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Recent advances in metamaterial integrated photonics

Pavel Cheben, Jens Schmid, Robert Halir, José Manuel Luque-González, J.

Gonzalo Wangüemert-Pérez, Daniele Melati, Carlos Alonso-Ramos

► To cite this version:

Pavel Cheben, Jens Schmid, Robert Halir, José Manuel Luque-González, J. Gonzalo Wangüemert-Pérez, et al.. Recent advances in metamaterial integrated photonics. Advances in Optics and Photonics, 2023, 15 (4), pp.1033. 10.1364/AOP.495828. hal-04354811

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https://hal.science/hal-04354811v1

Submitted on 16 Oct 2024

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# Recent advances in metamaterial integrated photonics

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PAVEL CHEBEN,<fnref n="1" />,* JENS H. SCHMID,<fnref n="1" /> ROBERT HALIR,<fnref n="2" /> JOSÉ MANUEL LUQUE-GONZÁLEZ,<fnref n="2" /> J. GONZALO WANGÜEMERT-PÉREZ,<fnref n="2" /> DANIELE MELATI,<fnref n="3" /> AND CARLOS ALONSO-RAMOS<fnref n="3" />

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**Abstract:** Since the invention of the silicon subwavelength grating waveguide in 2006, subwavelength metamaterial engineering has become an essential design tool in silicon photonics. Employing well established nanometer-scale semiconductor manufacturing techniques to create metamaterials in optical waveguides has allowed unprecedented control of the flow of light in photonic chips. This is achieved through fine-tuning of fundamental optical properties such as modal confinement, effective index, dispersion and anisotropy, directly by lithographic imprinting of a specific subwavelength grating structure onto a nanophotonic waveguide. In parallel, low-loss mode propagation is readily obtained over a broad spectral range since the subwavelength periodicity effectively avoids losses due to spurious resonances and bandgap effects. In this review we present recent advances achieved in the surging field of metamaterial integrated photonics. After briefly introducing the fundamental concepts governing the propagation of light in periodic waveguides via Floquet-Bloch modes, we review progress in main application areas of subwavelength nanostructures in silicon photonics, presenting the most representative devices. We specifically focus on off-chip coupling interfaces, polarization management and anisotropy engineering, spectral filtering and wavelength multiplexing, evanescent field biochemical sensing, mid-infrared photonics, and nonlinear waveguide optics and optomechanics. We also introduce a nascent research area of resonant integrated photonics leveraging Mie resonances in dielectrics for on-chip guiding of optical waves, with the first Huygens' meta-waveguide recently demonstrated. Finally, we provide a brief overview of inverse design approaches and machine learning algorithms for on-chip optical metamaterials. In our conclusions, we summarize the key developments while highlighting the challenges and future prospects.

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© 2023 Optica Publishing Group

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## 1. Introduction

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Optical metamaterials are artificial composite materials, usually periodically nanostructured at the subwavelength scale to have desired properties not found in their constituents. Because of the subwavelength periodicity, diffraction does not occur, and light propagates through the metamaterial as if it was a homogeneous optical medium with effective optical properties determined by the constituent materials and the nanostructure topology. While the term “metamaterial” has been in common use only since the early 2000’s, the concepts of investigating and exploiting the effects of subwavelength structuring on material optical properties can be traced all the way back to early studies with radio waves and wire grids by Heinrich Hertz in the late 19th century [1]. In 1956, S. Rytov published an influential theoretical study showing that subwavelength stratified structures behave as birefringent homogeneous media with an effective refractive index that depends on the layer materials and thicknesses [2]. Subsequently, there have

<footnote n="1">¹National Research Council Canada, 1200 Montreal Road, Ottawa, ON K1A 0R6, Canada</footnote>

<footnote n="2">²Telecommunication Research Institute (TELMA), Universidad de Málaga, CEI Andalucía TECH, Louis Pasteur 35, 29010 Málaga, Spain</footnote>

<footnote n="3">³Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 10 Bv. Thomas Gobert, 91120 Palaiseau, France</footnote>

<footnote>*pavel.cheben@nrc.ca</footnote>
been decades of research on this topic in the field of subwavelength diffractive optics [3].

Over the last 20 years, much of metamaterial research has focussed on creating materials with properties that stem from optical resonances of particles and nanostructures. Initially, metallic particles with plasmonic resonances were used to demonstrate, for example, exotic properties such as negative refractive index [4]. However, due to the relatively high optical losses typically observed in plasmonic structures, over the last decade there has been increasing interest in all-dielectric metamaterials, often in 2D configurations known as metasurfaces for flat optics components [5], building upon foundational work of Lalanne et al. in blazed binary gratings from the late 1990s [6]. On the other hand, photonic crystals, which are also based on periodic dielectric structures, were increasingly used in integrated photonics after being initially proposed by S. John and E. Yablonovitch in the late 1980s [7, 8]. Photonic crystal structures exploit optical phenomena caused by photonic bandgap effects. The bandgap arises from Bragg grating reflections in the waveguide plane, which are generally avoided in metamaterials by choice of a smaller subwavelength periodicity; however, in some implementations the distinction between the two research areas can be blurred.

Concurrently to these developments, silicon photonics research underwent explosive growth, as the technology matured to become a leading photonic integration platform [9–11]. The growing importance and proliferation of silicon photonic circuits for telecommunications, datacom and optical interconnects has resulted in the establishment of public access foundries that have enabled researchers to prototype silicon nanostructure designs of components and circuits with a clear path towards commercial applications. In recent years, the same development can be observed for silicon nitride photonics. New applications for these technologies are now being explored, including integrated quantum photonics, sensing, imaging and beam steering for lidars or free-space communications, to name a few.

It may seem like an almost natural consequence of the scientific and technological evolution that the new field of subwavelength or metamaterial integrated photonics, which can be considered a marriage of metamaterials and silicon photonics research, came into being in the mid-2000's. Early foundational work in this area included the introduction of new fiber-chip coupling concepts [12, 13] and the demonstrations of antireflective waveguide facets [14, 15] and lenses [16]. Soon after, a number of original devices incorporating subwavelength structures was developed, including mode transformers [17], waveguide crossings [18], fiber-chip edge [19] and surface grating [20–22] couplers, athermal waveguides [23, 24], wavelength multiplexers [25], ultra-fast optical switches [26], directional [27] and multimode interference [28, 29] couplers, and biochemical sensors [30]. In the following years, integrated metamaterial photonics flourished and became an integral part of silicon photonics research while also breaking into commercial applications. A wide range of silicon photonic components was shown to be enabled or benefit in performance from the inclusion of metamaterials. The impressive progress that was achieved has been summarized in several review papers [31–36].

In this article we will review the recent advances in metamaterial integrated photonics. We begin with a section introducing the basic concepts and theoretical framework required for the reader to understand the following chapters, including a review of the propagation of light in periodic waveguides, homogenization theory and metamaterial waveguide design guidelines. Subsequently, in Sections 3 - 7 we describe progress in major application areas of subwavelength integrated photonics, such as components for off-chip coupling, polarization control, spectral filtering, chemical or biological sensing and photonics for the mid-infrared spectral region. Waveguide dispersion engineering and control of photon-phonon interactions in subwavelength structures for the enhancement of non-linear effects are the topic of Section 8. The use of Mie resonant particles as building blocks for on-chip waveguides has only recently been demonstrated and is opening the door to a new and exciting branch of integrated metamaterials research. The early work in this area is reviewed in Section 9. The important topic of numerical design
optimization of the complex metamaterial components is discussed in Section 10, including the design of aperiodic free-form metamaterials and the role of machine learning enhanced design methods. We conclude our review by providing an outlook and summary.

## 2. Subwavelength grating metamaterial waveguides

### 2.1. Light propagation in periodic waveguides

Consider the periodic structure shown in Fig. 1(a), where the direction of propagation is along the z axis. The waveguide core consists of material blocks of refractive index $n_{wg}$ and size $a \times w \times t$, spaced with a period $\Lambda$, embedded in a cladding material with refractive index $n_{cl}$, and supported by a substrate with refractive index $n_{subs}$; the ratio $a/\Lambda$ is often referred to as duty cycle. Light propagation through such a periodic waveguide can be described via Floquet-Bloch modes [37]. If only one Floquet-Bloch mode is propagating along the structure the electrical field is given by:

$$ \mathbf{E}(x, y, z) = \mathbf{F}(x, y, z) \exp(-ikz). \qquad (1) $$

Here $\mathbf{F}(x, y, z)$ is the Floquet-Bloch mode profile which is defined over a single period so that $\mathbf{F}(x, y, z + \Lambda) = \mathbf{F}(x, y, z)$. The exponential term accounts for propagation and attenuation via the complex wavevector $k = k_r + ik_i$. The real part of the wavevector is related to effective index of the Floquet-Bloch mode as $k_r = (2\pi/\lambda)n_F$, so that the guided wavelength can be written as $\lambda_g = \lambda/n_F$. Note that for a mode propagating in the positive z direction the real part of the wavevector is always positive, $k_r > 0$. If the waveguide is passive the imaginary part is always zero or negative, $k_i \le 0$. The Floquet-Bloch mode profile $\mathbf{F}(x, y, z)$ and the propagation constant $k$ generally have to be obtained numerically and both open source and commercial software packages, based on eigenmode expansion or finite difference time domain methods, are available for this purpose [38–40].

Depending on the ratio between the period of the structure, $\Lambda$, and the guided wavelength $\lambda/n_F$ the waveguide can operate in different regimes, as illustrated in Fig. 1(b). In this figure the vertical axis represents angular frequency, $\omega = 2\pi c/\lambda$, where c is the speed of light in vacuum. Positive values of the horizontal axis are used for the real part of the wavevector, and negative values for the imaginary part.

<figure><img src="image_3.png" /><figcaption>Fig. 1. a) Geometry of a periodic waveguide. For the sake of clarity the upper cladding, with refractive index $n_{cl}$, is not shown. b) Schematic dispersion diagram of the waveguide; shaded areas indicate different operating regimes and the light line is shown dashed.</figcaption></figure>

• In the bandgap, or Bragg regime, half the guided wavelength matches the pitch,

$$ \Lambda = \lambda/(2n_F), \qquad (2) $$
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so that $k_r$ does not change with the operating frequency. Since light is gradually reflected back by the successive and constructively interfering back-reflections, forward propagating light is attenuated and hence $k_i$ is negative [see Fig. 1(b)]. This regime is the foundation for the design of spectral filters, which are described in section 5.

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* For frequencies above the bandgap, or, equivalently, a pitch $\Lambda > \lambda/(2n_F)$, the periodic structure radiates light into the cladding and substrate. This region is of particular interest for grating couplers, detailed in section 3, which operate below the light line, but leak part of the guided wave in the form of radiation. This radiation amounts to losses of the guided mode so that $k_i$ is generally negative in this region. It is, however, possible to achieve loss-less propagation ($k_i = 0$) under certain conditions. The first is by requiring that the cladding refractive index be about three times smaller than effective index of the guided mode [41]. Even without this requirement loss-less propagation is possible if the individual waveguide segment size [see Fig. 1(a)] is carefully designed to resonate. This special case is referred to as Huygens waveguiding, and is examined in section 9.

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* Below the bandgap, i.e. when the pitch is $\Lambda < \lambda/(2n_F)$, there are no diffraction effects, and light propagates through the ideal periodic structure without losses ($k_i = 0$) much like it would in an equivalent homogeneous waveguide. In this subwavelength regime the optical properties of the equivalent homogeneous waveguide can be readily controlled by the geometry of the structure and can be described, often to a sufficient degree of accuracy, with simple mathematical approaches.

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## 2.2. Subwavelength gratings as uniaxial crystals

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In this section we describe a simple mathematical model for the optical properties of periodic waveguides. The underlying idea is that the periodic waveguide shown in Fig. 1(a) can be considered a “cut-out” of the infinite laminar material shown in Fig. 2(a) [42]. In the subwavelength regime this laminar structure can be accurately described as a uniaxial crystal, i.e. an equivalent homogeneous anisotropic material [Fig. 2(b)], so that the subwavelength structured waveguide itself can be modeled as a homogeneous waveguide made out of this equivalent material [Fig. 2(c)]. Note that the uniaxial crystal is described by a permittivity tensor

$$ \epsilon/\epsilon_0 = n_{\text{eq}}^2 = \begin{bmatrix} n_{\parallel}^2 & 0 & 0 \\ 0 & n_{\parallel}^2 & 0 \\ 0 & 0 & n_{\perp}^2 \end{bmatrix}, \qquad (3) $$

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where $\epsilon_0$ is the vacuum permittivity. We note that the equivalent index $n_{\text{eq}}$ describes the infinite laminar medium shown in Fig. 2(a) whereas we use $n_F$ to refer to the effective index of a Bloch-Floquet mode propagating through a subwavelength waveguide as shown in Fig. 1(a).

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Let us examine light propagation through this laminar structure more closely. Referring to Fig. 3(a), we consider a plane wave that propagates with a wavevector $\mathbf{k}$ forming an angle $\theta$ with respect to the normal of the material interfaces. For TE polarization the electric field $\mathbf{E}$, and hence the displacement field $\mathbf{D} = \epsilon\mathbf{E}$, lies within the plane of the chip, i.e. in the $x-z$ plane in Fig. 1(a), and equivalently in the plane shown in Fig. 3(a), whereas for TM polarization they are orthogonal to this plane. In this infinite structure the dispersion relation of the wavevector $\mathbf{k}$ can be described analytically [43, 44]; Matlab and Python implementations of the corresponding formulae are available online [45].

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To illustrate the behavior of this laminar structure let us apply the equations from [43–45] to alternating layers of silicon ($n_{\text{wg}} = 3.476$) and silicon dioxide ($n_{\text{cl}} = 1.444$). This laminar Si/SiO$_2$ material was proposed for the first time in [46], for birefringence control in planar
<figure><img src="image_4.png" /><figcaption>Fig. 2. a) An infinite laminar medium composed of alternating layers of the waveguide core material, with refractive index $n_{wg}$, and the cladding material, with refractive index $n_{cl}$. b) Equivalent homogeneous anisotropic material in the subwavelength regime. c) A metamaterial model of the periodic structure in Fig. 1(a) in the subwavelength regime.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. a) Propagation of a plane wave through an infinite laminar structure composed of two dielectric materials. b-c) Cartesian components of the wavevector when the structure is partially in the bandgap (b) and when it acts as a subwavelength metamaterial (c). The shaded part of the plots represent the imaginary part of the wavevector while the rest of the plot shows the real part. As customary $k_0 = 2\pi/\lambda$.</figcaption></figure>

Specifically, let us consider a 50% duty cycle ($a/\Lambda$), an operating wavelength $\lambda = 1550$ nm, and two different pitches: $\Lambda = 250$ nm and $\Lambda = 150$ nm. For the larger pitch [Fig. 3(b)] we observe that loss-less propagation ($k_i = 0$) is possible for most propagation angles, but for propagation near $\theta \sim 0$ the structure is operating in the bandgap (or Bragg regime). The real part of the equivalent index is constant and fulfils the Bragg condition (2) in the material, $n_{eq} = |\mathbf{k}|/k_0 = \lambda/(2\Lambda) = 3.1$. Furthermore the imaginary part becomes non-zero, indicating that forward propagating light is attenuated due to backreflection. Clearly this periodic structure cannot be modeled via a homogeneous waveguide, because it operates in the Bragg regime for certain propagation angles. However, as the pitch is reduced [Fig. 3(c)], the bandgap disappears because the effective index never reaches the Bragg threshold: in analogy with equation (2) we find $n_{eq} < \lambda/(2\Lambda)$, and the imaginary part of the wavevector is zero. Since this holds for all propagation angles $\theta$ the structure can be approximated by an equivalent uniaxial crystal.

Focussing on the subwavelength metamaterial regime [Fig. 3(c)], we observe that for propagation angles $\theta \sim 0$ the structure behaves identically for TE and TM polarization. The displacement field and the electric field are parallel to the interfaces of the materials for both polarizations, and therefore they experience the same equivalent index, which we will refer to as $n_\|$. For our particular example $n_\| \sim 2.8$. However, as the propagation angle increases the behavior becomes markedly different. For TM polarization electric field and the displacement field remain parallel to the interfaces and hence the wavevector stays constant, just as in an
184 isotropic medium:

$$ \frac{|\mathbf{k}_{\text{TM}}|}{n_{\parallel}} = k_0. \qquad (4) $$

185 By contrast, for TE polarization the orientation of the displacement field and the electric field changes with respect to the material interfaces as $\theta$ is increased, eventually becoming perpendicular for $\theta = 90^{\circ}$, experiencing a smaller effective index which we will refer to as $n_{\perp}$. Note that $n_{\perp} \sim 1.9$ is significantly smaller than $n_{\parallel} \sim 2.8$. This behavior is characteristic of uniaxial crystals and can be modelled by the elliptical dispersion relation

$$ \left( \frac{|\mathbf{k}_{\text{TE}}| \cos \theta}{n_{\parallel}} \right)^2 + \left( \frac{|\mathbf{k}_{\text{TE}}| \sin \theta}{n_{\perp}} \right)^2 = k_0^2. \qquad (5) $$

190 Note that within the coordinate system of Fig. 2(a) we have $|\mathbf{k}_{\text{TE}}| \cos \theta = k_z$ and $|\mathbf{k}_{\text{TE}}| \sin \theta = k_x$.

191 Thus, light propagation through a periodic structure in the subwavelength regime can be readily described via equations (4) and (5), with $n_{\parallel}$ and $n_{\perp}$ depending on both the pitch $\Lambda$ and the operating wavelength $\lambda$ through a set of complex yet analytical equations [43–45].

192 In the special case of the deep subwavelength regime, i.e. when $\lambda \gg \Lambda$ approximate expressions for $n_{\parallel}$ and $n_{\perp}$ are given by Rytov’s formulas [47]:

$$ \begin{aligned} n_{\parallel}^2 &\approx \frac{a}{\Lambda} n_{\text{wg}}^2 + \left(1 - \frac{a}{\Lambda}\right) n_{\text{cl}}^2, \\ n_{\perp}^{-2} &\approx \frac{a}{\Lambda} n_{\text{wg}}^{-2} + \left(1 - \frac{a}{\Lambda}\right) n_{\text{cl}}^{-2}. \end{aligned} \qquad (6) $$

196 On the other hand, when the operating near the bandgap, the “cut-out” approach becomes less accurate and full FDTD simulations of the periodic structure are preferable [42].

## 2.3. Design considerations

199 When working with subwavelength metamaterials some practical considerations need to be taken into account. The first is the choice of an appropriate period ($\Lambda$): smaller periods yield values of $n_{\parallel}$ and $n_{\perp}$ that vary less with wavelength, but also require smaller feature sizes which are more challenging to fabricate. Considering again alternating layers of silicon ($n_1 = 3.476$) and silicon dioxide ($n_2 = 1.444$) around a wavelength $\lambda = 1550$ nm, a commonly used pitch is $\Lambda = 200$ nm which enables the synthesis of a wide range of $n_{\perp}$ and $n_{\parallel}$, as shown in Fig. 4.

<figure><img src="image_6.png" /><figcaption>Fig. 4. Metamaterial equivalent refractive indexes, $n_{\perp}$ and $n_{\parallel}$, as a function of duty cycle and for different operating wavelengths, calculated with the script available at [45].</figcaption></figure>
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A second considerations arises when using small duty cycles which translate into smaller equivalent indexes of the metamaterial and hence weaker waveguiding, that can potentially lead to leakage to the substrate. It was shown in [48] that these effects can be neglected if the resulting Floquet-Bloch mode index $n_F$ is larger than 1.65 for 2 µm thick box layers; this criterion can be relaxed to $n_F > 1.55$ for 3 µm thick box layers. A final considerations concerns only wide multimoded subwavelength waveguides in which the mode is tightly confined: in this case jitter in the placement of the individual segments of the order of 5 nm can result in substantial transmission losses of about 3 dB/100 µm [49]. This effect arises because the jitter perturbs the periodicity on which the propagation of the Floquet-Bloch mode is fundamentally based.

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### 3. Off-chip nanophotonic coupling interfaces

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Off-chip coupling interfaces are of fundamental importance in optoelectronics as every photonic chip needs to be optically connected with the external world. Availability of efficient off-chip couplers is a basic prerequisite for utilization of integrated photonic devices in practical applications. Coupling to planar waveguides circuits was identified as a major challenge since the early years of integrated optics [50]. This issue became particularly pressing in silicon photonics [11, 51] as the cross-sectional area of a silicon wire waveguide is more than two orders of magnitude smaller than that of a standard SMF-28 optical fiber, resulting in a large mode size disparity. Different types of fiber-chip couplers have been developed for silicon photonic waveguides, with edge couplers and surface grating couplers adopted as two main configurations [52–54]. The fiber-chip couplers often employ SWG metamaterials to improve coupling performance. The first configuration – the edge coupling – is based on in-plane topology in which the optical axis of the fiber and the planar waveguide are aligned to each other. A common strategy is to progressively decrease the mode effective index along the coupler via an inverse [55] or an SWG [19] taper, the latter being schematically illustrated in Fig. 5a. This expands the waveguide mode until it matches the spot size of the optical fiber (either lensed, or cleaved) near the chip edge. The edge couplers typically achieve a high coupling efficiency down to sub-decibel level and a broad wavelength bandwidth exceeding 100 nm. As such, they are often favored for applications demanding a wide spectral range or polarization insensitivity, such as wavelength division multiplexed (WDM) communications, nonlinear frequency generation and optical spectroscopy. An important limitation of the edge couplers is that they can only be located at the periphery of a chip and require individual chip dicing, i.e. they cannot be used for wafer-level testing.

<figure><img src="image_7.png" /><figcaption>Fig. 5. Schematics of fibre-chip couplers incorporating SWG metamaterials. a) Edge coupler with an SWG taper. b) Surface coupler with the grating strength controlled through the SWG engineering.</figcaption></figure>
The second coupling configuration, depicted in Fig. 5b, utilizes a surface grating coupler with a diffraction grating etched into the planar waveguide in longitudinal direction (x) to couple light between the waveguide and an optical fiber positioned above the grating [56]. The grating strength can be controlled by SWG engineering in transverse (y) direction. The radiation angle θ is given by the grating equation:

$$ \theta = \arcsin \left( \frac{n_F}{n_{cl}} + \frac{m\lambda}{n_{cl}\Lambda} \right) \qquad (7) $$

where n<sub>F</sub> is the effective index of the mode propagating in the waveguide grating (Floquet-Bloch mode), n<sub>cl</sub> is the refractive index of the upper medium into which light is radiated, the integer m is the diffraction order, λ is the wavelength, and Λ is the grating period. The radiation angle at the central wavelength is often selected such that the fiber axis is inclined (Fig. 5b, angle α) from the vertical direction to achieve the in-plane momentum matching for the first diffraction order, while preventing the second order diffracted beam from back-coupling to the planar waveguide. Since a standard cleaved SMF-28 fiber is typically used, the mode size is about 10 µm which is large enough for good alignment tolerances, accommodating in-plane misalignment errors of about 2.5 µm with a penalty less than 1 dB. This permits the use of assembly instruments with higher throughput and reduced cost for packaging the chips. The surface couplers also allow wafer-scale testing with easy access to different locations within the optical circuit, and can readily accommodate high-volume fabrication and packaging processes. These advantages come at the cost of a limited spectral range, which is typically less that 40 nm at 1 dB efficiency drop, as a result of the grating diffraction angle variation with wavelength (Eq. 7). The coupling loss is also comparatively higher, about 2-4 dB for state-of-the-art silicon photonic foundries. The readers can refer to more details in the recent review articles on this topic [53, 54].

Since the first demonstration of a subwavelength grating fiber-chip coupler in 2006 [13, 19], SWG nanostructures have been leveraged to develop off-chip nanophotonic couplers with unprecedented performance. In this section, we provide a brief review of the latest progress in SWG metamaterial couplers, with an emphasis on the approaches adopted to achieve high coupling efficiency, broad wavelength range, and polarization insensitivity. Different application scenarios will be discussed, including fiber-chip coupling and off-chip coupling to free-space and bulk optics resonators.

### 3.1. Subwavelength grating fiber-chip edge couplers

SWG fiber-chip edge coupler constituted the first demonstration of a subwavelength engineered silicon strip waveguide [13, 19], opening the field of metamaterial integrated photonics [33] and as such is of particular relevance to this review. The coupler principle is based on a gradual modification of the waveguide effective index and the corresponding mode size transformation. This is accomplished by varying the duty ratio of the subwavelength grating formed in the composite waveguide by interlacing the core and cladding materials [19, 57, 58], as shown in Fig. 6a. Since the two materials (e.g., Si and SiO<sub>2</sub>) are combined at the subwavelength scale, loss and wavelength resonances due to diffraction effects are suppressed and the mode confinement is directly controlled by the volume fractions of Si and SiO<sub>2</sub>. In order to avoid formation of the Bragg (bandgap) resonances, the grating period is chirped to keep the effective wavelength-to-period ratio approximately constant along the taper. A high coupling efficiency of 92% (0.32 dB) and a polarization dependent loss as low as 0.05 dB was achieved for a broad spectral range exceeding 100 nm in a telecom C-band, using a lensed optical fiber with a mode field diameter of 3 µm [58]. Metamaterial engineering obviates the need for the local waveguide height adjustments that would generally be required for polarization independent operation of conventional solid core tapers in CMOS foundries. Additional advantages of SWG metamaterial couplers are in increased design flexibility and improved robustness to fabrication imperfections
compared to conventional tapers. The SWG couplers exhibit superior performance both at the nominal design point and at the worst-case point within fabrication tolerance range [59].

<figure><img src="image_8.png" /><figcaption>Fig. 6. a) SEM images of different parts of the first demonstrated SWG fiber-chip edge coupler. Reprinted with permission from [19]. ©The Optica Publishing Group. b) IBM fiber-chip coupler top-view schematics of the design at different locations along the taper with the corresponding SEM images of the experimental structure. ©IEEE. Reprinted, with permission, from Barwicz *et al.*, IEEE JSTQE 25, 1-13 (2018) [59].</figcaption></figure>

A larger mode field diameter (10 µm) is required for coupling with a standard SMF-28 fiber. This also helps to achieve a robust alignment tolerance at assembly and allows the packaged device to withstand environmental stresses and strains it is exposed to during operation. However, increasing the mode field diameter causes radiative leakage to the silicon substrate through the buried oxide (BOX) layer [60], the thickness of which is limited to about 2-3 µm in SOI wafers commonly used in silicon photonics. IBM [59] demonstrated a metamaterial coupler similar to ref. [58] but integrated with optical fiber V-grooves to allow for a larger mode size while suppressing the mode leakage to the silicon substrate. The coupler top-view schematics at different locations and the corresponding SEM images are shown in Fig. 6b. The coupler was fabricated in a 300 mm CMOS production site using immersion 193 nm stepper lithography. Fiber-chip transmittance measurements confirmed a coupling loss of 0.7 dB (TE polarization) and 1.4 dB (TM) from a standard SMF-28 optical fiber to a silicon wire waveguide at telecom O-band. The coupler is currently available as a commercial Product Design Kit (PDK) at GlobalFoundries [59].

An alternative coupling approach to mitigate leakage to substrate is shown in Fig. 7a. The coupler utilizes a thick upper cladding with engineered refractive index to shift the mode field distribution upwards, thereby reducing the field overlap with the buried oxide and the silicon substrate [61]. This is accomplished by incorporating in the SiO₂ upper cladding thin high-index silicon nitride layers to increase the cladding effective refractive index. A subwavelength grating with a variable duty ratio is employed to gradually reduce the refractive index of the Si₃N₄ layers towards the chip edge, resulting in an adiabatic mode expansion, as illustrated in Fig. 7b. Coupling efficiency of 90% (-0.41 dB) was calculated for a cleaved SMF-28 fiber. At the same time, the large mode size affords a robust alignment tolerance of 2.2 µm for a 1dB efficiency penalty. A similar strategy, i.e. utilizing the upper cladding with thin layers of high-index
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material, was adopted by TeraXion Inc. [62]. The light is transferred near the coupler tip to a composite waveguide comprising an array of thin silicon nitride strips embedded in a 6-µm-thick silicon oxynitride cladding. This effectively pulls the mode towards the upper cladding region, which minimizes leakage to the silicon substrate. A coupling loss as low as -0.5 dB was reported for TE polarization at 1550 nm, using a high-numerical aperture (HNA) fiber with a mode field diameter of 6.6 µm.

<figure><img src="image_9.png" /><figcaption>Fig. 7. a) Concept of the edge coupler with an SWG engineered upper cladding. 220-nm-thick silicon wire waveguide is tapered from 270 nm nominal width on the chip to 150 nm width near the chip edge, along a taper length of 260 µm. At the same time, the metamaterial refractive index of two thin Si₃N₄ layers embedded in the upper SiO₂ cladding is gradually increased towards the chip edge by increasing SWG duty ratio, over a length of 50 µm for the upper Si₃N₄ layer (green) and 200 µm for the bottom Si₃N₄ layer (blue). This geometry results in a mode size expansion along the coupler so that optimized mode overlap with optical fiber is achieved. b) Upper panel shows calculated mode evolution along the coupler (sideview, i.e. plane perpendicular to the chip), with optical fiber near the chip facets and cross-sections A, B, C and D located at different positions along the taper. Bottom panel shows transverse mode distributions at different cross-sections (A, B, C and D) of the taper. Adapted from [61] ©Optica Publishing Group. c) 3D schematics and d) SEM image of the SWG trident edge coupler. Reprinted with permission from [63].</figcaption></figure>

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Yet another approach to simultaneously achieve a polarization independent coupling and robust alignment tolerance is based on a suspended Si-Si₃N₄ hybrid photonic platform [64]. The light is coupled from the optical fiber into the top silicon nitride waveguide and subsequently to a silicon waveguide underneath, while the refractive indexes of both waveguides are independently controlled by SWG engineering for optimal vertical coupling. The measured coupling loss for TE/TM polarization is -0.86 dB/ -0.94 dB with SMF-28 fiber. Recently, SWG metamaterials have also been incorporated in a trident geometry [63, 65, 66], i.e. coupler with three separate SWG tips near the facet, as illustrated in the schematic of Fig. 7c and the SEM image of Fig. 7d. The SWG trident structures may alleviate substrate leakage loss and accommodate a larger mode size. The coupler was fabricated on an SOI wafer with 220 nm silicon, 3 µm BOX and 3.5 µm SiO₂ cladding, and the coupling loss using SMF-28 fiber was -2.22/-2.53 dB for the TE/TM
## 3.2. Metamaterial surface grating couplers

Realizing fiber-chip surface couplers with efficiency and bandwidth comparable to the edge couplers has been an important objective in silicon photonics research. SWG metamaterials have proven to be exceptionally useful to this end. The first SWG-enhanced grating coupler was proposed in 2005 [12] via implementing a reflective subwavelength grating onto a conventional silicon waveguide grating to increase coupler directionality. This is achieved by recombining in-phase two diffraction orders into a single beam radiated into free space, with an estimated improvement in coupling efficiency from 30% to 96% near 1550 nm. In the subsequent developments, SWG nanostructures were hybridized with a conventional waveguide diffraction grating creating an effective medium to control the grating strength and thereby maximizing fiber-chip coupling efficiency [20–22]. An SEM image of the coupler is shown in Fig. 8a, with a diffractive grating formed along the longitudinal (x) direction and a non-diffractive SWG structure superposed in the transverse (y) direction [22]. By adjusting the duty cycle of the SWG along the longitudinal direction, the optimal grating strength distribution is achieved using only a single etch step. Such continuously apodized grating generates a near-Gaussian radiated field that closely matches the mode of the SMF-28 fiber, with a mode overlap of about 95%.

<figure><img src="image_10.png" /><figcaption>Fig. 8. a) SEM image of the first fiber-chip surface coupler with the grating strength apodized by SWG metamaterial [22]. A detail view is shown in the inset. b) Schematics of the zero-order surface grating coupler. Inset: SEM image of the fabricated coupler. Reprinted with permission from [67].</figcaption></figure>

Leveraging this SWG apodization method, the first sub-decibel surface grating coupler was demonstrated for a standard 220 nm SOI platform [68]. This was achieved by a multi-parameter optimization of the grating structure in combination with a backside metal reflector for constructive interference between the beam reflected at the BOX-substrate interface and the upward diffracted beam. The silicon substrate was locally removed under the couplers by using a lapping tool and ICP etching. A 10-nm-thin Al mirror layer was deposited underneath the BOX, i.e. in close proximity (3 µm) from the top device layer. The experimental peak fiber-chip coupling efficiency in the C-band was 0.69 dB (85.3%) with a 3dB bandwidth of 60 nm. Alternatively, a double SOI wafer can be used instead of the backside metal reflector, yielding a calculated efficiency of 0.42 dB [69].

More recently, SWG apodization was implemented by CEA-Leti in their CMOS process using deep-ultraviolet 193 nm dry lithography with optical proximity correction [70]. SWG index engineering permitted to increase the minimum feature size to 120 nm compared with 50 nm for a conventional design, the latter being beyond reach for 193 nm dry lithography. The adoption
of 193 nm immersion lithography by silicon photonic foundries will enable even more refined
coupler designs with minimum feature sizes of approximately 50 nm, which is expected to allow
fiber-chip coupling losses below -1dB for packaged devices [70].

An important method to increase coupler efficiency is to utilize a blazed geometry with
vertically asymmetric grating teeth diffracting light preferentially in one direction [71]. This can
be achieved by using L-shaped grating segments while simultaneously incorporating a short SWG-
index matching section at the waveguide-grating junction to minimize back-reflections [72, 73].
Recently, Benedikovič et al. [74] reported an L-shaped coupler design with SWG metamaterial
trenches optimizing both the grating directionality and radiated field profile. The calculated
fiber-chip coupling efficiency is 95% (-0.25 dB) at 1550 nm for a minimum feature size of
120 nm, compatible with deep-UV lithography. An apodized bi-level grating coupler was also
designed using a complex wavevector band structure technique, and experimentally demonstrated
in a monolithic SOI CMOS process achieving 92% (-0.36dB) coupling efficiency [75]. Grating
couplers with L-shaped segments are further discussed in Section 10.3, with an example shown
in Figure 32e.

SWG structures have also been successfully employed to increase grating coupler spectral
bandwidth. The operational bandwidth is limited by the grating diffraction angle variation with
wavelength, given by Eq. 7. In refs. [76, 77] it has been shown that controlling waveguide
dispersion by using SWG can help reduce this diffraction angle variation, extending the 1dB
bandwidth to about 100 nm albeit the coupling loss was comparatively high, in the range of 4-5
dB. A fundamentally new strategy to broaden coupler bandwidth while minimizing the loss was
recently demonstrated by Sanchez-Postigo et al. [67, 78]. The key idea is to reduce the coupler
angular variation with wavelength by setting *m* = 0 in Eq. 7, in other words, operating in zeroth
order. This cancels the explicit variation with *λ*, thereby increasing the bandwidth. In this case
the angular variation with wavelength only arises from the dispersion of *n*<sub>F</sub> and *n*<sub>cl</sub>, which can be
controlled by SWG dispersion engineering. The coupler schematic and SEM image are illustrated
in Fig. 8b. When placing a high-index medium with *n*<sub>cl</sub> > *n*<sub>F</sub> on the top of the waveguide, the
zeroth-order beam is coupled into the high-index medium, with the radiation angle given by Eq.
1 for *m* = 0. A single-crystal silicon prism is used as the high index medium. The prism shape is
required to avoid total internal reflection at the superstrate-air interface so that the beam can exit
from the high-index medium to the air and then to the fiber. A quarter-wave anti-reflective coating
is deposited at the prism facet facing the fiber to minimize Fresnel reflections. Subwavelength
grating is used to engineer the waveguide dispersion for increased wavelength bandwidth, while
the prism is slightly tilted to vary the gap between the waveguide and the prism. This apodizes
the coupling strength for an optimized overlap of the radiated field with the SMF-28 fiber. The
experiments confirmed a 1-dB bandwidth broader than 90 nm and a coupling loss of less than 1
dB for TE polarization, which is the highest coupling efficiency-bandwidth product (CE · BW)
yet reported for fiber-chip surface couplers.

SWG structures have also been employed to reduce grating coupler polarization sensitivity
[79, 80]. As it follows from Eq. 7, polarization insensitive operation can be realised by making
the waveguide mode effective index *n*<sub>F</sub> identical for TE and TM polarizations. This has been
demonstrated by using an SWG structure to control the effective indices of the TE and TM
modes in the grating region [79]. Zou *et al.* [80] reported a two-dimensional grating coupler
with reduced polarization dependent loss by optimizing a separation between SWG cylinders
forming the grating cell. Last but not least, SWG engineering has been employed for designing
perfectly vertical couplers [81–83] and for expanding the operation range of grating couplers into
extended short wavelength infrared [84] and mid- and long-wavelength infrared [85, 86].
### 3.3. Radiation to free space and coupling to microresonators

Recently, there have been intense research and development activities in on-chip optical phased arrays (OPAs) capable of high-speed shaping and scanning of optical beams in free space based on electronic phase control without mechanical moving parts [87]. Such active off-chip coupling interfaces are required for LiDARs and 3D scanners enabling machines, autonomous vehicles, drones and robots to map and interact with the surrounding environment. In on-chip OPAs, surface grating couplers are used as optical antennas to couple light from planar waveguides to free space propagating beams [88]. SWG engineering has been proven to be an effective method to mitigate challenges in designing antennas for both one-dimensional (1D) [89, 90] and two-dimensional (2D) [91, 92] OPAs.

In 1D OPAs, weakly modulated waveguide grating antennas with lengths of several millimeters are required to obtain the highly collimated beams needed for high resolution scanning. It has been very challenging to make such long antennas in silicon waveguides because the high index contrast does not allow accurate control of grating strength. Recently, Ginel-Moreno et al. demonstrated a new type of silicon waveguide grating with a largely increased antenna length [89, 90, 93]. This is achieved by using a metamaterial waveguide core formed by a periodic SWG, which evanescently couples the propagating Floquet-Bloch mode to the radiative segments separated by a gap $g$ from the waveguide core, as shown in Fig. 9a. A 2-millimeter-long antenna was fabricated on a conventional 220 nm SOI platform using single-etch process, with a minimum feature size of 80 nm compatible with deep-UV immersion lithography. A far-field beam divergence as small as 0.1° and a wavelength scanning sensitivity of 0.13°/nm were measured [90].

<figure><img src="image_11.png" /><figcaption>Fig. 9. a) 3D schematic view of a weakly radiating surface antenna in an SWG waveguide. Reprinted with permission from [90] ©The Optica Publishing Group. b) SEM image of an OPA section, comprising an array of long SWG-engineered waveguide antennas. Reprinted with permission from [93]. c) Proposed nanophotonic antenna with far field broadened by transversally-interleaved subwavelength grating. Reprinted with permission from [92]. d) Schematic diagram and SEM image of the SWG metamaterial taper for coupling light to whispering gallery mode resonators. Reprinted with permission from [94] ©The Optica Publishing Group.</figcaption></figure>

Conversely, 2D OPAs require compact strongly modulated antennas to allow for densely
packed arrays comprising hundreds or even thousands of elements. Minimizing the separation between the emitters also reduces the number of diffracted orders, hence ideally steering the beam in a single direction over a broad angular range [87]. Grating strength can be enhanced and antennas shortened by using a thicker waveguide core, e.g. 300 nm SOI platform [74]. Recently, Khajavi *et al.* reported a design of an SWG engineered antenna with L-shape segments in 300 nm SOI, simultaneously achieving high diffraction efficiency and directionality. The calculated diffraction efficiency is 89% (-0.5 dB) for a compact antenna size of 7.6 µm × 4.5 µm with a far-field beam width of 11°/ 52° [91] in the longitudinal/transverse directions. However, the comparatively small beam width in the longitudinal direction restricts the OPA's steering range. This challenge has been resolved in the subsequent work, again by using subwavelength engineering [92]. The key idea is to broaden the far-field by controlling the near-field phase with a transversally-interleaved nanostructure, as schematically shown in Fig. 9c. Two gratings with different longitudinal periods (600 nm and 730 nm), hence with different radiation angles (-21° and 7°) are transversally interleaved with a subwavelength pitch of 125 nm, within the antenna width of 1.75 µm. This results in an exceptionally broad far-field beam width of 52° × 62°. The antenna has an ultra-compact footprint of 3.1µm × 1.75µm and the calculated efficiency reaches 82% for a structure implemented on a double-SOI substrate [92].

Yet another appealing application of SWG engineering is coupling light between planar waveguides and bulk optics resonators. Fig. 9d shows a schematic and an SEM image of an SWG taper, allowing versatile coupling between silicon waveguides and different types of whispering gallery mode (WGM) resonators with a wide range of sizes (300 µm – 3.6 mm) and refractive indices (1.42 – 2.21). An inverse taper geometry with a continuously varying metamaterial index is used to shape the optical field distribution and wave vector of the guided mode to optimize coupling with WGM resonators. Up to 99% light coupling efficiency was demonstrated for microspheres and microdisks made of silica, lithium niobate, and calcium fluoride, with quality factors $Q$ approaching $2×10^7$ [94].

# 4. Anisotropy engineering and polarization management

As detailed in section 2.2 subwavelength gratings behave as anisotropic metamaterials. In integrated optics this inherent anisotropy of SWGs was first exploited for the design of ultrabroad-band multimode interference couplers [95], but is also an attractive property for the design of polarization handling devices [96]. In the following sections we will analyze some outstanding SWG-based polarizers, polarization splitters and polarization rotators, as well as multimode and polarization independent devices.

## 4.1. Polarizers

Integrated polarizers, i.e. devices that let pass through either TE or TM polarization, can directly exploit the anisotropy of SWGs. A straightforward approach to achieve this is using SWGs to engineer modal confinement, so that the more weakly confined polarization is radiated (leaked) from the waveguide. Fig. 10(a) shows a TE-pass polarizer based on a 3-µm-radius, 180° bend with an exterior SWG cladding parallel to the waveguide core [97]. This cladding orientation corresponds to $\theta = 90°$ in Fig. 3(a) and preserves guiding for the fundamental TE mode, which is polarized perpendicular to the material interfaces, hence experiencing the reduced refractive index $n_{\perp}$ in the cladding. However, the fundamental TM mode is polarized along the material interfaces and thus experiences the larger refractive index $n_{\parallel}$ in the cladding, which in combination with the sharp bend radius leaks the mode out of the waveguide. An analogous design for a TM-pass polarizer is shown in Fig. 10(b), where the waveguide core is composed of a subwavelength multislot waveguide [98]. In this case the lower $n_{\perp}$ refractive index perceived by the fundamental TE mode in the core produces leakage, while the TM mode is well guided with the larger $n_{\parallel}$ refractive index. The mode converters shown in Fig. 10(b) are required to
<figure><img src="image_12.png" /><figcaption>Fig. 10. a) A TE-pass polarizer based on an SWG cladding proposed in [97]. b) A TM-pass polarizer with multi-slot subwavelength waveguide core [98]. c) Using tilted SWG waveguides yields a TM-pass polarizers that converts the fundamental TE₀ mode into the backward propagating TE₁ mode. Reprinted with permission from [99] ©The Optica Publishing Group.</figcaption></figure>

reduce transition losses between the strip input/output waveguides and the multislot waveguide. Both polarizers achieve measured sub-decibel losses and polarization extinction ratios above 20 dB covering a 400 nm bandwidth. A different approach for a TM-pass polarizer is presented in Fig. 10(c): a periodic waveguide in which every second strip is tilted still behaves as a SWG for TM polarization because a change in the propagation angle $\theta$ produces no noticeable effect on the wavevector $k_{TM}$ [see Fig. 3(c)]. However, for TE polarization an asymmetric Bragg reflector is formed, which reflects the forward propagating fundamental TE mode into the second order TE mode. In simulation this device exhibits insertion losses below 0.5 dB, 20 dB polarization extinction ratio and a 150 nm bandwidth.

## 4.2. Polarization splitters and rotators

Integrated polarization beam splitters (PBS) separate the fundamental TE and TM modes travelling in the same input waveguide into separate output waveguides, as shown schematically in Fig. 11(a). These devices enable the realization of polarization diversity circuits that are crucial to overcome the inherent polarization dependence of high-index-contrast platforms such as silicon-on-insulator. PBS have to provide optimum performance for both polarization states, which is notoriously difficult to achieve over broad bandwidths. In the following we review how the unique anisotropic properties of SWGs have been used to overcome this limitation; the performance of the individual devices is summarized in table 1.

Placing longitudinal SWG strips between parallel waveguides is known to reduce coupling between them for TE polarized light (electrical field perpendicular to the material interfaces) [104]. In [100] the authors used this effect to reduce TE coupling between two parallel SWG waveguides, as illustrated in Fig. 11(a). TM polarization is parallel to the interfaces of both the SWG waveguides and the SWG strips between them, so the whole structure exhibits the same equivalent refractive index. It thus acts as a homogeneous multimode interference (MMI) device, coupling TM polarized light to the cross output over a broad bandwidth. As shown in Tab. 1 this approach yields excellent performance in a bandwidth of 210 nm. However, it requires sub-100-nm feature sizes which can be challenging to fabricate.

A subwavelength structure with a constant tilt angle (see inset in Fig. 11(b)) can achieve
<figure><img src="image_13.png" /><figcaption>Fig. 11. a) A PBS using SWGs with different orientations. Reproduced with permission from [100]. b) A multimode interference coupling optimized for polarization splitting via a tilted SWG. Reproduced with permission from [101]. ©The Optica Publishing Group. c) A polarization splitter and rotator. Reproduced with permission from [102]. ©The Optica Publishing Group. d) A PBS based on tilted SWGs that implements an on-chip Glan-Thompson prism. Reproduced with permission from [103].</figcaption></figure>

polarization splitting with larger feature sizes: the tilt angle affects TE polarized light, but has only a negligible impact on TM modes [105]. This effect has been exploited in a directional coupler composed of two SWG waveguides which are identical except in the tilt angle, so that TM polarization couples to the cross port, while TE coupling is suppressed [106]. A similar approach has been proposed in the emerging lithium-niobate-on-insulator platform [107]. In a multimode interference coupler a judicious choice of the tilt angle yields image formation at different outputs for TE and TM polarizations [see Fig. 11(b)], while offering a broader bandwidth than directional coupler based approaches. Nevertheless, the polarization beam splitters described so far rely on some kind of modal interference mechanism whose wavelength dependence ultimately limits the operating bandwidth.

To further increase the bandwidths of polarization splitters, mode evolution designs have been explored [108, 109]. In this configuration two homogeneous parallel waveguides are tapered and separated by an SWG structure, with material interfaces oriented either perpendicular or parallel to the direction of propagation. For the perpendicular orientation the TE mode travelling through the input waveguide gradually evolves into TE mode of the cross waveguide, because it experiences the higher equivalent refractive index $n_\parallel$, whereas the TM mode remains in the input waveguide. For the parallel orientation it is the TM mode that adiabatically couples into the cross waveguide, while the TE mode stays in the original waveguide, because it experiences the lower equivalent refractive index $n_\perp$. Both configurations can achieve bandwidths in excess of 250 nm in simulation; the experimental bandwidths are somewhat lower, as shown in Tab. 1. On a fundamental level, in these mode evolution based devices the operating bandwidth is mostly limited by weak modal confinement at longer wavelengths.

One approach to overcome the aforementioned bandwidth constraints is by judiciously cascading polarization splitting directional couplers in a pair of unbalanced Mach-Zehnder interferometers [110]. The phase response of the interferometer arms combined with asymmetric coupling yields bandwidths up to 310 nm in simulation. A different approach is shown in Fig. 11(d), where tilted SWG waveguides are used to implement an on-chip version of a Glan-
Thompson prism [103]. In this device polarization splitting is achieved by polarization-dependent total internal reflection yielding a 415 nm bandwidth [see Tab. 1]. For TM polarization the difference between the effective indices of the vertically and horizontally oriented SWG stripes (shown in yellow and green in Fig. 11(d)) is negligible, so that light propagates straight through the device. TE polarized light propagating in the tilted SWG input waveguide experiences a strong index contrast when incident on the horizontal SWG strip. For a large enough input angle $\theta_{in}$ it thus undergoes total internal reflection and is redirected to the upper output.

<table><caption>Table 1. A table comparing the performance of some SWG-based polarization beam splitters. PER – polarization extinction ratio, IL – insertion losses, BW – bandwidth. Simulation results are marked by *, all other results are experimental. †Limited by measurement equipment.</caption><thead><tr><th>Reference</th><th>Year</th><th>Configuration</th><th>PER (dB)</th><th>IL (dB)</th><th>BW (nm)</th></tr></thead><tbody><tr><td>[100]</td><td>2019</td><td>TE: Suppressed coupling; TM: Multimode interference</td><td>&gt;20</td><td>&lt;1</td><td>210</td></tr><tr><td>[106]</td><td>2020</td><td>Tilted SWG directional coupler</td><td>&gt;15</td><td>&lt;1</td><td>&gt;70</td></tr><tr><td>[101]</td><td>2020</td><td>Tilted SWG multimode interference</td><td>13</td><td>1</td><td>120</td></tr><tr><td>[108]</td><td>2020</td><td>Mode evolution</td><td>&gt;20</td><td>&lt;1</td><td>230</td></tr><tr><td>[109]</td><td>2021</td><td>Mode evolution</td><td>&gt;20</td><td>~1.2</td><td>150<sup>†</sup></td></tr><tr><td>[107]*</td><td>2021</td><td>Directional coupling; LNOI</td><td>&gt;20</td><td>1</td><td>85</td></tr><tr><td>[110]*</td><td>2021</td><td>Cascaded Mach-Zehnder interferometers</td><td>&gt;20</td><td>0.5</td><td>310</td></tr><tr><td>[103]</td><td>2023</td><td>Tilted SWG Glan-Thompson polarizer</td><td>&gt;20</td><td>&lt;1.7</td><td>415</td></tr></tbody></table>

Remarkably SWGs can also be used for broadband polarization rotation: introducing a subwavelength structure in a conventional ‘cut corner’ polarization rotator provides control over the dispersion of the two hybrid modes, potentially enabling bandwidths in excess of 400 nm [111]. A broadband device that performs both polarization splitting and rotation is shown in Fig. 11(c): the fundamental TE mode travels through the device unchanged, but the TM mode experiences both mode rotation and splitting [102]. In the rotator section the fundamental TM mode is adiabatically converted into the first order TE mode. This rotation is enabled by the use of vertically asymmetric structure with a silicon dioxide substrate and an air (acting as upper cladding): the asymmetry facilitates coupling between TE and TM modes with similar refractive indices, and the slowly tapered waveguides ensures that this coupling is complete. In the mode splitting section, the first order TE mode evolves into the fundamental TE mode of a parallel waveguide, thereby completing the polarization splitting and rotating process. In both sections the SWG structures enhance the mode conversion, yielding a shorter device with an operating bandwidth up to 250 nm.

### 4.3. Multimode and polarization independent devices

Polarization independent operation is desirable in a variety of applications in which the input state of polarization is not known a priori or may change over time. In high-index-contrast platforms polarization independent behavior is, however, difficult to achieve because the waveguides exhibit strong birefringence. Fig. 12(a) illustrates how SWG metamaterial anisotropy can enable polarization independent power splitting in the silicon platform [112]. The device works as a 1 × 2 multimode interference coupler, forming a double image of the input field at the position of
the two output waveguides. The imaging distance is, however, considerably longer for TM than for TE polarization. The four longitudinal SWG strips on either side of the input waveguide are used to compensate this difference. The TE mode does not penetrate into these strips because their equivalent refractive index $n_⊥$ is smaller than that of the waveguide core $n_‖$ [see Figs. 3(a) and (c)]. For TM polarization the longitudinal strips exhibit the same equivalent index $n_‖$ as the

<figure><img src="image_14.png" /><figcaption>Fig. 12. a) A polarization independent power splitter [112]. b) The ‘bricked’ metamaterial topology. Reproduced with permission from [113]. c) A multimode waveguide crossing. Reproduced with permission from [114]. d) A polarization independent MMI implemented with a bricked SWG. e) A metamaterial engineered multimode waveguide bend. Reproduced with permission from [115].</figcaption></figure>

waveguide core, so that this polarization ‘sees’ a multimode waveguide in this input region, and thus effectively experiences a longer device. Simulation and measurement reveal that this device covers more than 400 nm of bandwidth with sub-decibel insertion losses [112].

A different approach to achieve polarization independence is by using novel metamaterial topologies such as a ‘bricked’ structure shown in Fig. 12(b). This metamaterial consists of SWG strips arrayed along the propagation direction with a pitch $Λ_z$ which are in turn segmented in the transversal direction with a pitch $Λ_x$. This structure behaves as a biaxial crystal with optical properties that can be tuned by controlling the distance $Δz$ by which the transversal segments are shifted [113]. For 50% duty cycles in both directions the permittivity tensor of this metamaterial is given by:

$$ \varepsilon(\overline{\Delta z})/\varepsilon_0 = \begin{bmatrix} n_{xx}^2(\overline{\Delta z}) & 0 & 0 \\ 0 & n_{\|}^2 & 0 \\ 0 & 0 & n_{zz}^2(\overline{\Delta z}) \end{bmatrix}, \qquad (8) $$

where

$$ n_{xx}(\overline{\Delta z}) = \frac{1}{2} \left[ (n_{\|} + \sqrt{n_{\|}n_{\perp}}) + (n_{\|} - \sqrt{n_{\|}n_{\perp}}) \cos(2\pi\overline{\Delta z}) \right] \qquad (9) $$

$$ n_{zz}(\overline{\Delta z}) = \frac{n_{\perp}n_{\|}}{n_{xx}(\overline{\Delta z})} \qquad (10) $$

with $\overline{\Delta z} = \Delta z/\Lambda_z$, and $n_⊥$ and $n_‖$ are the equivalent material refractive indexes with $\Delta z = 0$, which can be computed as described in section 2.2. Note that while $n_{xx}$ and $n_{zz}$ both depend on the shift $\Delta z$, $n_{yy}$ does not, because TM polarized light is always parallel to the interfaces. The additional degree of freedom afforded by the amount of shifting has enabled the design of a polarization independent 2 × 2 multimode interference coupler with a simulated bandwidth of
160 nm in the O-band [116]. An experimental demonstration of this device, shown in Fig. 12(d), achieving polarization independence over a 60 nm bandwidth has been recently published [117].

Handling multiple waveguide modes simultaneously is a key requirement for mode division multiplexing techniques [118]. Fig. 12(c) shows a multimode waveguide crossing for TE polarized modes that utilizes SWG anisotropy to reduce crosstalk [114]. To illustrate the operation of the device let us consider light propagating from left to right in Fig. 12(c). A SWG taper is used to transform the homogenous input waveguide into a SWG waveguide with equivalent index $n_\parallel$, as depicted in Figs. 3(a) and (c). The central region exhibits a slightly higher equivalent index, due to the added longitudinal strips, but, critically, is surrounded by a cladding of longitudinal strips with an equivalent index $n_\perp$ that is much lower, thereby providing strong lateral confinement that suppresses crosstalk for light travelling from left to right. Due to the symmetry of the structure the role of waveguides and claddings is reversed when light travels from bottom to top, achieving the same overall effect. As a result the device behaves as a near ideal waveguide crossing for up to three modes over a 80 nm bandwidth [114]. Curves in multimode waveguides are similarly challenging because the slight distorsion of the mode fields in the bent waveguide can produce significant modal crosstalk. MacKay et al. recently demonstrated a metamaterial engineered waveguide bent with a gradual curvature radius, shown in Fig. 12(d), which achieves transmission of five TE and four TM modes with a signal-to-cross-talk ratio better than 14.4 dB [115].

## 5. Spectral filters and waveguide multiplexers

Optical filtering and multiplexing are of fundamental importance in many scientific and industrial fields, including telecom and datacom applications and optical signal processing. The first practical filters and multiplexers in SOI platform were integrated ring resonators and arrayed waveguide gratings (AWGs) [119–121]. While the first silicon-based ring resonators had a few nanometers bandwidths ($Q$-factors ~200-300), continuous advances in photonic chip manufacturing technologies have enabled photonic filters with quality factors of several million [122], required for key applications such as nonlinear optics [123], biochemical sensing [124] and ultra-precise gyroscopes [125]. In this section we review how subwavelength grating materials are being exploited to enhance the performance of integrated photonics filters, focusing on waveguide Bragg gratings, ring resonators and contra-directional couplers.

### 5.1. Bragg grating filters

Bragg gratings are wavelength dependent reflecting structures created by periodically modulating the refractive index along the propagation direction. The behavior of a Bragg grating is mainly determined by its spatial period, $\Lambda_B$ and grating strength, $\kappa$. The relation between the period and the central wavelength of the band reflected is calculated as, $\lambda_B = 2n_F\Lambda_B$. The grating strength controls two important figures of merit of the filter: the reflectance per unit length and the filter bandwidth. The most commonly used Bragg grating configuration in SOI is the sidewall grating schematically shown in Fig. 13(a). In this structure, the grating strength is controlled by the modulation depth of the sidewall corrugation, $\Delta W$. This implies that to achieve narrow filtering bandwidths, which require small values of $\kappa$, very small corrugations have to be etched into the waveguide sidewalls. For example, sub-nanometer filter bandwidths in standard 500 × 220 nm SOI waveguides necessitate corrugation depths smaller than 10 nm, which are very challenging to fabricate [126]. This problem can be solved by using multiple-etch-step geometries or cladding modulated filter topologies. Multiple-etch-step geometries allow for sub-nanometer filters with corrugations of ~40 nm at the expense of a more complex fabrication process [127, 128]. Cladding-modulated structures [129] physically separate the periodic perturbation from the waveguide core, thereby enabling ultra-weak filters, but they are less suitable for the implementation of strong filters. SWG-assisted Bragg gratings overcome
<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 13. Schematic representation of different integrated Bragg grating topologies including (a) sidewall, (b) longitudinal, (c) transversal, and (d) cladding modulated gratings. The grating strength is adjusted by tuning (a) the corrugation at each side of the waveguide, $\Delta W$, (b) the difference between duty cycles DC₁ and DC₂, (c) the width difference between alternating silicon segments, $\Delta W$ and (d) the separation, $s$, width, $\Delta W$, and length of the lateral blocks $\Lambda_B \cdot DC$.</figcaption></figure>

these limitations, facilitating the design of Bragg filters with a wide range of grating strength using only one etch step.

The first demonstration of an SWG-assisted Bragg grating was reported by Wang *et al.* [130,131]. The device comprises an SWG waveguide in which every second silicon segment is enlarged, as illustrated in Fig. 13(b). Double duty cycle periodic waveguides provide an elegant solution for engineering the index modulation in Bragg gratings, by lithographically controlling the equivalent refractive indices of the two metamaterials the waveguides are composed of. This technique proves particularly valuable in SOI Bragg gratings, where the index contrast between silicon and silicon dioxide is too strong for some applications. With this approach, the authors showed a measured 3-dB bandwidth of ~0.5 nm for a filter length of 1.12 mm. The strategy of altering every second segment was later exploited by Pérez-Galacho *et al.* [132], but alternatively widening the segments [Fig. 13(c)]. The authors reported a filter bandwidth of 1.1 nm with a rejection ratio of ~40 dB using a 1 mm long filter. Additional degrees of freedom to engineer grating strength have recently been demonstrated. Oser *et al.* [133] experimentally demonstrated that the combination of SWG waveguides with shifted asymmetric corrugations yields sub-nanometer bandwidths. Specifically, the authors reported 0.8 nm bandwidth filters with rejection ratios exceeding 40 dB for a symmetric corrugation shifted Bragg filter and a 0.6 nm bandwidth with a rejection level of 16 dB for an asymmetric corrugation shifted Bragg filter; both devices were 1 mm long.

These recent SWG-assisted Bragg grating filters have demonstrated sub-nanometer bandwidths with high extinction ratios. To achieve even narrower spectral bandwidths, Ctyroky *et al.* [134]
combined the cladding modulation strategy with an SWG metamaterial waveguide core [Fig. 13(d)]. This topology is particularly interesting as it allows both the maximum and minimum strength of the Bragg grating to be readily adjusted, substantially widening the grating modulation range. The maximum strength of the filter is controlled via the equivalent refractive index of the SWG core, delocalizing the mode field to enhance the interaction between the mode and the lateral loading blocks which form the Bragg grating. By separating the loading blocks the minimum grating strength exponentially decreases because of the evanescent nature of the mode profile in the lateral cladding. Additionally, a longitudinal shifting Δz between the lateral segments in each period has demonstrated to reduce the grating strength as

$$ \kappa(\Delta z) = \kappa(0) \cos(\pi\Delta z/\Lambda_B). \qquad (11) $$

The experimental demonstration of this Bragg grating topology [135] yielded 3-dB bandwidths as small as 150 pm for a filter length of 4.7 mm. The synthesis of complex spectral transmittance profiles requires a wide range of grating strengths and thus fully exploits the potential of the cladding modulated SWG-assisted Bragg gratings. Pereira-Martin et al. demonstrated this by designing a complex spectral filter comprising 20 arbitrarily spaced narrow notches [136]. Another interesting application requiring a wide range of grating strengths are delay lines implemented with Bragg filters. Sun et al. [137] used a cladding modulated grating to implement a 60 ps delay over an operating bandwidth of 41.7 nm. Furthermore, engineering the anisotropy of the SWG waveguide core by using tilted SWG segments can yield polarization independent Bragg filters, as demonstrated recently by Chen et al. [138, 139].

## 5.2. Ring resonators

Ring resonators are wavelength-selective optical cavities comprising a waveguide loop and a coupling region that transfers part of the light between the loop and an adjacent waveguide,

<figure><img src="image_19.png" /><figcaption>Fig. 14. Schematic and top view of the SWG racetrack resonator proposed in [140], with equal coupling coefficients for TE0 and TE1 modes. b) Schematic of the SWG ring resonator proposed in [141], where the shape of the silicon segments has been optimized to compensate the effect of the curvature. Reproduced with permission from [141].</figcaption></figure>
as schematically shown in Fig. 14(a). The behavior of ring resonators is governed by its optical path length (OPL) and quality factor (Q-factor). The OPL determines the position of the resonances as $\lambda_{res,m} = \text{OPL}/m = n_{\text{eff}}L/m$, and their separation, i.e. the free spectral range, FSR = $\lambda_0^2/(n_g L)$ where $n_{\text{eff}}$ and $n_g$ are the effective and group indices of the mode propagating in the ring waveguide, L is the length of the resonator and m is an integer. The Q-factor is related to the resonance width, $\Delta\lambda = \lambda_0/Q$. In this section we discuss the design of SWG-assisted ring resonators, focusing on the optimization of the coupling mechanism and the quality factor. SWG ring resonators for sensing purposes are described in section 6.

The first SWG-assisted ring resonator was proposed by Wang et al. [130]. The authors studied resonators in which both the ring and the coupling waveguides were SWG structures, reporting Q-factors similar to those of conventional SOI rings, ~8,800 and ~20,000 with bending radii of 10 µm and 15 µm respectively. An important advantage of SWG ring resonators is the field delocalization in SWG waveguides, allowing for larger gaps or shorter coupling lengths. Indeed, Ye et al. [140] exploited SWG engineering in a dual-mode racetrack resonator to reduce the coupling region length of the racetrack to 20 µm [Fig. 14(a)]. An equivalent conventional structure would require a coupling region of 95 µm [142]. In order to increase Q-factor, Wang et al. [141] proposed SWG ring resonators where the shape of the silicon segments is optimized to compensate the effect of the waveguide bend [Fig. 14(b)]. The trapezoidal configuration of the silicon segments implements a gradual equivalent refractive index profile, featuring a higher refractive index in the inner section of the ring and a lower refractive index at the outer edge. This intra-modal index engineering equalizes the optical path length for light travelling along the inner and outer parts of the ring, thereby reducing curvature losses and consequently enhancing the quality factor. With these trapezoidal SWG structures, the Q-factor was increased to ~45,000 for a 10 µm radius, three times larger than the non-trapezoidal SWG structures. Moreover, the authors were able to demonstrate ring resonators with a compact 5 µm radius and a Q-factor up to ~11,500. A similar structure comprising a racetrack resonator with trapezoidal SWG slot waveguides was also reported with a Q-factor of ~14,000 for both polarizations [143]. Recently, SWG ring resonators have been demonstrated in a silicon nitride prototyping foundry [144] with Q-factors up to ~200,000 for a 400 µm bending radius, demonstrating the promising prospects of SWG metamaterial structures in the silicon nitride platform.

## 5.3. Contra-directional couplers

A contra-directional coupler is a 4-port reflective device comprising two asymmetric waveguide arms brought into close proximity with longitudinal periodic perturbation of the refractive index in the region between the waveguides. Compared to conventional 2-port Bragg gratings the advantage of contra-directional couplers is the separation of the transmitted and the reflected signals in two different waveguides, obviating the need for external circulators or the extra 6 dB losses of Y-branch architectures. The refractive index perturbation couples contra-directional waves between the two waveguides at a wavelength $\lambda_{CDC}$, but also produces Bragg reflections in each waveguide at wavelengths $\lambda_{B1}$ and $\lambda_{B2}$. Specifically, the behavior of a contra-directional coupler is governed by the grating period, $\Lambda_{CDC}$, the perturbation strength, $\kappa$, and the effective indices of the fundamental modes in each waveguide, $n_{F1,2}$. The period sets the central wavelength of the coupler,

$$ \lambda_{CDC} = (n_{F1} + n_{F2})\Lambda_{CDC}, \qquad (12) $$

and also determines the Bragg reflections in each waveguide as $\lambda_{B1} \sim 2n_{F1}\Lambda_{CDC}$ and $\lambda_{B2} \sim 2n_{F2}\Lambda_{CDC}$. The grating strength controls the bandwidth and reflectance per unit length of the device. With their unique ability to synthesize equivalent refractive indices, SWGs have opened new venues in the design of contra-directional couplers, by providing control over both the waveguide asymmetry and the grating strength.
<figure><img src="image_20.png" /><figcaption>Fig. 15. (a), (b) Schematic of an SWG-assisted contra-directional coupler with the grating strength apodized by controlling (a) the gap between waveguides [146] and (b) the lateral corrugation of the SWG segments [148]. (c) Schematic of a single SWG-assisted contra-directional coupler (top) and cascaded devices (bottom) that multiplex two wavelengths, two polarizations, and two waveguide modes [149]. Reproduced with permission from [146] and [149].</figcaption></figure>

The first SWG-assisted contra-directional coupler was reported in 2016 by Naghdi et al. [145]. In this work, the authors used an SWG and a conventional strip waveguide. The SWG metamaterial waveguide has a substantially smaller effective index than the strip waveguide, thus providing a strong waveguide asymmetry ($n_1 \sim 2, n_2 \sim 2.26$). The inherent periodic modulation of the refractive index of the SWG waveguide with $\Lambda = 378$ nm yields $\lambda_{B1} \sim 1500$ nm and $\lambda_{B2} \sim 1700$ nm, resulting in a bandwidth of about 200 nm between the intra-waveguide Bragg reflections, with contra-directional coupling at $\lambda_{CDC} \sim 1565$ nm. The authors reported various devices with 3-dB bandwidths between 10 nm and 20 nm reaching extinction ratios up to 30 dB between the stop and pass bands in the transmitted signal. Strong sidelobes were observed in the spectral response, reducing sidelobe suppression ratio to 5 dB. In the subsequent work, Liu et al. [146] and Naghdi et al. [147] reported spectrally tailored SWG-assisted contra-directional couplers with reduced sidelobes. The grating strength was apodized by gradually changing the separation between the waveguides [Fig. 15(a)] achieving experimental sidelobe suppression ratios of 27 dB [146] and 19 dB [147] for a 100 µm length device. With this performance, SWG-assisted contra-directional couplers proved suitable for applications in wavelength division multiplexing and spectral notch filters with high rejection ratios.

Large extinction ratios are feasible with SWG-assisted contra-directional couplers as they allow for larger grating strengths compared to conventional waveguides. Charron et al. [150] demonstrated various SWG-assisted contra-directional couplers with extinction ratios larger than 30 dB and bandwidths of 8, 20 and 30 nm. In these devices the sidelobes were quite strong because a uniform (non-apodized) grating was used. A Gaussian-apodized SWG-assisted contra-directional coupler was reported by Yun et al. [151] using SWG waveguides with a lateral corrugation [Fig. 15(b)], with a rejection ratio in excess of 40 dB and a sidelobe suppression of 19 dB. To further suppress the sidelobes the apodized structure was cascaded, reaching sidelobe
suppression ratios up to 50 dB and insertion loss below 1 dB.

A 4-channel WDM architecture based on cascaded SWG-assisted contra-directional couplers was proposed by Naghdi et al. [152]. A 3 dB bandwidth of 6.7 nm was measured, suitable for coarse WDM, with insertion loss below 1.8 dB. Another 4-channel WDM multiplexer leveraging cascaded SWG-assisted CDCs was proposed by Chen et al. [148], with 15 nm channel spacing, sub-decibel insertion loss and crosstalk -15 dB. The strategy of cascading contra-directional couplers for wavelength multiplexing has been extended by the same authors to polarization and mode division multiplexing [149]. They reported a two-wavelengths, two-polarizations and two mode division multiplexing device shown in Fig. 15(c) with overall insertion loss of 6.6 dB in a back-to-back configuration.

## 6. Evanescent field sensors

For many diseases early diagnosis is crucial to improve treatment, outcomes and patient well-being. This requires sensing tools capable of detecting, often from a single patient sample, the presence of very small concentrations of different types of biomarkers (in the order of pg/ml) [153–155]. Silicon photonic evanescent field sensors are considered one of the most promising candidates for the development of lab-on-a-chip platforms [156–158]. To reach their ultimate sensitivity and detect small concentrations, in addition to reducing the background noise of the readout system, it is necessary to optimize the overlap of the evanescent field with the analyte [159, 160]. The powerful capability of engineering the spatial distribution of the mode profile provided by the use of SWG waveguides makes them very attractive for this purpose.

We generally distinguish two types of sensing: bulk and surface, as illustrated in Fig. 16(a) and (b). Bulk sensing measures the refractive index variations that occur in the bulk solution flowing over the sensing waveguide and is usually not able to discriminate between different biomolecules in the analyte. Surface sensing evaluates the variations that occur in the area closest to the surface of the sensor, approximately within 20 - 500 nm, and does have the capacity to discriminate between biomolecules. In this modality, the top and sidewall surfaces of the sensing waveguide core are first functionalized with specific bioreceptors for the recognition of the molecular species of interest. When target molecules bind to the bioreceptors, a layer of thickness "t" is formed on the surface of the waveguide. As a measure of the ability of the sensing waveguide to respectively convert refractive index variations of the upper cladding and increments in the adsorbed molecular layer thickness into change of the mode effective index (see Fig. 16), the bulk and surface waveguide sensitivities ($S_w$) are defined: $S_{w,bulk} = \partial n_{eff}/\partial n_c$ [RIU/RIU], $S_{w,surface} = \partial n_{eff}/\partial t$ [RIU/nm].

By implementing the waveguide into a specific sensing architecture this change in mode effective index is converted into a physical magnitude that can be easily measured. Sensing architectures can be classified into two broad categories: interferometric and resonant, depending on whether the variations of the effective index ($\partial n_{eff}$) result in a variation of the optical phase ($\partial\varphi$) or a change of the resonance wavelength ($\partial\lambda_o$) of an optical cavity or spectral filter. In a Mach-Zehnder Interferometer with a sensing arm length $L$ the architecture sensitivity is $S_{a,MZI} = \partial\varphi/\partial n_{eff} = (2\pi/\lambda_o)L$. This scheme suffers from phase ambiguity and sensitivity fading, since the signals from both arms are combined and read out by a photodiode which produces a photocurrent signal proportional to $\cos(\partial\varphi)$ [161]. Consequently, the same power reading may correspond to two different phase shifts. To overcome these drawbacks an MZI with coherent phase read-out can be employed [162] (see Fig. 16.c). By combining the signals from the reference and sensing arms into a phase-diversity network such as a 2x4 or 2x3 MMI, and detecting their corresponding outputs, a complex IQ-signal proportional to $e^{j\partial\varphi}$ can be obtained by linear combinations of the output powers, thus recovering the phase without ambiguity nor fading [158] [159]. In addition, MMI imperfections and fiber array misalignments can be compensated by calibration. In a waveguide ring resonator with a group index $n_g$ the architecture
788 sensitivity is $S_{a,ring} = \partial\lambda_o/\partial n_{eff} = \lambda_o/n_g$ [163, 164].

<figure><img src="image_21.png" /><figcaption>Fig. 16. Schematic representation of evanescent field sensing: a) bulk sensing and b) surface sensing. The refractive index of the adsorbed molecular layer depends on the surface functionalization process, i.e. surface modification and immobilization of high-affinity binder [165], [166], as well as on the type of analyte molecules (typically around n ≈ 1.45 for proteins). The increase of the adsorbed layer thickness "∂t" depends on the number of target molecules (e.g. virus) that bind to the bioreceptors (e.g. antibodies or Y-shaped proteins). c) Schematic of an interferometric sensing architecture with coherent phase read-out. Reprinted with permission from [158].</figcaption></figure>

789 The overall sensitivity of the waveguide implemented in the sensing architecture is given by 790 $S = S_w S_a$, and the limit of detection, that is the smallest possible perturbation that the sensor 791 can detect, is LOD = $3\sigma/S = 3\sigma/(S_a S_w)$, where $\sigma$ is the system noise. To factor out the 792 dependence on the measurement set up and facilitate a fair comparison between different sensors, 793 an intrinsic LOD or iLOD is commonly used in resonant-type sensors [163]. It is defined as the 794 required change of the refractive index to produce a shift in the resonance wavelength equal to 795 one linewidth ($\Delta\lambda_{3dB}$), thus resulting in

$$iLOD = \frac{\Delta\lambda_{3dB}}{S_a S_w} = \frac{\lambda_o}{Q S_a S_w}, \qquad (13)$$

796 where Q is the quality factor of the resonator. This expression highlights the importance of 797 reducing optical losses, which in liquid-based sensors are dominated by water absorption (52.5 798 dB/cm at $\lambda=1.55$ µm). In the presence of high analyte absorption, an increase in the light-solution 799 interaction and hence in the waveguide sensitivity does not translate into a decrease in the LOD, 800 because the quality factor is decreased by the same ratio. This is the main motivation to operate 801 in the O-band ($\lambda=1.31$ µm) [155, 164, 167, 168], where water absorption losses are about 10 802 times lower (6.6 dB/cm). Equivalent expressions in terms of signal to noise ratio can be obtained 803 for interferometric sensors, where the arm length that minimizes the LOD is $L \sim 1/\alpha$, with alpha 804 the field attenuation factor [159].
## 6.1. Subwavelength grating sensing waveguides

In order to achieve high sensitivities, waveguides must provide a large overlap between the mode field and the analyte. Both the bulk and surface waveguide sensitivities depend on the geometry and dimensions of waveguide, as well as the polarization state and the material platform. Silicon photonic wire waveguides operating with TM polarization were initially used for their simplicity and remarkable waveguide sensitivity ($S_{w,bulk}$ = 0.37 RIU/RIU, $S_{w,surface}$ = $0.8 \times 10^{-3}$ RIU/nm [169, 170]), due to the strong field intensity produced in the upper cladding in close proximity of the top surface of the waveguide core (see Fig. 16). For applications where the use of TE polarization is required, slot waveguides provide a significant improvement in sensitivity, twice that of the photonic wire operating in TM polarization ($S_{w,bulk}$ = 0.83 RIU/RIU, $S_{w,surface}$ = $1.6 \times 10^{-3}$ RIU/nm [171]). The proposal of SWG waveguides for sensing applications made it possible to optimally engineer the light-matter interaction and further enhance the waveguide sensitivity [30].

Figures 17(a) and (f) show a cut at $y = H/2$ of the electric field intensity across a period ($\Lambda$) for an SWG waveguide with typical dimensions ($W = 500$ nm, $H = 220$ nm, $\Lambda = 250$ nm), operating in TE polarization. The Floquet-Bloch mode field profile reveals that the benefit of using an SWG waveguide lies not only in the delocalization of the field, but also in the fact that it allows the analyte to penetrate into the gaps where the mode has a high electric field strength. By judiciously choosing SWG waveguide dimensions, bulk waveguide sensitivities close to or even greater than unity can be achieved while maintaining negligible substrate leakage loss and feasible feature sizes [158], also in the O-band where the smaller SWG pitch restricts the range of manufacturable grating duty cycles [164]. Since the first report on SWG-based sensing [30], research has focused on gaining further insight into specific aspects of the problem: the geometry of the waveguide, the sensing architecture in which it is embedded, the polarization effects, the type of sensing (bulk vs surface), the fabrication platforms [161], the fabrication tolerances of the guiding structures [155], the difficulties of the functionalization process [172], or the integration of the microfluidics with other components (lasers, detectors, etc.) [168]. Researchers have developed statistical models based on perturbation theory to precisely study the interaction of the analyte molecules and SWG waveguide evanescent field [173] and investigated the trade-off between field confinement and scattering loss [174].

Several SWG waveguide geometries have emerged, Figure 17(a)-(e) shows those that have attracted the most interest. For these specific geometries, the field profile of the Floquet-Bloch mode and an SEM image of the structure used in the experimental validation are shown. The slot-SWG waveguide [175] depicted in Fig. 17(b), obtained by a sub-wavelength segmentation of the conventional slot waveguide [176], attains a waveguide sensitivity ($S_w$) of 0.83 and an architecture sensitivity ($S_a$) of -599 nm/RIU when embedded in an MZI architecture interrogated with a wavelength sweep [Fig. 17(l)]. These values are similar to the 490 nm/RIU achieved with SWG waveguides in microring resonators [177] shown in Fig. 17(k). On the other hand, the multibox-SWG waveguide illustrated in Fig. 17(c), which can be seen as a generalization of the slot-SWG, is obtained by using multiple rows of silicon blocks [178]. In addition to the delocalization of the field, this geometry allows to significantly increase the contact surface of the analyte around each silicon block, thus enhancing the surface sensitivity. Specifically, bulk and surface sensitivities of 580 nm/RIU and 1900 pm/nm were experimentally demonstrated in ring resonators [178]. In a phased-shifted Bragg grating resonator [see Fig. 17(m)], a significant improvement of the LOD was achieved owing to an increased Q-factor since avoiding waveguide bends reduces radiation losses [179].

An SWG topology that has recently been studied intensively and is likely to become particularly suitable for sensing applications is the fishbone-SWG [Fig. 17(d)]. Circumventing the use of isolated silicon blocks alleviates the main practical problem faced by segmented SWG structures when used for biosensing. Normally, SWG waveguides are covered in a silicon dioxide upper
<figure><img src="image_22.png" /><figcaption>Fig. 17. a)-e) Geometries of different SWG waveguide topologies proposed for sensing applications. f)-j) Electric field spatial distribution of the TE-like Floquet-Bloch mode for one period, corresponding to the waveguide geometries shown in a)-e), for upper cladding water (n = 1.32), λ = 1.55 µm, and TE polarization. k)-o) SEM images of the sensors where the SWG waveguides shown in a)-e) were embedded in k) microring resonator, l) slot-SWG with a Mach-Zehnder interferometer, m) phase-shifted Bragg grating resonator, n) microring resonator, o) differential Mach-Zehnder interferometer. Figures k), l), m), n) and o) reprinted with permissions from [177] [175] [179] [155] [180], respectively.</figcaption></figure>

cladding. However, etching windows in the oxide to allow contact with the solution can damage the silicon blocks, even cause their peeling off from the buried oxide layer. Recently, Puumala *et al.* reported a comprehensive study of the fishbone SWG topology [155], demonstrating a performance comparable to the state of the art SWG sensors (in C-band, S<sub>bulk</sub>= 438 nm/RIU and iLOD = 7.1×10<sup>-4</sup>), while offering better manufacturability and lower risk of structural damage. A good compromise between performance and robustness can be achieved by judiciously selecting the width of the central spine, labeled as W<sub>fb</sub> in Fig. 17(d).

SWG waveguides have also been employed in bimodal interferometric sensing architecture, as illustrated in Fig. 17(e). In this case, instead of using two separate arms as in a conventional MZI, a single arm is exposed to the analyte through which two modes with effective indices n<sub>eff,1</sub> and n<sub>eff,2</sub> propagate. Since their evanescent fields have different decay rates in the upper cladding region [Fig. 17(j)], variations in the refractive index of the analyte affect them differently, thus resulting in a differential waveguide sensitivity [181]. This scheme has the advantage of reducing the sensor footprint approximately by a factor of two and increasing its robustness to environmental fluctuations, particularly thermal and mechanical [180]. It has been demonstrated that by engineering the dispersive properties of the first two TE modes in SWG waveguides, high bulk sensitivities in terms of spectral and phase shifting can be obtained — 2270 nm/RIU [182] and 10620×2π rad/(RIU·cm) [183] — when operating in the SWG and slow-light regime, respectively [Fig. 17(o)]. However, the LOD is still one order of magnitude lower compared to conventional silicon nitride waveguides (10<sup>-7</sup> RIU [181]). As the two modes of the bimodal waveguide are generated from the fundamental mode of the input waveguide and converted back to the fundamental mode of the output waveguide, this causes ripples in the output signal, thus
degrading the overall performance of the sensor [180]. In addition, not being able to access these
signals separately, prohibits utilization of coherent read-out schemes, thus requiring complex
techniques for the unambiguous extraction of the phase [184, 185].

The lower refractive index of the aqueous solution (1.32) compared to that of the BOX (1.44) implies that as the field is delocalized, the mode shifts towards the BOX, decreasing the interaction with the analyte. To mitigate this issue, two over-etching techniques have been proposed [186] [187], which, although they achieve bulk sensitivities close to unity, nevertheless complicate the fabrication process and do not substantially improve the LOD.

In the SWG waveguides discussed above, the field delocalization is controlled by judiciously engineering the metamaterial synthesized in the waveguide core. The modal field expands in both vertical and horizontal directions, which, for certain applications, may limit the ability to concentrate the field in the areas of the waveguide where it is of most interest. Furthermore, the maximum evanescent field is limited by leakage into the silicon substrate caused by the finite BOX; this effect is aggravated by the vertical asymmetry of refractive indices. To overcome these limitations, a dual-metamaterial SWG waveguide has recently been proposed [188]. By defining subwavelength-spaced hole patterns with different sizes in the core and lateral cladding, it is possible to independently control the field delocalization in the vertical and horizontal directions. First experimental results on microring resonators in the near-infrared band show the versatility offered by this new SWG topology, with Q factors of the order of 30,000 in temperature sensing experiments with air cladding.

## 6.2. Design guidelines for SWG sensing waveguides

The larger design space provided by SWG waveguides can be exploited to improve the sensor performance, but this may also increases the complexity of the design process. Hence it is important to understand the effects of the new parameters, as well as the restrictions they impose. If we consider the simplest of the SWG sensing waveguides, the segmented SWG [Fig. 17(a)], in addition to the thickness (H), width (W), and polarization, there are two new design parameters: the period (Λ) and the duty cycle (DC) of the SWG. The first needs to be selected such that the waveguide operates in the subwavelength regime, i.e., sufficiently far away from the Bragg condition. The duty cycle needs to be chosen respecting fabrication constraints, particularly in terms of minimum feature size (typically on the order of 70 nm for deep-UV immersion lithography). For a typical 250 nm period, this restricts duty cycles to a range between 0.3 and 0.7. Further design restrictions are imposed by the possible leakage losses to the substrate caused by the delocalization of the field and the finite BOX thickness. To minimize substrate leakage loss in an SOI platform, it is necessary to use a Floquet-Bloch mode effective index above 1.55 for a BOX thickness of 3 µm, and above 1.65 for a BOX thickness of 2 µm, as demonstrated experimentally by Sarmiento et al. [189].

Because the sensitivity can be expressed as the product of two terms (S = S_a S_w), it can be maximized by optimizing each factor separately. In the case of a resonant sensor, the design of the sensing waveguide is particularly important, since it affects both the waveguide sensitivity (S_w) and the architecture sensitivity (S_a = λ_o/n_g). As an example, Fig. 18.(a) shows the waveguide and architecture sensitivities for an SWG waveguide in a resonant Bragg-type architecture at the wavelength of 1.31 µm [164]. The curves were calculated for a wide range of widths, heights, duty cycles and for both polarizations, although for the sake of simple illustration of the principle only those for the TE polarization and waveguide width 300 nm are shown. The behavior observed in the architecture sensitivity is as expected, increasing monotonically as the mode is delocalized and the group index decreases. On the other hand, the waveguide sensitivity shows a maximum as the duty cycle decreases. This is a consequence of two competing effects: delocalization of the field increases the overlap with the analyte however for a weak waveguide the mode is displaced towards the BOX. The maximum occurs for an intermediate optimized duty cycle. Fig. 18.(b)
<figure><img src="image_23.png" /><figcaption>Fig. 18. Exemplary curves for the design of SWG sensing waveguides. Reprinted with permission from [164]. a) Waveguide sensitivity (solid, left axis) and architecture sensitivity (dashed, right axis). b) Bulk sensitivity (solid, left axis) and effective index of the fundamental Floquet-Bloch mode (dash, right axis). The curves were calculated for a segmented SWG waveguide implemented in a Bragg-type resonant sensing architecture operating in O-band ($\lambda = 1.31 \ \mu m$).</figcaption></figure>

shows the product of the two factors or the total sensitivity, which makes it easy to determine the optimum waveguide design. The effective index of the fundamental Floquet-Bloch mode is also plotted to ensure that the optimum design is at a point where leakage losses are negligible. In fact, it is observed that the optimum designs for the different thicknesses are located just at the point where leakage losses can still be neglected if a 3 $\mu m$-thick BOX is used. It is worth mentioning that in simulation, this evanescently-coupled Bragg filter sensor operating in the O-band exhibits a bulk sensitivity of 507 nm/RIU, a quality factor of $4.9 \times 10^4$ and an iLOD of $5.1 \times 10^{-5}$, which constitutes a 10-fold enhancement compared to state-of-the-art resonant-type sensors.

Importantly, to obtain this type of curves it is not necessary to perform an electromagnetic analysis of the entire sensing structure. It is sufficient to calculate the effective index of the fundamental Floquet-Bloch mode in a single unit cell, and its derivatives with respect to $n_c$ and $\lambda$, thus greatly reducing the computation time. This systematic approach to the design of SWG sensing waveguides was initially proposed in [158] and has also been adopted in other works [155, 164, 187], due to its simplicity and efficiency.

## 7. Mid-infrared subwavelength grating waveguides, devices and circuits

The mid-infrared (mid-IR, 2-20 $\mu m$ wavelength range) is a particularly relevant spectral window for sensing and spectroscopic applications as it contains the absorption fingerprints of many chemical and biological molecules [190, 191]. However, exploiting this spectral range, which is an order of magnitude wider than the near-IR (1.3-1.6 $\mu m$), remains an open challenge. To this end it is necessary to develop a material platform with a wide transparency in the mid-IR and compatible with large-scale fabrication, key for the widespread deployment of sensing applications. At the same time, wideband photonic devices are needed to implement multi-target detection systems operating over a broad range of the mid-IR spectrum. For the realization of complex sensing circuits, flexible control of light confinement in the waveguides would be beneficial. This can allow, for example, delocalizing the optical mode to increase light-matter interaction in the sensing waveguides, or tightly confining the optical mode in the interconnection waveguides to minimize the circuit footprint. These challenges can be addressed through modal confinement control and dispersion engineering in SWG devices.
<figure><img src="image_24.png" /><figcaption>Fig. 19. (a) Schematic view of a suspended silicon waveguide with SWG cladding. (b) Electric field distribution of the principal component of the TE mode ($|E_x|$) of the suspended waveguide, calculated for a wavelength of 7.5 µm, along the planes A (left panel) and B (right panel) defined in (a). (c) Effective index as a function of the wavelength for the TE-polarized waveguide mode. Field distribution and effective index are calculated using 3D-FDTD simulations for a waveguide with $t_{Si}$ = 700 nm, $W_c$ = 2.5 µm, $W_{cladd}$ = 7 µm, $l_a$ = 150 nm and $l_g$ = 150 nm.</figcaption></figure>

## 7.1. Suspended SWG waveguides for the mid-IR

The use of the silicon-on-insulator (SOI) waveguides in the mid-IR is limited by the strong absorption of the buried oxide (BOX) layer for wavelengths above 4 µm [191, 192]. This drawback can be circumvented by using suspended silicon membrane waveguides with SWG lateral cladding operating in the full wavelength transparency range of silicon (1.1-8 µm) [193]. Figure 19(a), shows the schematic view of the suspended SWG waveguide. A solid waveguide core (width $W_c$) is laterally cladded by two SWG lattices (gap and anchoring arm lengths $l_g$, $l_a$, respectively) with reduced equivalent index, the latter effectively confining the mode. At the same time, the SWG holes allow the penetration of the etchant (hydrofluoric acid solution) and the removal of reaction products during selective BOX layer removal, while the anchoring segments confer mechanical stability once the Si membrane has been liberated [193]. In addition, the SWG cladding provides new degrees of freedom to control modal confinement [188, 194, 195] and dispersion [196], advantageous for sensing and nonlinear applications.

As an example, Figure 19(b) shows the electric field distribution of the TE mode in a suspended SWG waveguide for a wavelength of 7.5 µm. The cuts A and B are through the SWG gaps and anchoring pillars, respectively. The waveguide has the core width $W_c$ = 2.5 µm, the cladding width $W_{cladd}$ = 7 µm, the silicon thickness $t_{Si}$ = 700 nm, and the SWG cladding with $l_a$ = 150 nm and $l_g$ = 150 nm. As shown in Fig. 19(c), the TE mode has an effective index ≥2 for wavelengths below 7.5 µm, indicating a good confinement within silicon. The waveguide propagation loss can be estimated as the sum of the contributions from material absorption ($\alpha_m$), leakage to the substrate ($\alpha_s$) and to the lateral slabs ($\alpha_l$) [197]. Silicon is transparent for wavelengths below 8 µm [198], while germanium allows low absorption loss up to 14 µm wavelength [199]. Leakage to the substrate and lateral slab are governed by the modal confinement in the vertical and horizontal directions, respectively, which are controlled by the thickness of the guiding silicon layer and the air gap underneath, and the SWG equivalent index [193, 197, 198].

Figure 20(a) shows the operating wavelengths of different types of suspended SWG waveguides developed for the mid-IR wavelength range over the last decade. First proposed in 2014 this
<figure><img src="image_25.png" /><figcaption>Fig. 20. (a) Different types of suspended SWG waveguides operating in the mid-IR, reported since 2014 (references indicated in brackets). First SWG-cladding suspended (b) silicon strip waveguide (reproduced with permission from [193] ©The Optica Publishing Group.), (c) germanium strip waveguide (reproduced with permission from [197] ©The Optica Publishing Group.), and (d) silicon slot waveguide (Reproduced from [194], with the permission of AIP Publishing).</figcaption></figure>

waveguide, shown in Fig. 20(b), achieved propagation loss of 3.6 dB/cm at a wavelength of 3.8 µm [193]. Lower propagation losses and longer operating wavelengths were subsequently demonstrated: waveguide loss of 2 dB/cm near 5.5 µm was reported in [200], and around 3 dB/cm at 6.6 µm and 7.67 µm were shown in [195, 198, 201]. Suspended SWG-cladded germanium waveguides, shown in Fig. 20(c), were recently reported with losses of 5.3 dB/cm near 7.67 µm [197]. Figure 20(d) shows a slot waveguide which increases the interaction of the optical mode with the waveguide surroundings [194]. In this case, the overlap with the air cladding is 54.4% and two-fold improvement compared to strip waveguide (an overlap of 24.6%). The slot waveguide propagation loss was determined as 2.8 dB/cm at 2.2 µm. Non-suspended slot waveguides with SWG core have been used for 5 Gbit/s data transmission at 2 µm wavelength [202]. The 2 µm window offers a promising alternative to expand the wavelength range of communication systems, due to the comparatively small silica absorption and low optical fiber loss.

## 7.2. Devices and circuits

Subwavelength grating waveguides have enabled engineering of modal confinement and dispersion, leading to the development of a wide range of high-performance devices in the mid-IR, including off-chip coupling interfaces [193, 198, 200, 203], power splitters [200, 204, 205] and resonators [194, 206], as shown in Fig. 21.

Efficient coupling of light between optical fibers and photonic chips is key to achieving integrated systems with high performance. Subwavelength gratings have been used to enhance performance of both surface grating couplers and edge couplers, as it is discussed in detail in Section 3 of this review. Figure 21(a), shows the first SWG grating coupler for the mid-IR, implemented in silicon-on-sapphire, with a coupling efficiency of 5 dB for TE polarization at 2.75 µm wavelength [203]. Grating couplers for suspended silicon and germanium waveguides
<figure><img src="image_26.png" /><figcaption>Fig. 21. SWG-engineered mid-IR devices. Fiber-chip grating coupler operating at the wavelengths of (a) 2.75 µm (©2023 IEEE. Reprinted, with permission, from [203]), (b) 3.8 µm (reproduced with permission from [193]. ©The Optica Publishing Group) and (c) 7.67 µm (reproduced with permission from [198] ©The Optica Publishing Group). (d) Free-space-chip facet coupler operating in 5-6 µm band. Reproduced with permission from [200]. (e) 2x2 MMI the wavelength range of 3.1-3.7 µm wavelength range. Reproduced with permission from [204] ©The Optica Publishing Group. (f) Y junction for the 5-6 µm band. Reproduced with permission from [200] ©The Optica Publishing Group. (g) Directional coupler (6.4-7 µm range) tunable by a microelectro-mechanical system. Reproduced with permission from [205]. (h) Ring resonator implemented with suspended slot waveguides, working near 2.2 µm wavelength. Reproduced from [194], with the permission of AIP Publishing. (i) Disk resonator with SWG cladding (2 µm wavelength). ©2023 IEEE. Reprinted, with permission, from [206].</figcaption></figure>

operating near 3.8 µm and 7.67 µm have also been reported [193, 197, 198], see Figs. 21(b) and 21(c). The edge coupler shown in Fig. 21(d) was used for coupling between suspended Si waveguides and free-space beams, with coupling efficiency of 10 dB near 5.5 µm wavelength [200].

Power splitters, including y-junctions, MMIs and directional couplers, are important building blocks in mid-IR devices for sensing where broadband operation is often required. SWG dispersion engineering in non-suspended SOI waveguides has been used to implement a 2x2 MMI with a wide bandwidth of 600 nm near 3.4 µm [204]. Y-junctions have been demonstrated with broad bandwidths for SWG-clad suspended waveguides, showing insertion loss below 1 dB at 5.5 µm [200]. Dynamic control of the coupling ratio in power splitters and combiners allows implementing tunable and re-configurable devices [207]. Tunable splitting ratios of directional couplers at 2 µm wavelength have been achieved using SWG suspended silicon waveguides integrated with MEMS [205].

Ring resonators are often used in silicon photonic sensors as they enhance light-matter interaction while maintaining compact footprints [208]. Figure 21h shows a ring resonator based on a suspended slot waveguide with a quality factor of 12,600 near 2.2 µm [194]. Non-suspended micro-disks with SWG structures have also been demonstrated yielding a sensitivity of ~390.4 nm/RIU and quality factors $Q \sim 1100$ at 2 µm [206].

The development of SWG waveguides and devices in the mid-IR has allowed proof-of-concept demonstrations of complex circuits, such as a toluene sensor [195] and a Fourier transform
<figure><img src="image_27.png" /><figcaption>Fig. 22. Mid-IR circuits incorporating SWG-engineered devices. (a) Toluene vapour sensor with limit of detection of 75 ppm, operating in the 6.4-6.8 µm wavelength range. Reproduced with permission from [195]. (b) Fourier-transform spectrometer with a resolution of 13 cm⁻¹ for wavelengths between 6.4 µm and 6.8 µm. Reproduced with permission from [200] ©The Optica Publishing Group.</figcaption></figure>

spectrometer [200]. The former comprises an integrated sensing circuit on a silicon chip which is enclosed in a gas chamber; an external narrowband laser source scans the absorption peak of toluene around 6.6 µm. The integrated sensing circuit, shown in Fig. 22(a), combines fiber-chip grating couplers, Y-junctions and two 28-mm-long suspended spiral waveguides, achieving a detection limit of 75 ppm with a response and recovery times of 0.8 s and 3.4 s, respectively. While scanning a specific absorption feature using a narrowband laser is an effective strategy for single-target detection, multi-target detection requires the analysis of broader mid-IR spectra. Integrated Spatial-Heterodyne Fourier-transform (SHFT) spectrometers have shown remarkable bandwidth and spectral resolution with improved resilience to fabrication imperfections [209–213]. The first on-chip SHFT spectrometer incorporating SWG waveguide delay lines was reported by Bock *et al*. in 2013 [212]. More recently, SWG suspended Si waveguides have been used in an SHFT spectrometer operating at a wavelength of 5.7 µm wavelength, the longest yet reported for a silicon on-chip spectrometer [200]. The device comprises 19 Mach–Zehnder interferometers, with light being injected and extracted from the chip using free-space beams, lenses and facet couplers. The device achieves a spectral resolution of 13 cm⁻¹ and a free spectral range of 100 cm⁻¹ in a wavelength range of 5.3-5.7 µm.

## 8. Nonlinear and opto-mechanical devices

Nonlinear light-matter interactions in integrated photonic devices have been exploited for many applications [214–218]. Kerr and Brillouin scattering processes are particularly important to realize complex photonic circuits for optical communications, sensing and quantum processing. These nonlinear phenomena are governed by phase-matching and energy conservation laws, and strongly depend on light confinement within the nonlinear material and modal overlap between the
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pump and scattered signals. Kerr and Brillouin effects have been successfully exploited on chip using both strip and rib waveguides [215,216]. Still, these geometries provide limited flexibility for the engineering of the bandwidth and efficiency of the nonlinear scattering. Subwavelength metamaterial engineering has emerged as a powerful strategy to control nonlinear phenomena in nanophotonic waveguides as it introduces additional degrees of freedom to engineer optical confinement and dispersion. By changing geometrical parameters like the SWG duty cycle, it is possible to control the effective metamaterial index to engineer the optical confinement. Periodic waveguides exhibit strong dispersion near the band edge [219]. The position of the band edge can be designed, for example, by changing the grating period, enabling dispersion control. In this section we briefly introduce the fundamentals of Kerr-type and optomechanical Brillouin interactions in integrated photonic devices and discuss how SWG metamaterials have been used to optimize and exploit these nonlinear effects.

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## 8.1. Dispersion engineering for Kerr-based supercontinuum generation

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Third-order Kerr nonlinearities have been widely exploited to realize frequency conversion and spectral broadening of light [214,220]. Supercontinuum generation (SCG) is a particularly promising phenomena allowing the generation of ultrawideband optical spectra from the nonlinear widening of optical pulses (Fig. 23), with applications in telecommunications, spectroscopy, optical coherence tomography and sensing [215]. SCG is a complex process that results from the interplay between many nonlinear phenomena, including self-phase modulation, four-wave mixing and soliton compression and fission. The bandwidth and efficiency of these phenomena is tightly related to phase-matching conditions and waveguide dispersion. In integrated waveguides, SCG is typically produced by soliton fission, a process where high-power femtosecond pulses are compressed and new wavelengths are generated by nonlinear conversion from the pump. Highly efficient conversion can be achieved by the phase-matching between the optical pump and wavelengths in the normal dispersion regime, also referred to as dispersive waves [214]. A dispersive wave is a form of Cherenkov radiation [221] resulting from the splitting of a soliton. Dispersive wave excitation is a nonlinear resonant process where a pulse propagating in a waveguide with higher-order dispersive effects is converted into a shifted frequency. The wavelength of the dispersive waves is determined by the waveguide dispersion and the phase-matching with the nonlinear pulse while the maximum bandwidth achievable with SCG depends on the spectral position of the dispersive waves. In addition to SCG, dispersive waves play a major role in applications in frequency conversion and comb generation [214, 221]. Anomalous group velocity dispersion (GVD) at the pump wavelength is generally required for SCG although recent promising demonstrations have exploited also the normal dispersion regime [222]. Nanophotonic waveguides provide strong modal confinement, with mode areas near $1~\mu\text{m}^2$, that result in a substantial reduction of the power required to trigger the SCG process. Integrated strip and rib waveguides have been used for SCG in the visible, near- and mid-infrared [215]. The GVD is generally engineered by optimizing the waveguide width and height but this approach offers only limited dispersion control.

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Subwavelength metamaterial engineering has been theoretically investigated to control the GVD of silicon waveguides [223,224]. The flexible design of the effective refractive index of the metamaterial and the strong dispersion induced by operating near the Bragg condition can be employed to tailor the waveguide GVD for optimized nonlinear interactions. For example, SWG engineering of the waveguide core as shown in Fig. 24(a) allows one to achieve wideband normal or anomalous GVD between 1.3 $\mu$m and 3.5 $\mu$m wavelength, covering the near-infrared and the shortwave-infrared regions [223]. A calculated anomalous GVD of 10 ps/nm/km in a remarkably wide region between 1.8 $\mu$m and 3.3 $\mu$m wavelength was made possible by the optimization of the dimensions of an SWG-cladded waveguide [223,224].

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When operating close to the Bragg wavelength, waveguide gratings can achieve GVD values
<figure><img src="image_28.png" /><figcaption>Fig. 23. Supercontinuum generation results from spectral broadening of high-power short optical pulses propagating in a nonlinear medium. Interplay between GVD and self-phase modulation compresses and fissions the pump pulses generating a wideband output spectrum. SWG metamaterial engineering allows flexible control of waveguide GVD, key for optimizing the nonlinear spectral broadening.</figcaption></figure>

orders of magnitude larger than strip and rib waveguides, resulting in significant pulse compression over millimeter lengths [219]. This effect has been used in silicon-rich nitride waveguides to compress a picosecond pump pulse, allowing substantial widening of the SCG [225]. In this case, the grating is implemented by adding a periodic array of pillars on the two sides of a strip waveguide, as illustrated in Fig. 24(b). The grating-cladded waveguide is followed by a conventional strip section. The strong anomalous dispersion induced by the grating results in soliton compression and fission that broadens the bandwidth of the SCG in the strip waveguide. This approach allowed the experimental demonstration of a four-fold enhancement of the SCG bandwidth compared to the case without a grating-induced pulse compression, covering the 1.2 µm to 1.65 µm wavelength range.

SCG expanding more than two octaves between the wavelengths of 1.55 µm and 7.5 µm has been recently demonstrated using suspended silicon waveguides with SWG cladding, as depicted in Fig. 24(c) [196]. SWG-based GVD engineering is used to independently control the position of the short-wavelength and long-wavelength dispersive waves allowing a remarkable flexibility in engineering the SCG bandwidth.

Multimodal quasi-phase-matching (QPM) has been extensively used in periodic waveguides to enhance the efficiency of nonlinear phenomena like second harmonic generation and difference frequency generation [226]. Likewise, QPM in silicon nitride waveguide gratings has been used to shape SCG via the selective excitation of dispersive waves. [227]. As shown in Fig. 24(d), the width of the waveguide is modulated periodically, inducing additional dispersion terms in the phase-matching equations. Optimization of the grating dimensions enables QPM between the pump soliton propagating in the fundamental waveguide mode and the dispersive waves in the second order waveguide mode. Dispersive wave excitation enhances the intensity of SCG for specific wavelengths that are determined by the grating periodicity. This additional dimension in SCG design space allows nonlinear wavelength generation for bandwidths that cannot be reached by aperiodic waveguide geometries. This approach was used to experimentally demonstrate up to 20 dB improvement in nonlinear signal generation for specific wavelengths in the 500-700 nm wavelength range.

SWG metamaterial engineering has been proposed to achieve ultrawideband phase-matching and energy conservation among different modes in a waveguide. The SWG geometry has been optimized to yield a parabolic index profile along the waveguide cross-section [228]. The optimized SWG waveguide supports several optical modes with equally spaced frequencies and shared wavevectors. This configuration, that is not feasible with step-index waveguides, provides energy conservation and phase-matching among different waveguide modes at 1.55 µm and 2
<figure><img src="image_29.png" /><figcaption>Fig. 24. (a) SWG silicon waveguide used for GVD engineering [223]. (b) Grating-cladded silicon-rich nitride waveguide allowing pulse compression and soliton fission for enhanced SCG bandwidth. Reprinted with permission from [219]. (c) Suspended SWG-cladded silicon waveguide used for the demonstration of SCG extending between 1.55 µm and 7.5 µm wavelengths [196] ©The Optica Publishing Group. (d) Silicon nitride grating waveguide optimized for dispersive wave excitation based on multimodal QPM. Reprinted (Fig. 1a) with permission from [227] ©(2023) by the American Physical Society. (e) SWG waveguide with parabolic index profile supporting different optical modes with equally spaced frequencies and shared wavevectors, i.e. fulfilling energy conservation and phase matching conditions. ©2023 IEEE. Reprinted, with permission, from [228].</figcaption></figure>

1129 µm wavelengths, with a simulated bandwidth exceeding 400 nm.

1130 **8.2. Subwavelength control of photons and phonons for optimization of Brillouin scattering**
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1132 Brillouin scattering is a third-order nonlinear process entailing the coherent coupling between optical photons and acoustic phonons. In integrated waveguides, this process can be orders of magnitude stronger than the Kerr effect, with great potential for applications in microwave signal processing in the optical domain and quantum signal processing. The optomechanical coupling is mediated by the photoelastic effect and the moving boundary phenomena [216, 217]. The photoelastic and moving boundary effects are governed by the material properties and the waveguide geometry, respectively. The photoelasticity refers to the changes in the electric susceptibility when material is subject to strain [216]. The moving boundary effect is caused by the displacement of the waveguide interfaces due to radiation pressure [229]. The Brillouin gain ($G_B$) is related to the mechanical quality factor ($Q_m$), the photoelastic effect and the moving boundary effect as [230]:

$$G_B \propto Q_m \frac{\omega}{\Omega^2} \cdot |PE + MB|^2, \qquad (14)$$

1143 where $\omega$ is the optical frequency and $\Omega$ the mechanical frequency. The photoelastic factor (PE) is related to the overlap of the optical field with the photoelastic-induced index variation. The moving boundary term (MB) is calculated as the overlap between the optical field and the boundary movement induced by the radiation pressure.

1147 Near-IR photons and GHz phonons in nanoscale silicon waveguides have comparable wavelengths (about 1 µm). Therefore, SWGs can be exploited to engineer the properties (i.e.,
confinement, dispersion, etc.) of both photonic and phononic modes while enhancing the
moving boundary effect. Still, simultaneous confinement of photons and phonons in integrated
waveguides remains an open challenge. While silicon waveguides provide strong confinement
of optical modes, mechanical modes are poorly confined due to the high stiffness of the silicon
core compared to the buried oxide, as shown in Fig. 25(a). One of the promising solutions to
circumvent this limitation is the use of suspended silicon rib waveguides, as illustrated in Fig.
25(b). This approach allowed remarkable demonstrations of optomechanical coupling, including
the first silicon Brillouin laser [231]. Yet, in this geometry there is limited overlap of the field
distributions for the optical and mechanical modes, resulting in a modest Brillouin gain of 1000
W⁻¹m⁻¹.

<figure><img src="image_30.png" /><figcaption>Fig. 25. (a) Silicon wire waveguides provide tight confinement of optical modes. However, the higher stiffness of the silicon core compared with the buried oxide layer results in strong phonon leakage to silicon substrate. (b) Suspended rib silicon waveguides overcome phonon leakage by partially removing the buried oxide.</figcaption></figure>

Optomechanical confinement in silicon SWG waveguides was recently proposed by Schmidt *et al.* [232]. The optomechanical suspended waveguide, presented in Fig. 26, comprises a solid silicon waveguide core and a periodic array of silicon ribs forming an SWG cladding, i.e., providing lateral confinement for the optical mode. The ribs are periodically patterned with a partially etched transverse grating, with dimensions optimized to yield a complete stopband for the mechanical mode. Leakage of the mechanical mode through the rib structure is therefore suppressed by reflection. This geometry achieves a promising calculated Brillouin gain of 1750 W⁻¹m⁻¹ for an optical mode at a wavelength of 4 µm and a mechanical mode with a frequency of 3.35 GHz. Still, the proposed geometry requires three different etch depths, complicating fabrication. Subsequently, a calculated Brillouin gain of 3500 W⁻¹m⁻¹ was obtained for an optical mode at 1.55 µm wavelength and a mechanical mode with 9 GHz frequency in a suspended SWG silicon waveguide, requiring two etch depths [233]. In this case, the mechanical mode is confined by effective acoustic impedance mismatch between the solid waveguide core and the SWG cladding. Genetic optimization allowed optomechanical confinement with a single etch depth, further increasing the mechanical frequency up to 15 GHz and reaching a simulated Brillouin gain of 3300 W⁻¹m⁻¹. Further details on the optimization process are presented in section 10.

Subwavelength structuring of silicon has been recently used to demonstrate the simultaneous
tight confinement of optical and mechanical modes in non-suspended silicon waveguides [233].
Here, the optical mode is confined by effective refractive index contrast, as in conventional SWG
waveguides while the mechanical mode is confined by destructive interference mitigating phonon
leakage to the substrate. This approach allowed the implementation of optomechanical resonators
and the experimental demonstration of optical excitation and readout of mechanical modes with
a quality factor of 1120 and a frequency of about 0.6 GHz. This is the largest yet achieved
mechanical quality factor for an optomechanical resonator in SOI waveguides with no buried
oxide removed, one order of magnitude higher than the previous demonstrations in non-suspended
<figure><img src="image_31.png" /><figcaption>Fig. 26. (a) First proposal of an SWG optomechanical silicon waveguide. The optical mode is confined by effective metamaterial index difference. The mechanical mode is confined by stopband effect in the periodic cladding. Reprinted with permission from [232] ©The Optica Publishing Group. (b) SWG-engineered optomechanical waveguide achieving mechanical confinement by acoustic impedance mismatch between the waveguide core and periodic cladding. Reprinted with permission from [234] ©The Optica Publishing Group. (c) SWG-mediated tight confinement of optical and mechanical modes in non-suspended silicon waveguides [233]. Optical mode is confined by SWG index difference. Mechanical mode is confined by destructive interference of phonon leakage to the substrate.</figcaption></figure>

silicon structures [235]. For comparison, the mechanical quality factor of suspended silicon structures is typically in the range of 2000-3000, for ambient conditions [236, 237].

# 9. Resonant metamaterial waveguides

## 9.1. Mie resonant nanoparticles

From elementary electromagnetic scattering theory it is known that the optical properties of dielectric particles depend strongly on the ratio of incident wavelength to particle size. Rayleigh scattering, which is characterized by a dipole-like profile of radiated light and a $\lambda^{-4}$ dependence of the scattered intensity, is observed for particles much smaller than the wavelength of light. Dielectric particles with a larger size, on the order of the wavelength of the incident light, can support multipolar Mie resonances that fundamentally affect the particle's scattering properties. For the particular case of dielectric spheres, Mie theory [238] provides exact expressions for the scattered fields and cross-sections. The scattered field is expressed in terms of the electric and magnetic coefficients of an expansion in a series of spherical harmonics. These coefficients correspond to the contributions to the field from the various electric and magnetic multipoles. The lowest-order resonance is the magnetic dipole resulting in a peak of the scattering efficiency for a wavelength approximately equal to the index of refraction times the diameter of the particle, i.e. $\lambda \approx nd$. For decreasing wavelength, this peak is followed by the electric dipole resonance, magnetic and electric quadrupoles and other higher-order multipoles.

Similar Mie resonances are also observed in particles of different, non-spherical shapes, even though an exact analytical solution of Maxwell's equations is not generally available and numerical methods have to be used to calculate the scattering properties of such particles. The observation of Mie resonances in particles with shapes that are amenable to standard planar semiconductor nanofabrication has led in recent years to a large number of research works in the area of dielectric metasurfaces [240]. An example is the silicon nanodisk shown in Fig. 27(a) with its calculated scattering cross-section. Fig. 27(b) illustrates the radiation profiles of the electric and magnetic dipole resonances. Mie resonant particles have been extensively studied
<figure><img src="image_32.png" /><figcaption>Fig. 27. (a) Scattering of a silicon disk with a radius of 200 nm and a height of 260 nm. Electric and magnetic dipole resonances are clearly visible in the scattering cross section. (b) Schematic radiation patterns of electric and magnetic dipole modes. (c) Uni-directional forward scattering by superposition of electric and magnetic dipole resonance radiation patterns (Huygens' particle). Reprinted from [239].</figcaption></figure>

and used for applications such as lensing [241], information and image processing [242] machine learning [243], quantum state detection [244], lidar [245], lasing [246], photonic displays [247], quantum light sources [248] and spectroscopy [249]. Compared to plasmonic metamaterials, which generally suffer from significant optical losses due to the presence of free carriers in the metallic materials, all-dielectric metamaterials offer the potential to build high-efficiency functional nanostructures.

An interesting aspect of Mie resonant particles, and metamaterials based on them, is the potential to tailor their optical properties by choosing an appropriate combination of material, size and shape to achieve desired scattering profiles. A particularly important example is the design of particles with overlapping resonances of different multipolar orders. It was discovered by Kerker et al. [250] that backscattering is suppressed for so-called “Huygens’ particles” with spectrally overlapping magnetic and electric dipole resonances, now commonly referred to as the “Kerker condition”, which can be mathematically expressed as:

$$p = \frac{\sqrt{\varepsilon_r m}}{c} \qquad (15)$$

where *p* and *m* are the electric and magnetic dipole moments, respectively, $\varepsilon_r$ is the relative dielectric permittivity and *c* is the speed of light. This effect is illustrated in Fig. 27(c) and has been exploited in highly transmissive Huygens’ metasurfaces [251]. The subwavelength metamaterial waveguides discussed so far in this review can be regarded as arrays of silicon nanoparticles acting as Rayleigh scatterers, at least insofar as no resonant effects are observed or exploited in their design. We now turn our attention to the promising emerging research work focussed on using Mie resonant particles for waveguiding in integrated optics.
## 9.2. Waveguiding by chains of resonant particles

The guiding of light by chains of coupled nanoresonators, as conceptually shown in Fig. 28(a) [252], has been studied theoretically [253,254] and by simulations [255]. First experimental demonstrations are quite recent. In 2017, Bakker et al. [256] performed a near-field scanning optical microscopy (NSOM) study of light coupled into a chain of amorphous silicon nanodisks fabricated on a quartz substrate, as shown in Fig. 28(b). The intensity profile of the guided mode, excited by vertical illumination near the end of the chain, was measured and compared to the calculated results, as shown in Figs. 28(c) and (d). These waveguides operated in the lowest frequency band of the dispersion diagram, like a conventional SWG waveguide, at a wavelength of 960 nm, spectrally close to the magnetic dipole resonance of the individual particles. Light guiding was demonstrated over distances of 500 µm with a lowest measured propagation loss of 5.5 dB/mm. This relatively high loss is partially caused by the material absorption of the amorphous silicon at the operating wavelength. In a follow-up paper by the same group [257], waveguides comprised of silicon nanodisks were demonstrated at the optical telecommunications wavelength of 1550 nm on a standard silicon-on-insulator platform. Propagation losses were reduced significantly to 1-3 dB/cm, comparable to conventional silicon wire or SWG waveguides. In addition, these waveguides exhibited slow light behavior near the photonic bandgap. It should be noted that the waveguide was operated on the long wavelength tail quite far from the fundamental magnetic dipole resonance of the nanodisks at approximately 1260 nm. Resonance effects can therefore be expected to have a relatively weak influence on the behavior of this nanochain waveguide.

<figure><img src="image_33.png" /><figcaption>Fig. 28. (a) Conceptual schematic of light propagation along a chain of Mie resonant dielectric nanoparticles. Reprinted from [252]. (b) Experimental demonstration of optical waveguiding by a nanoparticle chain using near-field scanning optical microscopy (NSOM). (c) NSOM image of light propagating along the chain. (d) Calculated field profiles corresponding to image in (c). Reprinted from [256].</figcaption></figure>

## 9.3. Huygens' metawaveguides

A very recent study [258] has provided an experimental breakthrough by demonstrating for the first time the effects of resonance engineering of individual nanoparticles on the optical properties
<figure><img src="image_34.png" /><figcaption>Fig. 29. (a) Scattering cross section and profile (left panel) of a cuboid silicon Huygens particle in silicon dioxide matrix with dimensions w = 515 nm, l = 315 nm, h = 220 nm, with strong backscattering suppression in the telecom C-band due to overlapping electric and magnetic dipole resonances (right panel). (b) Dispersion diagram of a chain of silicon Huygens' particles (left) and multipole decomposition of the field profile of a single period of the waveguide (right) showing locking of the electric and magnetic resonances over the upper branch of the dispersion relation (red). Scanning electron micrographs and calculated fields of functional silicon Huygens' metawaveguide components: Bends (c) and Y-splitters (d). Reprinted from [258].</figcaption></figure>

of a nanochain waveguide. In particular, it was shown that for cuboid silicon particles with dimensions 315 nm × 515 nm × 220 nm surrounded by a silicon dioxide matrix, the electric and magnetic dipole resonances overlap at 1550 nm wavelength, as shown in Fig. 29(a) (right panel). This means that the Kerker condition for the suppression of backscattering is fulfilled and as a result the particles exhibit the highly asymmetric scattering profile shown in Fig. 29(a) (left) with a forward-to-backward scattering ratio of approximately 8. The dispersion diagram of a periodic waveguide composed of these silicon Huygens' particles was calculated, as shown in Fig 29(b). It was found that over a spectral region between approximately 1500 nm and 1600 nm wavelength, the electric and magnetic moments calculated over a single period were
locked together in equal magnitudes. This is the wavelength range covered by the first frequency band above the fundamental photonic bandgap and denoted "Huygens' band" in Fig. 29(b) (right). The Kerker condition is therefore preserved for the coupled array of particles over the approximately 100 nm wide spectral window covered by the Huygens' band. Experiments with the Huygens' metawaveguides fabricated on a standard silicon-on-insulator platform were presented that demonstrated light propagation over almost 8 mm distance with a loss of 4 dB/cm. Light was coupled from a conventional photonic wire waveguide to the metawaveguide via specially designed adiabatic couplers with mode conversion losses of 0.35 dB. The group index was found to be negative over the entire Huygens' band, consistent with the calculated dispersion diagram. It was shown that backscattering from facets or defects such as missing particles was strongly suppressed, providing experimental evidence that the resonance engineering of the particles affected the optical properties of the metawaveguide. Finally, some functional components based on the Huygens' waveguide were demonstrated. Waveguide bends, as shown in Fig. 29(c), with tight bend radii of 2.7 µm and 94% transmission indicate that the coupled resonances support strong light guiding beyond the limit expected from conventional waveguiding by total internal reflection. A waveguide Y-splitter based on a discontinuous transition within a single period of the metawaveguide, as in Fig. 29(d), was shown to have comparable performance to conventional adiabatic transition splitters with much larger footprint. The suppression of backscattering from defects or waveguide discontinuities is thought to be crucial for achieving good performance of such a strongly adiabatic mode converter device.

In summary, resonant metamaterial integrated photonics is an emerging field of subwavelength nanophotonics seeking to exploit engineering of Mie resonant nanoparticles to compose waveguide components. Recent experimental studies show that such resonant metawaveguides can be created on CMOS compatible standard silicon-on-insulator platforms at typical telecommunications wavelengths and they can be integrated seamlessly with conventional on-chip waveguide circuits. This will open the door for resonant metawaveguides with tailored light-matter interaction and novel optical properties to be rapidly developed. The novel waveguides can be expected to be exploited in many existing and emerging applications of silicon photonic integrated circuits, such as telecommunications, sensing, beam steering and quantum optics [259].

# 10. Metamaterial devices designed with optimization and machine learning algorithms

The use of metamaterials greatly expands the toolbox available to the photonic designer, allowing an unprecedented control over the optical properties of the materials in integrated photonics. Most of the devices discussed in the previous sections are based on periodic or semi-periodic nanostructures whose unit cells are realized with simple geometrical shapes, for example cylinders or cuboids. Their design is often performed conventionally (i.e., without advanced algorithms) and relies on the possibility to develop effective models describing light propagation in the metamaterial, as we discussed in Section 2. Removing constraints on periodicity or introducing non-conventional shapes for the unit cells can substantially expand design capabilities. Recently, this strategy has been used to demonstrate increasingly compact structures and multi-functional devices. As a drawback, modeling the optical behavior of such metamaterials becomes challenging. At the same time, the lack of periodicity or simple shapes largely increase the number of design variables that must be handled simultaneously, rendering manual selection of parameter unfeasible. In order to effectively exploit this class of metamaterials new design approaches became critical and many different tools have been investigated. In this section, we refer to these approaches as "inverse design", insofar as they start from a desired functionality and aim to retrieve backwardly the device geometry that allows to achieve it. We will discuss some of the most common techniques based on optimization and machine learning algorithms and review recent examples of metamaterial-based integrated photonic devices designed with the
help of these tools. It should be noted that the same techniques can be equally applied to devices that do not make use of metamaterials, or to free-space optics structures such as metasurfaces. These examples are not discussed here and the readers can refer to these recent review articles for more information [260, 261].

### 10.1. Gradient-free optimization

Gradient-free algorithms have been among the first explored due to their versatility and relative ease of implementation. A set of parameters $x = x_1, ..., x_N$ describes the geometry of the device and a figure of merit $F(x)$ is defined for a specific design objective (e.g., the insertion loss, the spectral bandwidth, the reflectivity, etc.). The optimization problem can be defined as

$$ \begin{array}{ll} \underset{x}{\text{minimize}} & F(x) \\ \text{subject to} & H_i(x) = 0, \quad i = 1, ..., h \\ & G_j(x) \le 0, \quad j = 1, ..., g \end{array} \qquad (16) $$

with $H_i$ and $G_j$ constraint functions of the parameters. Multi-objective optimization, i.e., the simultaneous optimization of multiple figures of merit, is commonly achieved combining different metrics into a single figure of merit. The optimization algorithm implements a procedure (i.e., heuristic) that tries at each iteration to improve the value of $F$ without requiring knowledge of the gradient $dF/dx$. For example, genetic optimization mimics biological evolution starting from a random population of possible solutions $x^i$, selecting the ones resulting in the best values for $F$, mixing them and applying random variations in order to generate an updated population. This procedure is repeated until the particular set $\bar{x}$ that either maximizes or minimizes $F$ is identified. The advantage of this strategy is that, since in many practical cases the function $F(x)$ is not known analytically and its value needs to be obtained by numerical methods, the gradient $dF/dx$ could be tricky or computationally expensive to calculate accurately. Moreover, $F$ is often a highly non-convex function of $x$, with a large number of local maxima or minima. Exploiting gradient information to guide the optimization is very likely to trap the algorithm in the vicinity of one of these local optima. On the contrary, heuristics typically include some sort of randomization that makes them suitable for global optimization, i.e., in a high probability of finding the overall maximum or minimum of $F(x)$. However, gradient-free algorithms are often computationally inefficient. Since the design space is stochastically explored, a large number of evaluations of the figure of merit $F$ for different parameter sets $x_i$ is commonly needed, requiring an extensive computing time, especially if complex simulations are required, e.g., 3D finite difference time domain (3D-FDTD). Moreover, the size of the design space and hence the number of evaluations increase exponentially with the number of parameters (i.e., the size of $x$). Likewise, the number of iterations required by the algorithm to reach convergence often grows in large design spaces. This poor computational scaling, sometimes referred to as the *curse of dimensionality*, limits in practice the use of gradient-free optimization to problems with a few hundreds up to about a thousand of parameters.

In the photonics research, genetic optimization and direct binary search are among the most widely exploited heuristics, although particle swarm, ant colony, and simulated annealing have also been successfully used. [262] A wide range of metamaterial-based photonic devices have been designed using gradient-free algorithms, ranging from periodic SWGs to free-form metamaterials. Nuño Ruano et al. [263] proposed the suspended silicon SWG waveguide reported in Fig. 30(a). The unit cell of the periodic metamaterial waveguide was genetically optimized in order to achieve simultaneous confinement of near-infrared photons (optical modes) and gigahertz phonons (mechanical modes). To this purpose, the genetic optimizer was coupled to a multi-physics simulator for the calculation of both the optical and mechanical modes of the waveguide, and the related Brillouin gain, which was used as the figure of merit $F$. The
<figure><img src="image_35.png" /><figcaption>Fig. 30. (a) A suspended silicon waveguide with metamaterial cladding for optical and mechanical mode confinement [263]. (b) A three-port wavelength demultiplexer [264]. (c) A polarization splitter and rotator [265]. (d) A multi-mode optical hybrid [266]. Devices shown in (a) and (b) have been optimized with a genetic algorithm and in (c) and (d) using direct binary search.</figcaption></figure>

optimization involved 8 parameters describing the shape of the unit cell and it was achieved using a population of 50 individuals and 30 generations. The optimized structure shows a high Brillouin gain up to $3310 \text{ W}^{-1}\text{m}^{-1}$ and it is designed for a single etch step, which is an important practical advantage compared to previously proposed optomechanical waveguides requiring several etch steps. Guo et al. exploited a periodic SWG metamaterial to design a six-channel mode (de)multiplexer, a 90° optical hybrid, and a two-channel flat-top wavelength demultiplexer on a conventional 220 nm SOI platform [267]. The gradient-free, covariance matrix adaptation evolution strategy was used to design the shapes of the devices. Successive runs of the optimizer were performed by gradually increasing the number of parameters and the complexity of the structures through manual intervention, slowly guiding the optimization towards the desired global optimum. The complete optimization involved more than a hundred of parameters and required several hundreds of generations to converge. The use of SWGs provided devices with length comprised between 12 µm and 18 µm and measured insertion loss below 1 dB.

Gradient-free optimization has also been successfully used for non-periodic metamaterial structures, constituting one of the first applications of genetic algorithms to photonic design [268]. Liu et al. reported ultra-compact wavelength demultiplexers realized by placing a number of silicon pillars with a fix shape within the optimization area [264]. The position of each structure was optimized with a genetic algorithm, resulting in a non-periodic bi-dimensional metamaterial. The three-port wavelength demultiplexer shown in Fig. 30(b) was realized with 42 rectangular pillars with a size of $100 \times 200 \text{ nm}^2$, i.e. ensuring a good fabrication tolerance, for a compact area of only $1.4 \times 1.8 \text{ µm}^2$. The figure of merit was constructed as the weighted sum of the average of the transmission through each output port while minimizing the level of noise. The algorithm used a population of 30 individuals and converged within 200 generations. Channel passbands were centered at 1180 nm, 1460 nm, and 1700 nm, with experimental transmission ranging from 0.45 to 0.71.

An even larger design freedom can be achieved exploiting digital metamaterials. The design area is divided into a grid of square or circular pixels with typically a size of about 90 – 130 nm to guarantee compliance with fabrication specifications. This approach can effectively synthesize free-form metamaterials without the preliminary definition and parametrization of the shape of the unit cells. In this case, direct binary search is a well suited optimizer as it often
converges in a few iterations, commonly less than 10. Devices incorporating digital metamaterials have been demonstrated also using genetic optimization [269]. Digital metamaterials are a popular choice for the design of compact polarization control devices, with examples including a polarization beam splitter with experimental loss on the order of 1 dB and extinction ratio exceeding 10 dB [270] and polarization rotators with 1 dB losses and extinction ratios of up to 20 dB [269, 271]. Figure 30(c) reports a silicon polarization splitter and rotator demonstrated by Liu et al. [265]. The 7.92 × 1.8 µm² area was divided in 990 square pixels and the figure of merit for the optimization was constructed as the weighted sum of insertion loss, cross-talk, and extinction ratio for both TE and TM polarization states. The best performing device showed an experimental insertion loss of 0.2/1 dB for TE/TM polarizations, cross-talk below -20 dB, and polarization extinction ratio higher than 20 dB.

Digital metamaterials have also been used for designing multimode waveguides and devices. The compactness and good optical performance that have been achieved recently allowed to go beyond the demonstration of isolated devices, combining several components into complex multimode circuits entirely based on inverse-designed structures. For example, using silicon digital metamaterials Liu et al. demonstrated three-mode bends, crossings, and demultiplexers with micron-scale sizes, insertion loss below 1 dB, and cross-talk consistently below -20 dB across the 1500 nm - 1580 nm wavelength range [272]. These components were combined in a three-mode circuit capable of sustaining on-chip data transmission up to 112 Gbit/s for each mode. Exploiting a similar concept, Zhou et al. demonstrated the mode-demultiplexing optical hybrid illustrated in Fig. 30(d) [266]. The circuit includes a mode demultiplexer, single-mode bends and crossings, and 90° optical hybrids. It allows to demultiplex 4 modes from a multimode waveguide and perform coherent detection on all of the four modes. Each device is approximately square with side between 2.4 µm and 4.8 µm, comprising 400 to 1600 holes. The entire circuit occupies an area of only 20 × 30 µm², with total insertion loss of 5 dB and crosstalk of -15 dB.

## 10.2. Gradient-based optimization of free-form metamaterials

As a solution to the poor scaling of gradient-free optimization algorithms, the gradient $dF/dx$ can be computed at each iteration cycle and used to guide the optimization towards the maximum (or minimum) of the figure of merit F. The main advantage is that using the information on the gradient makes the optimization more efficient in terms of computational cost and it is hence a convenient choice when the design space is very large (e.g., more than 1000 parameters) and simulations are computationally intensive. As a drawback, in large design spaces with many maxima and minima, the starting point of the optimization strongly determines its final result which subsequently is only guaranteed to be a local optimum, potentially far from the desired overall global optimum [273]. Nevertheless, even when the global optimum is not reached, highly-performing local optima may still be good solutions from the practical point of view. First-order gradient descent and its various adaptations, and quasi-Newton methods such as Broyden-Fletcher-Goldfarb-Shanno (BFGS) are typical examples of gradient-based algorithms employed in photonics problems [274, 275].

An essential requirement for gradient-based algorithms to be efficient in large design spaces is for the gradient information to be easy and fast to compute. Simple numerical differentiation by means of finite differences can lead to a large number of simulations when the figure of merit depends on many parameters. The large success of gradient-based optimization in photonics is mainly due to the use of the adjoint method for gradient computation [274, 276], albeit other efficient approaches such as automatic differentiation have also been demonstrated [277, 278]. As shown in Fig. 31(a), being $AE = b$ the direct electromagnetic problem, with A the Maxwell's linear operator, E the electric field in the optimization region, and b a vector representing the
<figure><img src="image_36.png" /><figcaption>Fig. 31. (a) Schematic representation of the adjoint method. Using only two simulations (the direct and the adjoint) it is possible to compute the gradient of the figure of merit F with respect to the dielectric permittivity distribution. Gradient-based optimization and the adjoint method have been used to design devices based on free-form metamaterials, including (b,c) wavelength demultiplexers [279, 280], (d) fiber-chip couplers [281], (e) mode demultiplexers [282], (f) nonlinear devices [276], and (g) on-chip particle accelerators [283].</figcaption></figure>

1434 source, the adjoint problem can be formulated as

$$A^T E_{\text{adj}} = - \left( \frac{\partial F}{\partial E} \right)^T . \qquad (17)$$

1435 In Eq. (17), $E_{adj}$ is the electric field in the optimization region computed as a solution of the adjoint problem and F is the figure of merit. As a result, the adjoint method allows to use only two simulations (the direct and the adjoint one) to compute the gradient of the figure of merit F with respect to the spatial distribution of the dielectric permittivity and hence the design parameters x, making the computational cost of dF/dx almost independent on the number of parameters. 1436 1437

1438 1439 Optimization based on the adjoint method for gradient calculation has been successfully applied to many photonic devices with shapes described by tens of parameters [274, 275], and recently also to digital metamaterials with hundreds of pixels [284]. Additionally, its ability to address very large design spaces with great efficiency has been a cornerstone in achieving 1440 1441 1442 1443 truly free-form metamaterials. In what is usually referred to as topology optimization, the 1444 1445 permittivity distribution within the design area is represented using the mesh of the simulation, which may include hundreds of thousands of cells or more [282, 285] and is iteratively modified 1446 1447 1448 by the algorithm without requiring the definition of predetermined shapes or other geometrical restrictions, not even at pixel level as done in digital metamaterials. Arbitrary shapes can also be 1449 1450 conveniently represented using the level set method [274]. 1451

1452 1453 1454 1455 1456 1457 1458 Because of the use of gradient-based algorithms, strategies are required to avoid poor-performing local optima, including multiple restarts with different initial guesses which however affects the efficiency of the optimization, or the use of a manually designed device as a starting point which nevertheless implies a limited exploration of the design space. An effective approach that is commonly applied with good results is to perform a first optimization allowing permittivity in each point of the design area to assume any value between that of the waveguide core and that of the cladding (continuous optimization). In a second step, a binarization process is applied, e.g., through filtering and projection [286], and the resulting structure is then used as the initial guess for a second, discrete optimization. 1459

Irrespective to the chosen optimization strategy, an important issue for topology optimized
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metamaterials is represented by fabrication constraints and tolerances. While in digital meta-
materials minimum feature size can be reasonably controlled by defining the size and shape of
the pixels, topology optimized structures typically include small features that are beyond the
capability of the fabrication technology. Moreover, fabrication robustness needs to be included
within the optimization process to ensure the devices still exhibit an acceptable performance
for given fabrication errors. A first possibility is to act during the continuous optimization step
(where permittivity can assume any value between that of the core and the cladding) to guide
the optimizer towards structures that are already quasi-binary, for example using biases [287].
Alternatively, the objective function can be crafted to obtain optimized devices that maximize the
optical energy within the core regions, which can result in nearly binarized structures without
small features [280]. Another possibility is to act on the binarization and perform it progressively
with a varying projection function [278]. Lastly, it is also possible to directly introduce fabrication
constraints in the final discrete optimization step [288].

Early demonstrations of topology optimization in photonics focused on photonic crystal
structures [285] and since then several classes of integrated devices have been shown, primarily
in the SOI platform. Reducing device footprint compared to conventional geometries has been
the main target. For example, wavelength demultiplexers with areas below 3 × 3 µm² have
been achieved [287, 289]. Channel spacing is commonly of several tens of nanometers and
experimental insertion loss and crosstalk usually are about 2 dB and -10 dB, respectively. An
interesting example has been reported by Huang *et al.* who combined wavelength demultiplexing
and focusing in a single, multi-functional device, shown in Fig. 31(b) [279]. The device has a
footprint of 2.4 × 10 µm² and it is directly connected to a 10 µm fiber coupler on one side and
two 0.48 µm waveguides on the other, without requiring any additional taper. Light coming from
the fiber at wavelengths of about 1520 nm and 1580 nm is demultiplexed and focused in the
two waveguides with experimental losses of 2 dB (excluding the fiber-chip coupling loss) and
crosstalk of -25 dB and -12 dB for the respective wavelengths. The research efforts that focused
on including fabrication constraints and imperfections produced devices with steadily improved
performance. Zhang *et al.* recently reported the wavelength demultiplexer shown in Fig. 31(c),
demonstrating experimental losses as low as 1 dB and crosstalk of almost -20 dB [280].

Hammond *et al.* exploited topology optimization to address the challenging design problem of
vertical grating couplers with both single- and dual-polarization operation [281]. They employed
a dual-layer structure made of silicon and poly-silicon to brake the vertical symmetry, as shown in
Fig. 31(d). The optimization involved a comparatively large area of 10 × 10 µm² and resulted in
two different patterns for the two layers, approaching a 3D metamaterial structure. Experimental
measurements of the device, realized in a commercial foundry respecting standard design rules,
achieved insertion loss of -4.7/-7 dB for the single/dual-polarization design, with back-reflections
of -15 dB.

Multimode devices have also received a significant interest, such as the two-mode and three-mode demultiplexers reported by Frellsen *et al.* [282], as shown in Fig. 31(e). The best experimental results have been obtained for the two-mode design with an ultra-compact footprint of 2.6 × 4.2 µm², insertion loss of less than 1.2 dB, and extinction ratio larger than 14 dB in the C band. As in the case of digital metamaterials, the improvement in performance and reliability of topology-optimized multimode devices has allowed their incorporation in larger systems. Yang *et al.* recently reported on-chip communication using four modes and seven wavelength carriers generated by a spectrally flattened comb, reaching an aggregate capacity of 1.12 Tbit/s. The four modes were multiplexed and then demultiplexed using a topology optimized device with an area of 6.5 × 6.5 µm² with minimum feature size of 80 nm, showing experimental insertion losses smaller than 0.8 dB and crosstalk below -18 dB [290].

Recent applications of topology optimization also includes dispersion engineering [291] and
the design of nonlinear devices, as proposed by Hughes *et al.* through an extension of the
adjoint method [276]. As an example, they designed a compact photonic switch using Al₂S₃ and Kerr nonlinearity. As shown by the 2D simulations results in Fig. 31(f), by increasing the input power from 10⁻⁹ to 0.057 W/µm, light pulses were routed on chip to two different output ports. As another example, Fig. 31(g) shows the on-chip particle accelerator reported by Sapra et al. [283]. The device is based on a 3-µm-wide periodic structure pumped from the side by a pulsed laser exciting a slab waveguide mode. Moving electrons interact with the non-homogeneous electromagnetic field within the periodic structure and are accelerated by the Lorentz force [292]. Inverse design was used to ensure the periodic structure provides phase matching between the light pulse propagating in the slab waveguide and the transversely propagating electrons.

### 10.3. The role of machine learning in metamaterial design

<figure><img src="image_37.png" /><figcaption>Fig. 32. (a) Use of a neural network for forward and inverse photonic design [293]. (b) Schematic representation of how dimensionality reduction can be used to describe a device with a small number of parameters [294]. Machine learning has been recently introduced in the design of metamaterial devices, including (c) waveguide couplers [295], (d) universal transmission matrices [296], and (e) surface grating couplers [297]</figcaption></figure>

Machine learning includes numerous strategies and algorithms that aim to extract information from a provided set of data in order to carry out a specific task without the need to be specifically programmed for it. While many classes of algorithms can be used to achieve this objective, artificial neural networks are the most popular one [260]. Machine learning algorithms commonly require a preliminary phase of training during which the available data are used to tune the internal parameters of the algorithm, e.g., the weights in an artificial neural network, often through gradient-based optimization. Once trained, the algorithm is usually fast to run, even when performing rather complex tasks. Nevertheless training can easily require tens of thousands of data sets obtained by complex simulation, as in the case of photonic design problems. The overall efficiency of machine learning approaches should hence be evaluated on a case by case basis.

As schematically represented in Fig. 32(a), in the field of photonics machine learning has mainly been used to perform two types of task. The first one is the prediction of the optical
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response of a structure, a *direct* problem often referred to as *forward design*. In this case, the neural network acts as a fast *surrogate model* that can provide an approximate solution to the electromagnetic problem in a time that is orders of magnitude shorter compared to conventional simulation. These algorithms fall under the class of *supervised learning*. The training is performed using a set of structures with their corresponding optical responses (labels) and proceeds by minimizing the difference between the responses predicted by the neural network and the known ones. Various physical quantities can be predicted, e.g., the transmission spectrum [293, 298], the output intensity profile [296], or fundamental quantities as the internal fields that can then be used to extract a higher-level response [299, 300]. The same strategy can be used also to address non-electromagnetic problems, such as predicting how fabrication tolerances would alter the geometry of the structures in order to apply preventive corrections [301].

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Once trained, the surrogate model can be used in combination with an optimization algorithm to obtain a prediction of the figure of merit during each iteration. Direct parameter sweeps [298], gradient-free [302] and gradient-based algorithms [303] have been demonstrated. By using the surrogate model, the optimization can be performed at a much higher speed compared to the case where each response needs to be computed via numerical simulations. It should be kept in mind though that the predictor only provides an estimation of the figure of merit, which can be inaccurate or even completely wrong, particularly outside of the parameter range covered by the training data [260, 304]. A simulation-based final optimization is usually required.

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A second possibility to use neural networks is to directly address the *inverse design* problem as depicted in Fig. 32(a). The goal is to train a network that takes as an input a desired optical response and provides at the output a structure which gives such a response. This approach circumvents the need for an optimizer and it can be a highly efficient solution, in terms of computational time, to the design problem. Nevertheless, the solution of the inverse problem represents a significant challenge compared to the direct problem because the same optical response can often be produced by different structures (*one-to-many problem*). In such a case, the training of a neural network usually fails to converge. Different strategies have been proposed to mitigate this issue, including tandem network architectures [296, 305] and the use of generative networks such as generative adversarial networks or autoencoders [296].

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Yet another way machine learning can be used to empower the design of photonic devices is by means of dimensionality reduction, a methodology classified as *unsupervised learning* and schematically represented in Fig. 32(b). With unsupervised learning techniques, the training requires a set of data without any associated label, e.g., a set of devices without their corresponding optical response. The goal of dimensionality reduction is not to model the relation between a device and its response (in either the direct or inverse direction), but rather to simplify the representation of the device by reducing the number of parameters *x* used to describe it. This is possible because parametrizations typically used for photonic devices often include a high degree of correlation that can be removed with a minimal loss of information. Different algorithms can be used to perform this tasks, including principal component analysis [294], neural networks in the form of autoencoders [306], and also low-dimensional representations of a structure in the Fourier domain [307]. The use of a more efficient representation for the design space has many advantages. A small number of parameters can be swept in order to simulate the response of all possible combinations and exhaustively map the design space. In this way, multiple figures of merit can be efficiently evaluated and visualized, as well as structural differences and limitations of different design choices. Handling a smaller number of parameters also makes it easier to understand the role that each parameter plays in the response of the device, gaining physical insight on the device under analysis [308]. The same approach can be applied not only to the design space but also to the response space, i.e., to the parameters used to represent the optical response of a device. This is also an effective solution to address the *many-to-one problem* in inverse design [306, 309].
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A third class of machine learning tools that has found recent applications in photonics design falls in the category of *reinforcement learning*. This includes iterative algorithms which do not require labeled datasets for training but rather iteratively solve the optimization problem by a sequence of actions and their associated positive or negative rewards [295,310].

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Machine learning has also been successfully used in the domain of metasurfaces and these algorithms are among the latest computational tools also explored for the design of metamaterial-based photonic devices [303–305]. Da Silva Ferreira et al. [304] and Bor et al. [295] exploited two different approaches based on machine learning to address the problem of designing compact digital metamaterial couplers connecting two waveguides with different widths. In [304], the authors partitioned a 4.2-µm-long area into circular pillars of 300 nm in diameter, each of which could have a different refractive index from the available set. A neural network was trained to predict the coupler efficiency for a given distribution of the pillars. The training required a dataset of about 6000 samples and a comparatively long time equivalent to about 20 finite element simulations. Once trained, the network was able to predict the coupling efficiency five order of magnitude faster than the reference simulation. In [295], the 4 × 2 µm² area shown in Fig. 32(c) was divided in 100 × 100 nm² silicon pixels whose distribution was then optimized by means of a reinforcement learning algorithm to maximize the efficiency. Both approaches reached similar efficiencies of 85% and 87%, respectively. Tahersima et al. exploited a digital metamaterial to implement 1 × 2 power splitters with arbitrary splitting ratios, as shown in Fig. 32(a) [293]. 20 × 20 holes with a diameter of 45 nm were etched in an area of 2.6 × 2.6 µm². The authors successfully trained both a forward network predicting the spectral response at the two output ports of the device given the location of the holes, as well as an inverse network providing the hole distribution necessary to obtain a desired splitting ratio. The neural network required 20 000 preliminary samples for the training and it was subsequently used to directly design splitters with ratios ranging from 1/1 to 1/3.5 with an efficiency of 90%.

Dinsdale et al. proposed a strategy based on a tandem network configuration to inverse design universal optical components [296]. The device structure in Fig. 32(d) is based on a 6 × 33 µm² multimode waveguide with weak scattering perturbations. The forward part of the algorithm took top views of the structure to predict the electric field amplitude profiles at the waveguide output. An inverse network based on a variational autoencoder was used to implement the opposite function, i.e., provide a device structure giving a desired output profile. With this scheme the authors experimentally demonstrated arbitrary transmission matrices. Finally, dimensionality reduction was used by Kamandar Dezfouli et al. to design the vertical fiber grating coupler shown in Fig. 32(e) [294,297]. The grating unit cell incorporated both an L-shaped structure and a SWG metamaterial section and it was described by five parameters, i.e., the length of each section and the equivalent refractive index of the metamaterial. By using principal component analysis, this parametrization was compressed to only two parameters that could then be efficiently swept to identify designs with high efficiency of 75% and back-reflections below -20 dB. Thanks to the efficient parametrization it was also possible to acquire a detail mapping of the design space and identify the highest achievable efficiency depending on the minimum feature size allowed by the foundry.

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## 11. Conclusion and outlook

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In this paper, we reviewed recent developments, including both scientific advances and practical implementations of SWG metamaterial waveguides and devices in integrated photonics, focusing on silicon-on-insulator as the key platform for advanced integrated optical circuits. We examined the most promising directions of research, but also practical concerns, including application requirements and prospects for the industrial adoption of these structures. Incorporation of optical metamaterials on silicon chips opened unprecedented opportunities to control light propagation at the subwavelength scale and bolstered the performance of state-of-the-art integrated photonic
devices. Metamaterial structures have been shown to be beneficial in multiple applications, such as optical communications, biochemical sensing, nonlinear optics and quantum technologies.

Metamaterial nanophotonic couplers enable efficient interfacing of optoelectronic chips with the external world, including fibre-chip edge couplers, surface grating couplers and waveguide nanoantennas for optical beam forming and phased arrays. SWG couplers are currently utilized in industrial CMOS production sites using immersion deep-uv lithography, available as commercial Product Design Kits (PDKs). The intrinsic anisotropy of SWG metamaterials has been exploited in the design of broadband multimode interference couplers and optical hybrids for coherent communications and polarization handling devices, including polarizers, polarization splitters and rotators, and polarization-agnostic devices. On the other hand, polarization management with multiple waveguide modes remains elusive, especially when compact devices with broadband performance are required. Here, free-space-on-a-chip solutions that can intrinsically handle multiple modes provide a way forward. Another option is exploring smart inverse design techniques that can handle the complexity of optimizing devices for multiple modes.

Concurrently, the SWG's potential to accurately control waveguide refractive index and modal confinement is being leveraged in designing microring resonators, Bragg gratings, contra- directional couplers and complex spectral filters for applications ranging from wavelength division multiplexing to astrophotonics and quantum optics. Furthermore, adoption of SWG waveguides allows fine-tuning of the light interaction with biochemical molecules to substantially enhance the sensitivity of evanescent field sensors. This is achieved by optimizing the mode overlap with analyte molecules including within the SWG gaps where the electric field strength is increased. To this end, new waveguide topologies with large sensitivities are being developed, such as slot-SWGs, multi-box and dual-material waveguides, and fishbone SWG structures. SWG metamaterials are also gaining importance for enhancing nonlinear optical interactions in planar waveguides. Ability to control through SWGs waveguide group velocity dispersion and transverse index profile has been identified as a key benefit for optimizing the nonlinear spectral broadening in supercontinuum generation, difference-frequency NLO processes and second- harmonic generation. In a similar fashion, SWGs simultaneously allow engineering confinement and dispersion of both photonic and phononic modes, optimizing Brillouin interactions in optomechanical nanophotonic waveguides.

While many subwavelength based devices with excellent performance have been demonstrated over the past years in research labs around the world, a key challenge lies in moving these devices from "lab-to-fab", which will likely require foundry-side process optimization and advances in optical proximity correction [311]. Furthermore, most research has so far focused on metamaterial topologies that can be defined in a single lithographic step. The use of multiple etch steps, including creation of complex 3-dimensional metamaterials in several layers, has not yet been systematically explored, yet holds a strong potential for the development of ultra-high- performance devices. On the other hand, while practically all SWG-based devices have been developed in silicon-on-insulator waveguides so far, new implementations in different waveguide platforms are expected to emerge in the near future. The first such demonstration has recently been reported in silicon nitride waveguides [144] and compound semiconductor-on-insulator [312], and it is easy to foresee the inclusion of metamaterials in platforms such as thin-film lithium niobate, conventional III-V materials or even diamond photonics.

Most recently, a promising research direction emerged exploring Mie resonant particles for optical waveguiding. First Huygens' meta-waveguides have been demonstrated operating under Kerker condition by spectrally overlapping magnetic and electric dipole resonances, with low waveguide propagation loss, near-zero backward scattering and negative group index over a broad spectral range. These resonant structures can be fabricated directly on the silicon-on-insulator platform where light is seamlessly coupled with the conventional Si-wire waveguides and other nanophotonic components such as nanoantennas, resonators and metasurfaces. The resonant
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response of the individual Huygens' scatterers can be used to enhance light-matter interaction processes governing light emission, absorption, and ultra-fast and nonlinear optical phenomena. These advances are poised to lead to new functionalities, such as miniaturized modulators, nanoscale emitters and dispersion-engineered nanophotonic structures, ushering in a new field of resonant metamaterial integrated photonics.

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Last but not least, there is an overarching need for new simulation and design approaches to handle these diverse types of metamaterials and their on-chip integration. As multiple design variables and figures of merit need to be simultaneously handled, optimization techniques are becoming an indispensable design tool. This can be achieved using gradient-free algorithms based on different heuristics (genetic optimization, direct binary search, particle swarm, ant colony, and simulated annealing) as well as gradient-based algorithms, with the adjoint method being particularly important as it allows efficient handling of large design spaces with thousands of variables. At the same time, neural networks and machine learning tools will likely be increasingly used in metamaterial device design. These advanced algorithms can be employed to accurately predict fundamental physical quantities such as internal electromagnetic fields. That in turn makes it possible to reliably assess high-level device response, including calculations of preventive layout corrections to account for fabrication tolerances.

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Since its inception less than 20 years ago, the field of integrated metamaterials photonics has steadily grown in importance and become an essential part of modern silicon photonic component design. With the impressive recent advances discussed in this review and many new application areas for integrated photonic circuits on the horizon, we predict a bright future for this field of research and technology development.

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**Funding.** Funding is acknowledged from the Spanish "Ministeria de Ciencia e Innovación" under project TED2021-130400B-I00/AEI/10.13039/501100011033/Unión Europea NextGenerationEU/PRTR, as well as support by National Research Council of Canada, University of Malaga, University of Paris-Saclay and CNRS France.

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**Acknowledgments.** Discussions with Prof. Inigo Molina Fernandez, Prof. Alejandro Ortega Monux, Dr. Alejandro Sanchez Postigo, Prof. Winnie Ye, Prof. Isabelle Staude, Dr. Daniel Benedikovic, Dr. Abdelfetah Hadij-ElHouati, Dr. Dan-Xia Xu, Dr. Siegfried Janz, Dr. Jianhao Zhang, Dr. Maziyar Milanizadeh, Dr. Yuri Grinberg, Prof. Andrew Knights, Prof. Jonathan Bradley, Prof. Milan Dado, Prof. Laurent Vivien, Dr. Aitor Villafranca Velasco, Dr. Daniel Pereira-Martín, Dr. James Pond, Mr. Denizhan Sirmaci, Miss Shahrzad Khajavi, Mr. Pablo Ginel-Moreno, Mr. Radovan Korcek, Mr. Cameron Naraine, Dr. Saad-Bin-Alam, Mr. Alejandro Fernandez Hinestrosa and Mr. William Fraser, are gratefully acknowledged.

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**Disclosures.** None.

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**Data availability.** No data were generated or analyzed in the presented research.

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**Supplemental document.** None.

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