swarm optimization algorithm also can be applied to design metamaterial waveguides (see Fig. 11r1, s1), such as 2 × 2 power splitter 429 and polarization beam splitter 430 . Particleswarm optimizations are based on the movements of a population of candidate solutions (particles) in the search space. During optimization, the initially distributed particles continue moving towards the then-current optimum particle in the swarm, until the FOM is reached 431 .

Adjoint method
Design procedures (i) Tuning the permittivities of all pixels with a fixed shape continuously and individually to obtain an optimized analog pattern with "gray" pixels using adjoint method. (ii) Then, a forced biasing approach was applied to convert the analog pattern to a "quasi-digital" one in which the permittivities of most pixels are close to two boundary values. (iii) Fabrication-constraint brute-force quantization methods were designed to transform the "quasi-digital" pattern into an "N-ary digital" pattern, as illustrated in Fig. 11t1.
Foundational developments Distinctive from the TO or lever-set method, gradient information is generally hard to retrieve for digital meta-waveguides. Without gradients to efficiently update searching direction, convergence of these optimization algorithms is often considerably slower. Fortunately, recent research validated the feasibility of adapting the adjoint method into the inverse design of digital meta-waveguides. As shown in Fig. 11u1, a 3 dB power divider and a dualmode multiplexer were demonstrated 62 , using PhC-like subwavelength structure with cylindrical-shaped base pixels in design region. At the implementation stage of "N-ary digital" pattern, cylinders with different "gray" permittivities in the "quasi-digital" pattern are replaced with air cylinders with original radius, silicon cylinders, or air cylinders with N − 2 different radii on the basis of effective medium theory to minimize the performance degradation due to the digitalization process. Compared with the conventional DBS method, the introduction of adjoint method can improve design efficiency by about five times, while the performance optimization can reach approximately the same level.

Deep learning enabled inverse design in metamaterial waveguides
Artificial neural networks (ANN) are algorithms that utilize a series of nonlinear mathematical functions, in the form of neurons, to specify highly nonlinear mappings between inputs and outputs 148 . A training process, in which weights within the neurons are iteratively adjusted, is used to tailor these mappings. ANNs have been a topic of study for the last half century and have gained immense popularity in nearly every technical field of research over the last decade due to the emergence of the deep network framework, user-friendly software packages, and application-specific hardware 432,433 . The deep ANN framework, where many layers of neurons are utilized in the algorithm, has led to particularly versatile and expressive algorithms that can serve as universal functional approximators. Software packages, ranging from PyTorch to TensorFlow, have led to standardized and easy-to-program interfaces to implementing ANNs with arbitrary architectures 432 . Specialized hardware includes graphical and tensor programming units that can dramatically speed up the network training process. Furthermore, the culture of opening and sharing in the machine-learning community leads to the availability of many state-of-the-art algorithm resources easy to access and follow up online 432,434 (https:// github.com/iguanaus/ScatterNet, http://github.com/ yuruiqu/Transfer-learning_Data, https://github.com/ PRGatech/DimensionalityReduction).
In this section, we give an overview of inverse-design methods based on discriminative networks, generative networks, and reinforcement learning. While we will present examples from the broader context of nanophotonics design (see Fig. 12), the concepts can readily apply to metawaveguide design without loss of generality 148,432,435 .

Inverse design with discriminative ANNs
Discriminative ANNs are algorithms that can capture data relationships of the form y = f(x) When x specifies a device pattern and y is its optical response, the discriminative ANN serves as a surrogate electromagnetic (EM) solver, solving the forward problem 432 . To train a discriminative ANN, a training set of known device patterns and optical responses is first created using a conventional EM solver. The ANN is then trained to minimize error between the outputted and known optical responses, for given device pattern inputs. Mathematically, the loss function is defined to be the mean square error between the outputted and ground truth optical responses, and this function is minimized during the training process by applying back-propagation algorithm 148 . While the training data collection and ANN training are computationally expensive, a trained ANN can perform computations orders of magnitude faster than a conventional solver. A challenge to training an accurate and generalized ANN is curating a sufficiently large and diverse training set that properly represents the desired computation space. A basic and typical strategy is to use a random, uniformly distributed set of devices 432 . If there are known, statistically rare features pertaining to the device geometry or optical response, these features can be learned by the ANN by augmenting the training set with a disproportionally large number of examples of these data.
Discriminative ANNs can be used for inverse design in multiple ways. One is that the surrogate solver can be used in conjunction with conventional optimization methods, which include evolutionary strategies 436 , particle-swarm optimization 437 , and brute-force sweeping and searching 438,439 . Many of these strategies typically require hundreds to thousands of simulations performed in series, and the utilization of an accurate high-speed solver can reduce total optimization times from hours or days to seconds, one example is illustrated in Fig. 12a 439 . Another way is to use the trained forward ANN to facilitate the training of an inverse network, which attempts to output a device pattern given a desired optical response input 440 (shown in Fig. 12b). The reason why the tandem network is trained, as oppose to direct training of an inverse network, is that the inverse problem is a one-tomany mapping problem: there exist multiple device patterns that can produce the same optical response. This type of mapping cannot be captured by a discriminative network, which can only capture one-to-one or many-toone relationships. The trained forward ANN reduces the one-to-many mapping issue because it is not an exact surrogate simulator and only approximates the complex, degenerate design space. Tandem networks have been used to design thin film stacks 440 , color filters 441 , and topological photonic structures 442 . To tackle the one-tomany problem, several solutions have also been developed, including conditional generative adversarial networks and conditional variational autoencoders, where a latent vector is introduced in the ANN to address potential multiple solutions 433 .
Discriminative ANNs can also perform inverse design using the backpropagation method. Previously, backpropagation was discussed in of network training, where gradients to neuron weights were calculated to minimize the loss function. To perform inverse design with a trained discriminative ANN using backpropagation, the loss function is defined as the difference between the output and desired optical responses and backpropagation is used to calculate gradients to the input device pattern x to minimize the loss function. This method has been used to inversely design scatterers 443 and photonic crystals 444 . The concept can also be applied to more complex computational graphs that combine ANNs with analytic physical expressions 445 . Furthermore, a neural-adjoint method that adds the boundary loss to the loss function is proposed to improve performance 446,447 .

Inverse design with generative ANNs
Deep generative networks [448][449][450] are architecturally similar to discriminative networks except that one of the inputs to the network is a latent random variable z. As such, the output to the network is a distribution, typically of device patterns, which can be generated by sampling z. There are two general inverse-design concepts based on generative networks. The first is to train a generative network to produce distributions that mimic a training set of device patterns. For networks where the input is the desired optical response and z, the generator functions as an inverse network: z is sampled to produce an ensemble of candidate devices, which are then evaluated and filtered to identify devices with suitable performance. For networks where the input is a device operating parameter (i.e., operating wavelength or material index) and z, the generative network can be used for interpolation and output devices with operating parameters not found in the training set.
There are different ways for generative networks to learn training set distributions. One way is with variational autoencoders (VAEs), which consist of two parts: an encoder that maps device patterns onto a low dimensional latent space distribution, and a decoder that maps latent space data representations back to device patterns. The loss function includes a reconstruction term, which minimizes differences between the inputted and outputted device patterns to the full network, and regularization terms that tailor the latent space distribution to fit a Gaussian profile. As shown Fig. 12d, VAEs have been used to design analog power splitters with various splitting ratios 61 , and digital multimode interference waveguides were designed by training a VAE in tandem with a pretrained forward network 451 . Another way to train generative networks is with the generative adversarial network (GAN) framework, where the generative network is trained together with a discriminative classification network that attempts to distinguish whether an input device is fake (i.e., from the generative network) or real (i.e., from the training set). Over the course of training, the discriminator gets better at distinguishing real and fake devices while the generator produces more realistic looking devices. Upon the completion of training, the generator fools a trained discriminator by outputting a distribution of devices mimicking the training set. GANs have been used to design freeform metasurfaces that support tailored spectral responses 448 and high efficiency meta-gratings 449,450 ( Fig. 12e-g).
A second concept for inverse design with generators is based on the dataless training of generative networks to perform a population-based search for the global optimum 452,453 . These networks are termed global topology optimization networks (GLOnets) and the network training procedure works as follows. Initially, the outputted distribution from the generative network spans the full design space such that sampling z produces a batch of random devices. These devices are evaluated by an EM solver to quantify their performance and performance gradients (i.e., pattern modifications that improve performance), and these performance metrics are used to calculate a special loss function. Back-propagation is used to modify the network based on this loss function, such that the device distribution outputted by the network narrows and gets biased towards higher performance regions of the design space. This process is repeated and the device distribution from the network output distribution eventually collapses, ideally around the global optimum. Initial GLOnet demonstrations were used to optimize metasurface patterns, and the final freeform devices had efficiencies that were consistently better than those designed using gradient-based optimization 452 (Fig. 12h). GLOnets were also demonstrated to be compatible with reparameterization to enable the global search of metasurfaces with hard minimum feature size constraints 407 . Subsequent improvements to the stability and performance of GLOnets have been made in thin film stack optimization, through judicious selection of the network architecture 453 . These concepts showcase the great potential of hybrid algorithms to interface physics and data sciences.

Inverse design with reinforcement learning
In reinforcement learning, a neural network learns to specify a sequence of actions within an environment in a manner that maximizes a cumulative reward [454][455][456] . These concepts have been popularized with their application to games, such as Atari games or Go, and have had significant technological impact in fields ranging from robotics to communications 457 . By training a deep network to play the game many times, initially with random actions (i.e., exploration) and later with actions informed by past experiences, the network will learn to specify an optimal action that maximizes current and future rewards. The consideration of cumulative rewards during the training process allows the algorithm to balance tradeoffs between long-term and short-term reward gains 432 .
The framework of reinforcement learning naturally maps onto the iterative optimization process for nanophotonic devices. In this context, the state is the device pattern at a given iteration, the action is a modification to the pattern, and the reward combines the present and future device performance as quantified by the FOM. As the reinforcement learning algorithm learns from many trajectories through a design landscape and is based on maximizing cumulative reward, the device pattern modification suggested by the algorithm in a given optimization iteration is not simply the modification that produces the largest improvement to the FOM, which is the case in gradient-based optimization. Rather, the algorithm will suggest trajectories through a design landscape that can sidestep local optima in an attempt to maximize the performance of the final device. It is noted that the training process for reinforcement learning is computationally expensive, and the simulation of devices over many trajectories through a design landscape can take days on a conventional CPU, even for relatively basic problems. These algorithms are therefore more practically implemented when paired with accurate surrogate simulators. Reinforcement learning has been used to design periodic dielectric metasurfaces that support tailored spectral responses 454 , multi-layer structures 455 , perfect absorbers 456 , and metasurface holograms 457 (Fig. 12i). Table 1 summaries several representative metamaterial waveguides (usually with wavelength-scale dimensions) to briefly compare the inverse-designed methods discussed above. As the landscape of a metamaterial waveguide is generally non-convex, currently it is still not mathematically possible to guarantee global optimal solutions in inverse-design optimization problems 15,58 . Despite that certain global search or reinforcement learning algorithms 47,432,452 attempt to maximize the performance of the final device, but global optimum still not surely undertaken. Some active research area have been developed to explore the computational bounds on the true optimal value of an given optimization problem, which essentially derive from Lagrange duality 458 , local power conservation 459 or diagonal physics dual 460 based on the basic properties of the constraints and objective functions. Such bounds not only help to provide guidelines to the maximal device performance possible within physical limits, but also can be used to rule out constraints and objective functions for which no device pattern can achieve such good objective value.

Comparisons of different design methods
Gradient-based algorithms (e.g., TO or level-set method) have many advantages, such as faster convergence and powerful mathematics-handling capacity particularly with lager parameter space. However, fabrication constraints are usually necessary. Imposing a larger feature size constraint in the optimization may deteriorate the performance 461 . In some extreme cases, the inverse-design algorithm could not find photonic structures on-demand even without imposing fabrication constraint 462 , the theoretically achievable performance or the desirable functionality may be ultimately bound by material constraints.
Non-gradient-based algorithms (such as DBS, GA or PSO algorithms) are usually easy to implement and need few mathematical requirements (e.g. complex gradient calculations are not needed). However, the convergence is often considerably slower, particularly when the objective function or parameter space becomes complex. To mitigate this issue, parallelizing the algorithm and using larger clusters of processors would be necessary 463 . On the other hand, although digital devices inspired by these non-gradient-based optimization methods has developed rapidly and achieved remarkable successes in many fields, most of the optimization methods are less involve with more complex physics problems, such as some emerging applications in nonlinear optics 366,388,393 .
For deep learning methods, it is possible to accelerate the simulation and design of optical devices in an effective and efficient manner 438,464,465 . The choice of neural network architecture and training strategy depends largely on the type of photonic system being analyzed. For systems that are relatively low dimensional and can be described by fewer than approximately ten independent geometric parameters, a discriminative neural network can effectively serve as a surrogate solver for that system and can be used in various optimization strategies to perform inverse design. These networks do require a large, one-time computational investment into a training set, making the neural network approach appropriate only when it involves systems, such as color filters or meta-atoms, where a wide range of layouts and optical responses are of interest. If a library of useful device shapes is known and are useful guidelines for design, the GAN approach provides a direct route to learning and interpolating related shapes from this library. If the goal is to identify a particularly high-performance device from a computation space, GLOnets provides a strategy to effectively search for the global optimum, without the requirement of a training set.

Summary and outlook
In summary, optical waveguides have proven a canonical platform to integrate diverse functional subwavelength meta-structures to enable meta-waveguides with either novel functionalities or largely boosted device performance. Compared with conventional waveguides devoid of subwavelength structures, meta-waveguides can not only just guide and confine light, but also perform various functionalities underpinned meta-structures. Compared with photonic crystal waveguides, meta-waveguides can act as 'designer artificial media' (with distinctive waveguiding mechanism to photonic bandgap) that enable versatile and powerful control over light propagation with subwavelength precision. By allying functional subwavelength structures with dielectric and plasmonic waveguide platforms, versatile coupling interfaces, on-chip optical signal processors, photonic neural networks, multifunctional routers, mode convertors, sensors, quantum and nonlinear devices can be envisaged. By integrating meta-structures with optical fibers, massive applications in optical communications, imaging, biomedical sensing and labon-fiber technologies can be realized in a flexible, compact and multifunctional manner.
Despite the concept of meta-waveguides is still in its infancy, exciting progress are hatching with bright perspectives and profound potential applications. As conceptually illustrated in Fig. 13, the advancement of meta-  Fig. 13 Outlook on photonic integrated "meta-circuits". Diverse meta-waveguide-based photonic devices can be competitive substitutes for conventional integrated optical scenarios with compact footprint, enhanced efficiency and multifunctionality. A laser-pumped microring frequency comb is distributed to cascaded wavelength division multiplexers (WDM) enabled by inverse-designed metamaterial waveguides (WG). Data can be encoded by variable optical attenuators (VOA) and fed to programmable mode convertors, which can be assembled into integrated photonic networks to perform analog computing of matrix-vector multiplication (MVM) for parallel convolutional processings 177,523 . The reconfigurable mode convertor can convert mode Ψ A to Ψ B (Ψ is TE or TM) with various distinguishable levels or to different spatial mode channels in a multimode waveguide 541 . The optical data are then collected via meta-couplers and meta-fibers with high efficiency and fidelity for further processing. Metawaveguides can also facilitate on-chip optical signal processing, communications, display and sensing applications (shown as insets). Plasmonic meta-waveguides may shrink device footprint towards higher integration density. Meta-fibers can also find massive interconnects, signal shaping, biomedical sensing and micro-manipulation applications 232,235 . Device schematics are inspired from previous literatures 12,177,194,235,523,541-543 . waveguides can not only extend meta-optics to the realm of guided electromagnetic waves and waveguide technology, but also promise to reshape the landscapes of photonic integrated circuits and massive emergent applications. The awaiting challenges are outlined as the following. A brief roadmap is also presented as Fig. 14 for future research.

Challenges
Despite the exciting progress discussed above 12,16,17,466 , practical applications of meta-waveguides are still limited. Several challenges must be properly handled before such ideas may transfer from scientific demonstrations to viable real-word technologies in future.
First, reconfigurable meta-waveguides are highly desired. For most current cases, the device attributes are fixed after fabrication [35][36][37][41][42][43][44]70,[75][76][77] . Further advancement of photonic integrated circuits and other applications will demand multi-functionality and reconfigurability of diverse photonic devices in a way similar to their electronic counterparts 1 . To this end, alliance from material platforms would be necessary. For instance, the meta-structures on top of waveguides can be mold by phase-change materials 177 or twodimensional materials 40,467,468 to facilitate reconfigurable integrated devices. Lithium niobate is also recently emerging as an excellent electro-optic (EO) and nonlinear material platform 469,470 with new opportunities for creating meta-waveguides. Other tunable mechanisms such as electrical-32,471 , thermal- 174,472 , mechanical-, and all-optical control may be explored as well 473 .
Second, challenges in device design still exist. (i) For meta-waveguides designed by physical intuitions, a universal design framework is still elusive. Most explorations start from distinct physical models under different waveguide platforms with unique features 51,70,75 .
(ii) For inverse-designed meta-waveguides, in scientific research-wise, some academic questions still remain open 15 , such as the existence of solution under given optimization conditions and the ultimate computational boundary. In application-wise, fully automated photonic inverse-design kits and integrated software tools still await development.
Third, as shared by all emergent technologies struggling to migrate from lab to real-word, meta-waveguides also have to settle challenges in scalable fabrication and compatible integration. The meta-structures generally pose stringent requirements on fabrication resolution 12 . Fabrication constraints should be considered in design process to pursue "efficient-to-fabricate structures". Complementary metal-oxide-semiconductor (CMOS) compatibility to electronics needs to be addressed as well to eventually enable large-scale manufactures fitting the foundry's process design kits 1 .
Detailed challenges and potential research directions are discussed below, according to different design directions and distinctive waveguide platforms.

Challenges for dielectric meta-waveguides
For instance, the substrate light leakage during guided wave and free-space light couplings via metasurfacedecorated waveguides will reduce conversion efficiency. This issue could be solved by using waveguide-integrated distributed Bragg reflector substrate, layered structures, and topologically protected structures 474,475  • Largely miniaturized integrated devices beyond diffraction limit.
• New plasmonic meta-platform to explore SW physics. intrinsic dispersive characteristics of the meta-waveguides should be also addressed for broader band operations. In addition, most of the meta-waveguides introduced above are based on local metasurfaces 146 , where the desired device functionality is realized by independent scatterers ignoring the neighboring coupling of the metaunits 145 . These devices generally operate at single wavelength with phase-only manipulation, for instance, dielectric waveguides patterned with phase-gradient metasurface 35,[41][42][43]51 . To thoroughly handle the detrimental effect of neighboring coupling on device performance and overcome the physical limitations of the local meta-structures 146 , we may need to entail nonlocal metasurfaces 145 .
Nonlocal metasurfaces rely on interactions between adjacent meta-units, showing unique advantages for designing multi-wavelength and multifunctional devices. A novel group theory 145,146 is proposed to design metasurfaces supporting quasi-bound states in the continuum (q-BICs) 476 , showcasing excellent capability in multifunctional spectral/polarization control, wavefront shaping 145 and leaky wave phase-amplitude holograms using waveguide-integrated metasurfaces 28,80 .

Challenges for optical meta-fibers
One of the salient challenges for meta-structured optical fibers is scalable and cost-effective fiber manufacture with desired subwavelength structures 231,232 . Photonic crystal fibers with comparatively bigger feature size are already commercially on-shelf, yet relevant technologies to massively produce optical fiber meta-tips and metamaterial fibers are still under development. A rich library of templates can be transferred from free-space metasurface to fiber facets. However, currently demonstrated optical fiber meta-tips fabricated by slow and expensive processes such as FIB milling and EBL 75,313,314 are not suitable for commercial products. Photolithography and femtosecond laser writing on fiber facets and sidewalls require further amelioration on fabrication resolution 290,292,293,305 and apparatus developments for better sample mounting and alignments 101,313 .
For optical fiber meta-tips, nanoimprint and selfassembly are two promising inroads towards highthroughput fabrication. Despite a sub-15 nm feature size is realized in imprinted fiber facet 291 , the structural quality and robustness still need further improvement 311 . Self-assembly is another cost-effective but 'bottom-up' method 477,478 . However, this approach also faces challenges in integrating diverse functional materials on unconventional fiber substrate and the accurate control over assembled structures and locations 233 . Meanwhile, most meta-tips deposit lossy metal on fiber facets 75,100,231 . Other materials that guarantee high light-matter interactions with low loss and fabrication robustness still awaits explorations.
For metamaterial fibers, the current resolution of sidewall photolithography is still not adequate for subwavelength features 316 . Thermal drawling of specific preforms instead offers unique advantages for market exploration. However, challenges such as low spatial resolution and insufficient axial and radial uniformity still recline ahead 246,247 . Although the resolution of a single drawing may be compromised, iterative and multiple drawling can largely reduce feature dimension from microns to tens of nanometers 320,321 . The drawn fibers can be assembled subsequently to produce metamaterial fibers with specific transverse structure distributions 321 . When transferring these techniques to produce optical meta-fibers, structural quality and reproducibility will require further attentions.

Challenges for plasmonic meta-waveguides
One long-standing challenge for plasmonic waveguide is lacking an efficient and integrated bridge to couple freespace propagation light into highly confined plasmonic modes 70,332 . On-chip in-coupling gratings can be adopted to convert input light beam to surface plasmons along metal, which is further focused by a taper-like region and then coupled into plasmonic waveguides 479 . Although a large portion of input light has the opportunity to be guided into the waveguide, such solution also suffers from low-efficiency and large-footprint issues. The rapid development of meta-structures offers us a high-efficiency, mini-sized, high-integration and multifunctional platform to manipulate both PWs and SWs 70,[112][113][114][115] . However, an ideal meta-coupler to efficiently connect free-space optical mode and localized plasmonic waveguide mode is still under research 480 .
Plasmonic meta-waveguides can offer unique capabilities to transport and manipulate light in deep-subwavelength scale, which is especially desired for future highly integrated photonics 324,344 . Nevertheless, the large energy loss (including absorption loss and scattering loss) of highly confined plasmonic waveguide modes is another big issue that seriously hinders their practical applications 481 . To address this challenge, people have reported diverse hybridized meta-waveguides to concentrate more optical energies of plasmonic waveguide modes in low-loss materials (e.g., dielectric layers) instead of conventional lossy metals 340,[482][483][484] . Besides, developing new plasmonic materials (e.g., metallic alloys, doped oxides and semiconductors) also attains intense attentions of the community 481,485 . The propagation length of such hybridized or new-materialbased waveguide modes can be further improved 486 . In parallel, the low-cost and high-throughput fabrication techniques for creating high-quality plasmonic devices are always highly desired, which is the very foundation to suppress both kinds of systematic losses.

Challenges for Inverse-designed metamaterial waveguides
First, most optimization algorithms become less efficient when processing inverse problem with larger parameter space or big design area 15 . As device complexity increases, new electromagnetic simulation tools and optimization algorithms need to be explored, thereby improving the breadth of inverse-design problems. For Deep learning methods, many reported cases to date are still limited to simple nanophotonic optimization, such as free-space metasurface 439,449,452,457 . Challenges still exist in applying diverse artificial neural networks for designing waveguidebased devices 487 . Metamaterial waveguides generally have more complicated structures and the training dataset acquisition can be more time-consuming. Current explorations are mainly restrained in digital meta-waveguides with simple device functions 61,464,487 . For complex nanostructures with more degrees of freedom (e.g. analog metawaveguides) or devices with multiple sophisticated functionalities, some pre-processing stages or modification of the model input or output may be required 454,488,489 .
Second, there still lacks a comprehensive physical model to tutor how to choose the design region and initial pattern 153 , especially in rigorous math to answer whether the functionality can be effectively realized or what is the ultimate performance achievable for an optical optimization problem 459 . As the landscape of a meta-structure is generally highly non-convex, a good initial condition is required and the computationally tractable method towards the global optimum of the objective function usually does not exist 153 . Different initial patterns may lead to different local-optimum even using the same iteration algorithm. One has to use multiple random initial patterns to generate many "optimized" patterns and then select a relatively "best" one as the final pattern. For this purpose, several recent works have begun exploring physics-informed optimization methods based on prior physical model, helping to discuss the feasibility of inverse-design quantitatively 415,418,420,421 .
Third, a comprehensive and robust software platform 490,491 to develop diverse inversely designed integrated devices still needs further development. Standardization in design process is required to accommodate fabrication constraints in commercial photolithography for different types of robust nanophotonic devices 492 .

Perspectives
Meta-waveguides with subwavelength functional structures offer an exceptional playground to develop novel integrated photonic devices and explore nanoscale optical phenomena 493 . As precise and powerful control over light propagation can be enabled by the meta-structures, light-matter interaction can be further magnified to enhance efficiency. Various emerging device functions still awaits exploring 12 .

Playgrounds for nanoscale optical physics and versatile integrated devices
Although in an early phase of research, metaswaveguides have already shown great potentials on manipulating light on-chip with excellent versality 35,[50][51][52] . Their functionalities could be further extended for broader applications. For example, migrating different dynamic light control techniques achieved in free-space metasurfaces 471,473,494 to meta-waveguides, for various new chip-integrated applications such as augmented/ virtual reality (AR/VR) displays and optical ranging/ LiDAR 31,32,472 . Beside passive devices, meta-waveguides can be also synergized with active components like waveguide-based lasers (to configure light emitting) and photodetectors 495 (to enhance the light-matter interaction efficiency).
In addition, there could be further explorations on optical computing with metasurface incorporated waveguides 177 . Integrated photonics have proven ideal optical computing platforms for various tasks, including image processing and artificial intelligence, with unprecedented speed and lower power consumption 191,496 . It is possible that those capabilities can be implemented using either fixed or reconfigurable meta-waveguides for achieving complete on-chip operations with miniaturize device footprint and enhanced multifunctionality 177,497 . Meanwhile, integrated photonic platform may be also further developed to significantly miniaturize conventional freespace optical elements 498,499 . Advanced designs in integrated meta-gratings may also fuel further advancements of diverse applications both in imaging and information processing 192,497 . Moreover, recent advent of quantum silicon photonic chips also open the door towards practical quantum communication and computations with high scalability and compactness 497,500,501 . It is foreseeable that incorporation of meta-photonic structures with photonic integrated quantum chips can achieve more complex functions and bring them to the next level of miniaturization.
Furthermore, if we further engineer the imaginary part of the antennas refractive index (gain & loss) to exploit Parity-Time symmetry 502 or applying dynamic index modulations 503,504 , the functionalities of the metadevices can be further extended in a more interdisciplinary manner 502,505-508 .
Multifunctional fibers that can sense, image, and communicate As an important member of meta-waveguide family, optical meta-fibers have shown massive applications in telecommunication, biomedical sensing and so on, by leveraging the flexibility and versatility of meta-fiberbased devices 74,232 .
With the aid of meta-fibers 75,76 and multi-material fibers 267 , we can envisage multifunctional fibers that can sense 104,232,234,249,294 , image 74,246,276,277 , and communicate [96][97][98]293 leveraging versatile subwavelength structures. The optical fibers cease to be just a simple waveguide and become a new all-around technological platform where different kind of materials and structures at nanoscale are suitably integrated around, on top or inside the fiber itself, allowing for the realization of advanced platforms from life science applications and labon-fiber (LOF) technology 231 to optical computing 234 . The revolutionary idea here is to combine all the functional components of a generic optical system commonly employed in communication and sensing fields (light source, optical waveguide, photodetector, module to interact with the environment) into a single, flexible, multifunctional, and compact platform. This perspective will set a fundamental building block for the development of a new generation of portable, 'plug-and-play', autonomous optical fiber chips to sense, elaborate and transmit sensorial data in remote locations without needing connection to any bulky instrumentation. The realization of all-in-fiber active optoelectronics platforms will break new grounds in many strategic sectors such as photonic computing and imaging, wearable technology, 'internetto-things' systems and telemedicine 509,510 , by avoiding components such as fiber tapers, grating couplers and bulk lenses that may strongly affect the performances, reliability, and footprint of current lab-on-a-chip devices.
Reliable metasurfaces-assisted ultra-low-loss light delivery and collection may also push optical imaging to a higher level with profound medical and clinical impact. Novel 'intelligent needles' can be thus envisaged, for biomanipulation, tissue and liquid biopsies, loco-regional echography and drug delivery with potential impact in precision medicine scenarios and clinically relevant investigations 511 .

On-chip plasmonic meta-devices in deep-subwavelength scale
The capability of confining long-wavelength optical radiations into collective charge oscillations at deepsubwavelength scale has made plasmons as excellent information carriers beyond diffraction limit 150,330 . Compared with conventional dielectric waveguides, plasmonic meta-waveguides can enable ultracompact on-chip optical functional devices of subwavelength dimensions 16,17 . Moreover, a new class of plasmonic metawaveguide can enable intriguing plasmonic modes beyond those of conventional plasmonic waveguides, such as chiral-512 , non-diffraction-123 , complex-polarized- 334 , and conformal-SPPs 344 , providing us a versatile platform to manipulate near-field SWs.
Thanks to the unique characteristics of high speed in time and subwavelength scale in space, plasmonic metawaveguide-based devices and circuits may serve as competitive complements and ideal links between conventional electric and photonic devices. So far, fruitful results on functional plasmonic meta-devices are keeping emerging, such as laser sources 513 , logic gates 346 , and photodetectors 514 .
Meanwhile, the development of plasmonic waveguides based on diverse new materials also reveals a promising future direction for information communication, optical computation, near-field sensing, electro-optic modulators, and so on. For instance, plasmon polaritons in graphene/ hexagonal boron-nitride (h-BN) heterostructures exhibited a long intrinsic propagation length exceeding 10 µm (about 50 plasmonic wavelengths) 515 . Graphene can also be patterned with subwavelength structures to facilitate various metawaveguide devices for beam steering 516,517 , non-reciprocal SPP propagation 518 and topological edge plasmon 519 . In addition, plasmonic meta-waveguides carrying strongly enhanced local fields offer us a good platform for tunable light-matter interactions (e.g., nonlinear and Raman effects) when integrated with two-dimensional materials such as MoS 2 and graphene 348,520 . Other material platforms like doped oxides and semiconductors can be good substitutes to conventional noble metals for high-performance tunable plasmonic devices 481,521 .

Smart metamaterial waveguides for photonic computing and deep learning
The long-standing grand challenge in efficient and systematic design of diverse nanophotonic devices may be circumvent by inverse-design methods 15 , which sheds light on integrated optics design automation to enable large-scale sophisticated circuits. Inverse design is especially suitable for developing meta-waveguides-enabled devices, as forward device topology design is much complicated and require specific manpower expertise. As integrated software tools for inverse design are ramping up recently (such as SPINS 490 and Lumerical Inc. 491 ), we can envisage fully automated integrated photonic metadevices design kits and commercial EDA software in the future, where no special expertise in electrodynamics or integrated photonics is required in this 'design-by-specification' scheme. Standardization and system level integration will be entailed in later stage as helpful catalyzer to push integrated meta-waveguide devices one step closer to real-word applications.
As fruitful results are keep emerging recently with enhanced device performance and even previously inaccessible novel functionalities 387,422 , inverse-designed metamaterial waveguides may hatch as a significant complement to integrated photonics mainstreams designed by physical intuitions 395 . A new twist in optical analog computing may take place, by shrinking bulky systems into elements in mere wavelength-size 57,522 . The metamaterial waveguide itself can be designed as 'smart' media for photonic computing and artificial intelligence 497,523 , for realizing parallel mathematical operations (such as differentiation, integration, or convolution) 57,524 , object classification 59,525 , image analysis 526 , or feature detection 527 . In the meantime, inverse-designed metawaveguides can also find its place in deep learning and neuromorphic photonics 8,148 to facilitate chip-integrated high-speed photonic neural networks with low power consumption 60,528,529 . For instance, a metamaterial can be trained to perform vowel recognition for acoustic wave 60 . Analog computing can be thus implemented in a manner that relies on purposefully perturbing a given system in situ, instead of designing the material from scratch, showcasing applications in all-optical control 428 as well as programmable meta-inclusions in the microwave regime 530 . Programmable meta-units are also explored as physical weights in an end-to-end deep learning integrated sensing pipeline to enable joint learning of optimal measurement process and a matching processing algorithm with improved latency 531 . With the aid of tunable materials, for instance phase-change materials and EO materials, meta-waveguides can venture photonic integrated circuits into new territories by providing diverse integrated devices with multifunctionality and reconfigurability 177,532 .