
Recently, the introduction of nonlinear optical functions in integrated optics has sparked a lot of enthusiasm. Demonstrations (such as frequency combs^{1–4}, supercontinuum sources^{5–7} suited for scientific and industrial applications, and photon pairs sources for quantum applications^{8–10}) have shown the potential for integrated photonic platforms. Furthermore, largescale manufacturing capability and affordability have motivated the development of fully integrated, nonlinear optical devices, aiming at various applications, such as all onchip spectroscopy^{11}, onchip quantum computations and communications^{12}, efficient multiplexing for data communications^{13}, onchip metrology^{14}, biosensing^{15,16}, or LIDARs^{17}.
Integrating these different active functions in silicon photonics^{18}, has also attracted much attention. The possible compatibility with CMOS processes and the overall low cost of the platform indeed makes it very attractive to the industry, with devices directly suitable for dense optical circuit fabrication. These deep integration capabilities make the Sicompatible platform an interesting testbed for the transposition, on a miniaturized and integrated scale, of optical functions initially developed on more macroscopic scales by bringing together various cumbersome elements needing precise alignments. However, silicon photonics presents some impediments when considering the exploitation of nonlinear optical processes due to the weak nonlinear responses of the most common materials (i.e., SiO_{2}, SiN) compatible with its clean room processes, or to the presence of free carriers around 1.5 µm wavelength. In this Cband range, Si is highly nonlinear but suffers from Two Photons Absorption (TPA) due to its low bandgap of ~ 1.12 eV (at room temperature). Unfortunately, as seen in Table 1, most of the classical materials compatible with silicon integration, e.g., SiO_{2} and SiN, lack strength in terms of optical nonlinearities. For instance, the stochiometric SiN nonlinear refractive index (n_{2}) is almost two orders of magnitude lower than that of Si. Although SiN/SiO_{2} waveguides show great advantages in linear optics (e.g., a much lower index contrast than Si/SiO_{2} and low propagation loss (~ 3 dB/m^{19,20})), the stochiometric SiN nonlinear refractive index (n_{2}) is almost two orders of magnitude lower than that of Si.
Material Integrated optics Free space n_{2}/n_{2si}
(n_{2si}=7.2×10^{−18} m^{2}·W^{−1})Refractive index
(at 1550 nm)TPA (at 1550 nm) [m·W^{−1}] χ^{(3)} (m^{2}V^{−2}) @
1550 nmχ^{(2)} (m^{1}V^{−1})
(emission wavelength)Band gap (eV) Si 1^{21} 3.5 0.5×10^{−11} 1.6×10^{−21} − 1.12 Si_{3}N_{4} 0.034 ^{5} 1.92.0 None 2.3×10^{−22} − − SiO_{2} 0.003 1.44 None − − − WS_{2} 291.6^{22} − 1.58×10^{−9} 2.4×10^{−19} ^{23} 68×10^{−11} ^{24}(600 nm) 2.1 MoS_{2} 37.5^{25}
15.28^{26}− − 3.6×10^{−19} ^{23} 2.9×10^{−11} ^{27}
(780 nm)2 WSe_{2} − − − 1.0×10^{−19} ^{23} 0.4×10^{−11} ^{28}
(730 nm)1.75 MoSe_{2} − − − 2.2×10^{−19} ^{23} 5×10^{−11} ^{29}
(640 nm – 700 nm)1.7 BP − − − 1.6×10^{−19} ^{30} − 0.3−2 Graphene 2.08^{31}
5.69^{32}
10.69 ^{33}
2.78^{34}− −
1.0×10^{−19} ^{35}− Zerogap GO 16003750^{36} − − − − − Table 1. Second and thirdorder nonlinear optical parameters at the technically important telecom wavelengths for the most common 2D materials and the primary materials of the CMOScompatible platform used in Si and Sihybrid integration schemes. We distinguish the results obtained in integrated settings and free space, as shown by the discrepancies below. BP stands for black phosphorous, and GO for graphene oxide, n_{2Si} is the value of the Kerr refractive index for Si at 1550 nm.
Yet, impressive demonstrations have been presented throughout the years using various siliconbased materials to generate integrated optical functions, such as supercontinuum sources^{5–7}, frequency combs^{1–4}, and photon pair sources through spontaneous fourwave mixing. As the application fields developed, the requirements became more demanding. In some fields, such as highpower computing or data communications, the communities have pushed for more efficient devices showing high performance with low power consumption. Annealed SiN waveguides with loss levels of few dB/m have been demonstrated (e.g., allowing the generation of octavespanning supercontinuum processes^{5,6,37}), yet the needed pump power was still large^{1,4,38}. Moreover, long waveguides were needed to achieve sufficient nonlinear processes, thus making optical functions demanding for fully integrated devices.
Alternatively, materials such as Sirich waveguides^{39–42} or the use of pin junctions to engineer carrier density^{43} have shown some promising results but have some drawbacks. For instance, Sirich waveguides still present TPA, while the use of multiple pin junctions adds another layer of complexity to devices and circuits. Hence, the need for solutions to this limitation is still strong.
Another approach is to introduce highly nonlinear materials to the onchip platform to take advantage of the highly effective properties of these materials, thus boosting their nonlinear performance via the integration strategy. Following that idea, several materials have been tested throughout the past decade. Chalcogenides are a popular alternative to Sibased materials (e.g., GaAs^{44}, As_{2}S_{3}^{45,46}, GeSe^{47}, GeSbS^{48} or GaS^{49}). In the quest to identify disrupting nonlinear optical materials, twodimensional (2D) layered materials have been identified as extremely promising^{23,25–27,31,50}, for example, with thirdorder susceptibility up to two orders of magnitude higher than that of Si. The most widely known material in this category is graphene. Still, other 2D materials such as Transition Metal Dichalcogenide (TMDs, MX_{2} like crystals where M stands for a metal atom and X for a chalcogen like MoS_{2}, WS_{2}, WSe_{2}, MoSe_{2}) or BP are also receiving increasing attention^{51,52}. The principle behind hybrid integration (Fig. 1) consists in incorporating highperformance 2D materials into the existing integration platforms, thus increasing the overall effective nonlinear performance of the waveguides. Yet, several challenges arise from the hybrid integration approach, as 2D materials are still not fully understood. The impact of their addition to integrated structures needs to be handled carefully (such as dispersion, mode distribution, and nonlinear optical responses). This review presents recent progress in the field and our perspectives after briefly introducing the theoretical background of the hybrid integration of 2D materials for the different integrated optics platforms. The aim is to describe key steps, from the modeling and the design to the characterization steps. This will hopefully serve as a guideline for future works on the related fields, thus solving one of the obstacles met with integrated nonlinear photonics.
Fig. 1 Conceptual representation of the hybrid integration of 2D materials. Below are lists of the most common 2D materials with their bandgaps that have been integrated with waveguide structures. The inset spectra show the input (red) spectrum, and the output broadened spectra (blue for intrinsic waveguides and green for the 2D material integrated waveguides) during a typical pulse broadening experiment.

There is more to consider designing a hybrid waveguide structure than a more common waveguide due to the properties of the added material to the structure and the waveguides. Among all the different design factors, engineering the optogeometrical structures of the hybrid waveguide is one of the key parts of designing integrated nonlinear optical functions. Therefore, it is essential to estimate through eigenmode calculation the waveguide properties on which they are based^{53}. This mainly includes their effective index values, the transverse mode distributions, and the Group Velocity Dispersion (GVD) profiles at the operating wavelength. For 2D material integrated hybrid waveguides, the difficulty is to consider the modelling of a quasisurface 2D material correctly. However, the induced optical effect of a 2D material, mainly reflected in its surface conductivity, can be significant. This important physical quantity can be, in fact, described in two different ways: either by using the bulk surface conductivity $ {\sigma }_{b} $ in which the 2D material (or their heterostructures) is considered as a bulk material with an atomically thin thickness or, using a real surface conductivity ($ {\sigma }_{s} $) approach accounting for the actual 2D nature of the material. The two different methods yield the same results. However, in the first case, a typical Finite Difference Eigenmodes (FDE) calculation may require a very tight mesh to properly account for the boundary conditions set by the hybrid structure (monolayer 2D materials having thickness revolving around 1 nm and the overall structure having a dimension in the order of a few µm). This is why 2D materials can be preferentially described in waveguide mode calculations by the second approach, which needs a proper $ {\sigma }_{s} $ estimation. Several methods exist to perform the estimation of this parameter^{54}. For graphene, analytical solutions can be found. The Diraccone shape of its band structure indeed allows for analytical investigation^{5557}
$$ {\sigma }_{s}\left(\omega \right)=\frac{{e}^{2}{E}_{F}}{\pi {\hslash }^{2}}\frac{\tau }{1i\omega \tau }+\theta (\hslash \omega 2{E}_{F})+\frac{i}{\pi }ln\left\frac{\hslash \omega 2{E}_{F}}{\hslash \omega +2{E}_{F}}\right $$ (1) with ω the frequency, e the elementary charge, $ \hslash $ the reduced Planck constant, E_{F} the Fermi energy level, τ the relaxation time, and θ the Heaviside steplike function^{54}. Surface conductivity for graphene can be calculated (E_{F} = 0.6 eV and τ=0.25/2π) accordingly and is displayed in Fig. 2a. For other 2D materials, an analytical approach is not yet available due to their more complicated band structures. Hence, for 2D materials such as TMDs, experimental measurement data are required for the numerical modelling of the 2D materials (e.g., MoS_{2}, WS_{2}, MoSe_{2}, WSe_{2}^{58}). Fitting the data by a sum of Lorentzian, one can then calculate the relative permittivity $ {\varepsilon}_{r} $:
Fig. 2 Simulation examples of hybrid 2D material waveguides. a Linear surface conductivity of graphene calculated from Eq. 1 (Red curve for imaginary components black curve for real components). b Surface conductivity of two most common TMDs (i.e., MoS_{2}, WS_{2}) calculated from Eq. 3^{54,58}. c Typical Transverse electric (TE) mode distribution for a SiN planarized buried strip waveguide coated with MoS_{2} at 1550 nm optical wavelength. d Slice of the TE mode showing part of the mode overlapping the 2D material. The effective susceptibility tensor calculation of this structure takes into account the density of the mode with the nonlinearities of each part of the structure. e Evolution of the spectral envelope (in a logarithmic scale) along light propagation obtained from the simulation of pulse propagation in a SiN strip hybrid waveguide with a 14.8 μm long exfoliated WS_{2} flake transferred on top^{22}. The waveguide has a crosssection of h = ~ 330 nm and w = ~ 760 nm. The WS_{2} sheet is directly transferred on top of the waveguide, thus triggering the evolution of the light pulse along propagation. The area where the WS_{2} flake is located is also zoomed in the right panel.
$$ {\varepsilon}_{r}=\frac{\varepsilon\left(\omega \right)}{{\varepsilon}_{0}}=1+{\sum }_{k=1}^{N}\frac{{f}_{k}}{{\omega }_{k}^{2}{\omega }^{2}i\omega {\gamma }_{k}} $$ (2) where $ {f}_{k} $ stands for the oscillator strength, $ {\mathrm{\omega }}_{\mathrm{k}} $ the resonance frequency, and $ {\mathrm{\gamma }}_{\mathrm{k}} $ is the spectral width of the k^{th} oscillator. To access the surface conductivity, one can rely on a relation between surface conductivity and the material permittivity:
$$ \varepsilon\left(\omega \right)={\varepsilon}_{0}\left(1+\frac{i{\sigma }_{b}}{{\varepsilon}_{0}\omega }\right)={\varepsilon}_{0}\left(1+\frac{i{\sigma }_{s}}{{\varepsilon}_{0}\omega {h}_{eff}}\right) $$ (3) where h_{eff} is the effective atomic thickness of the thin layer 2D material, $ \varepsilon\left(\omega \right) $ is the permittivity, $ {\varepsilon}_{0} $ is the vacuum permittivity, $ {\sigma }_{b} $ is the bulk conductivity and $ {\sigma }_{s} $ is the surface conductivity. Fig. 2b displays the surface conductivity calculation for two of the most common TMDs (i.e., MoS_{2}, WS_{2}) using the data from Ref. 58. Retrieving the surface conductivity data is critical to model the optogeometrical properties of the added 2D material. Typical calculated TE mode distribution can be seen in Fig. 2c, d and is not significantly different from the uncladded waveguide (see Refs. 59,60). As such, no additional coupling losses are to be expected. Although 2D materials have a very limited effect on the passive properties of the waveguides^{59,60}, the additional phase shift from the 2D material is still required to be considered in several applications^{59,61}.
Subsequently, as discussed in the next subsection, a precise knowledge of the interaction between the 2D material and the transverse propagating mode might be required in the case of nonlinear applications, notably because of the gigantic nonlinear properties of the added material. 2D materials are yet likely to introduce also additional propagation losses^{61}, but as these losses are highly related to fabrication, transfer, and posttreatment methods (e.g., annealing), they are difficult to account for in numerical studies. In any case, worst case situations can be considered to assess the nonlinear performances of the devices embedding the 2D layers.

Nonlinear effects occur as a response to waveguiding structure materials during light propagation, including FourWave Mixing (FWM), SelfPhase Modulation (SPM), SecondHarmonic Generation (SHG), and ThirdHarmonic Generation (THG). Mathematically, nonlinear optical phenomena can be expressed as an expansion of the polarization function of the medium where $ \mathbf{E} $ is the electric field and $ {\chi }^{\left(n\right)} $ is the n^{th} order of the susceptibility tensor^{62}:
$$ \mathit{P}\left(t\right) ={\chi }^{\left(1\right)}\mathbf{E}\left(t\right)+{\chi }^{\left(2\right)}{\mathbf{E}\left(t\right)}^{2}+{\chi }^{\left(3\right)}{\mathbf{E}\left(t\right)}^{3}+... $$ (4) Additionally, thirdorder nonlinearities induce phase modulation properties in the medium. This can be described by considering the phasematching conditions^{63}:
$$ \kappa ={\Delta \beta }_{L}+{\Delta \beta }_{NL}=0 $$ (5) where $ {\Delta \beta }_{L}=\sum _{k=0}^{\infty }({1}/{k!}){\left({\partial \beta }/{\partial \omega }\right)}^{k}{(\omega {\omega }_{P})}^{k} $ and $ {\Delta \beta }_{NL}=2\gamma P $ , with $ P $ the peak pump power and $ \beta $ the optical mode propagation constant. $ {\Delta \beta }_{L} $ corresponds to the chromatic dispersion contribution to phase matching between the involved frequencies, a linear process derived from the optogeometrical parameters of the waveguide. $ {\Delta \beta }_{NL} $ is the corresponding nonlinear component induced by the nonlinear response of the material and generating another phase shift. Here, the coefficient $ \gamma $ is the effective nonlinear coefficient accounting for the strength of the nonlinear response occurring during propagation in the waveguide. Basically, the nonlinear waveguide behavior scales as the real part of this coefficient and thus needs to be as large as possible. The linear phase mismatch $ {\Delta \beta }_{L} $ originates from the waveguide mode dispersion profile. The geometry of the structure and the composite materials set optogeometrical parameters where properties and shapes must be studied in relation to each other. The Kerr nonlinear refractive index and the TPA coefficients of the waveguiding structure are generally introduced as n_{2} and β_{TPA} respectively, and can be related as follows^{64}:
$$ \frac{\omega }{c}{n}_{2}+\frac{i}{2}{\beta }_{TPA}=\frac{3\omega }{4{\epsilon }_{0}{c}^{2}{n}^{2}}\Gamma $$ (6) The effective nonlinear susceptibility tensor $ \Gamma $ directly expresses the contribution of every component of the hybrid structure. Every material is considered with its properties and weighted by the fraction of the mode overlapping with each part of the waveguide’s crosssection. Subsequently, $ \Gamma $ can be calculated as:
$$ \Gamma =\frac{{A}_{0}{\int }_{{A}_{NL}}^{}\overline{e}(r;\omega ){\chi }^{\left(3\right)}e(r;\omega )\overline{e}(r;\omega )e(r;\omega )dA}{{({\int }_{{A}_{\infty }}^{}{n}^{2}\left(r\right){\lefte(r;\omega )\right}^{2}dA)}^{2}} $$ (7) with A_{NL} the transverse area is occupied by nonlinear materials, $ {A}_{\infty } $the whole crosssection area, A_{0} the area of the waveguide’s core, $ e(r;\omega ) $the transverse electromagnetic mode distribution, and $ \overline{e}(r;\omega ) $its complex conjugate. This parameter is a complex number that can be separated into its real and imaginary parts: $ \Gamma =\Gamma \text{'}+j\Gamma \text{'}\mathrm{\text{'}} $^{64}. $ \Gamma \text{'} $ accounts for the refractive index variations, whereas $ \Gamma \text{'}\text{'} $ accounts for the nonlinear TPA. This parameter does not provide a useful comparison between different waveguide structures. A more informative factor is the effective nonlinear coefﬁcient $ \gamma $, mentioned in Eq. 5 which considers the mode group velocity v_{g}:
$$ \gamma =\frac{3\omega \Gamma }{4{\epsilon }_{0}{A}_{0}{v}_{g}^{2}} $$ (8) This factor carries the information of the nonlinear Kerr coefficient expressed with the real part of $ \gamma $ and the TPA with the imaginary part. The numerical integral over the whole waveguide crosssection from Eq. 7 must include the properties of different nonlinear materials. The 2D material can then be considered as a bulk layer with a thickness of h_{eff}. This is critical to consider the overlap between the transverse mode and the huge nonlinearities of the material. This simulation methodology thus makes it possible to study different typical hybrid waveguide geometries, regardless of the location or the number of 2D sheets, and to derive both the dispersion properties of the hybrid waveguides and their effective nonlinearities.
A more intuitive and complementary measure of the situations to be investigated consists simply in evaluating the total fraction of the power passing through the active material, i.e., the confinement factor in the active material^{60}. To do so, we derive the Poynting vector through the effective thickness of the layered material and the Poynting vector through the whole structure:
$$ \mathbf{S}=\mathbf{E}\times \mathbf{H} $$ (9) S is the Poynting vector, E is the electric ﬁeld, and H is the magnetic ﬁeld. To describe the propagating energy responsible for nonlinear effects occurring in the waveguide, the average power ﬂow through the wanted area is derived from S:
$$ ⟨P⟩=\left\langle{\iint \mathbf{S}.{u}_{z}ds}\right\rangle $$ $$ CF=\frac{{\left\langle{P}\right\rangle}_{2D}}{{\left\langle{P}\right\rangle}_{tot}} $$ (10) where <> denotes a time average and CF the confinement factor. $ {\left\langle{P}\right\rangle}_{2D} $ is thus the power flow through the 2D material only and $ {\left\langle{P}\right\rangle}_{tot} $ the power flow through the crosssection of the waveguiding structure. Knowing how much the 2D material contributes to the effective nonlinearity allows the calculation of its nonlinear Kerr refractive index from the experimental characterization of the hybrid waveguide (e.g., SPM or FWM measurements)^{25}. This means that this formalism can be used both ways, as a means of anticipating the effective performances of waveguide structures through simulation and optimization on the one hand^{60}, and as a way to trace back the contribution of the 2D materials from experimental data^{22} on the other hand.

From the effective nonlinear optical properties of waveguides, it is possible to describe the propagation of optical pulses numerically. Thirdorder nonlinear effects, free carriers, and linear passive losses are the main effects to be included in the model to match experimental results as closely as possible. Commonly used for that purpose, the propagation of nonlinear pulses can be modelled using the generalized nonlinear Schrödinger equation (GNLSE)^{63–65}:
$$ \begin{split}&\frac{\partial A}{\partial z}\frac{\alpha }{2}A\left(\sum _{k\ge 2}{\beta }_{n}\frac{{i}^{n1}}{n!}\frac{{\partial }^{n}}{\partial {T}^{n}}\right)A\left(\frac{{\sigma }_{a}}{2}+i{\sigma }_{n}\right){N}_{C}A\\=&i\gamma \left(1+\frac{1}{{\omega }_{0}}\frac{\partial }{\partial T}\right)\times \Bigg(\left(1{f}_{R}\right)A{\leftA\right}^{2}\\&+{f}_{R}A{\int }_{0}^{\infty }{h}_{R}\left(\tau \right){\leftA\left(z,T\tau \right)\right}^{2}d\tau \Bigg) \end{split}$$ (11) with A the complex envelope, α the attenuation constant, β_{n} the higherorder coefficients from the Taylor expansion of the propagation constant calculated around the central frequency $ {\omega }_{0} $. Free carriers are described using the following parameters: $ {\sigma }_{a} $ and $ {\sigma }_{n} $ are the free carrier absorption and refraction coefficients, and $ {N}_{C} $ is the free carrier density^{64,66}. The Raman response is considered with $ {f}_{R} $, the delayed Raman response to polarization^{63,67}, and $ {h}_{R}\left(t\right) $ the Raman response function. TPA can be expressed in this equation as the complex part of the nonlinear coefficient $ \gamma $^{64}. The different effects can be neglected or not, depending on the case. For instance, the Raman response is usually neglected for simulations in integrated optics for the Si or SiN platforms^{64,68}. Free carriers and TPA can be further ignored when using materials that do not trigger such effects in the frame of wavelengths used during the experiments (e.g., doped silica waveguides or SiN waveguides around 1.5 µm). Regardless, the GNLSE can be used for fitting purposes^{22,32,34,68}. Using the nonlinear coefficient γ as the fitting parameter, it is possible to retrieve the effective nonlinear performances of a waveguide. An extended literature review exists on numerical methods for integrating this equation, such as the splitstep Fourier method using a fourthorder RungeKutta scheme for the integration^{63,67,68}. Retrieving the effective nonlinear coefficient of a waveguide by fitting experimental spectra using Eq. 11 is an important link in the characterization of hybrid waveguides. When realizing pulsebroadening experiments in hybrid waveguides, it can be used to retrieve the effective nonlinear coefficient. Together with Eq. 7, this can be combined to retrieve the nonlinear refractive index of the 2D material itself. It must be pointed out that the longitudinal variation of the nonlinear coefficient needs to be addressed for more accurate results. Because current transfer methods rarely allow for full coverage of the device, it is frequently the case that optical pulses propagate in at least two different types of structures consecutively between light injection and collection in and out from the chip. Therefore, a longitudinal profile of the nonlinear coefficient along light propagation must be considered to perform realistic numerical modelling of experiments. This is illustrated here by Fig. 2e, where a flake of WS_{2} was deposited on a fraction of a waveguide. The reports of such approaches have shown promising results, with extracted Kerr index and TPA coefficient values for the 2D material comparable to those reported in the literature^{22}. Note that all our abovementioned theoretical discussions (e.g., modeling linear and nonlinear optical properties, pulse propagation in hybrid waveguides) on silicon waveguide examples apply to cylindrical silica optical fiber waveguides and can be extended to new frequency generation processes beyond the highlighted pulse broadening effect.

The main components of a hybrid waveguide are the photonic chip itself and the 2D materials. The processes have been well studied and optimized for the fabrication of integrated waveguides in the lab. Starting with the manufacturing, one can distinguish three main methods to fabricate 2D materials: Mechanical Exfoliation (ME)^{69,70}, Liquid Phase Exfoliation (LPE)^{71} and ChemicalVaporDeposition (CVD)^{72}. All these cases have advantages and disadvantages. For example, mechanical exfoliation usually delivers the best crystal quality, the uncertainty of the process of low volume makes it difficult to scale up at industrial levels. Other methods, such as LPE or CVD, offer much bigger volumes and are promising for industrial scale, but the quality of the materials remains to be improved.
Fabrication of the 2D materials is one side of the process. After, 2D materials need to be transferred on top of the integrated photonic chip. The great majority of the existing work on hybrid waveguides utilize direct transfer methods, such as wet transfer^{73,74} (fishing of the 2D material in suspension in a solution), dry transfer^{75} (deposition of the material using a polymer stamp) or semidry transfer^{76}. However, the field has recently seen a few publications attempting direct growth of 2D materials via CVD on the chip^{77,78}. This method could prove itself revolutionary as it would circumvent most of the problems arising in more conventional methods (polymer residue, small scale of the deposit,…), which however needs further investigation for deterministic hybrid waveguide fabrications at a large scale.

The denomination “2D material” designates a thin crystal layer with a thickness of just a few atoms. No material is completely 2D and does have an effective thickness h_{eff }(usually a few nm). This corresponds to the atomic thickness of the material in the case of monolayer or fewlayer crystals. The 2D material category includes graphene, a honeycomb structured layer of carbon atoms that was first demonstrated thanks to mechanical exfoliation^{70}. Its unique electronic and optical properties^{79–81}, sparked enormous interest in 2D materials, and prompted the discovery of numerous other thinlayer crystals with interesting physical properties^{82}. Other 2D materials (such as TMDs, BP, or hexagonal BoronNitride (hBN)) have also been widely studied. Fig. 1 presents typical “honeycomblike” structures of 2D materials that differ in the nature of their band gaps and their group of symmetry (e.g., D_{6h} for graphene, D_{3h} for typical TMDs^{83}). These variations can be exploited to get the best out of hybrid integration depending on the targeted optical functions (e.g., switch operation, broadband light sources^{84}).
2D materials show very promising performance in terms of optical nonlinearities^{23,51,84}, as illustrated in Table 1 with the characterized nonlinear response of various hybrid waveguides. Secondorder nonlinear effects (like SHG)^{85,86} and other nonlinear effects (such as Raman emission and saturable absorption) can also be reinforced thanks to 2D materials. However, assessing the nonlinear optical properties can prove challenging. A lot of discrepancies can be found between different references and characterization methods. This can be attributed to the influence of the layer fabrication methods and the quality of the thin layered crystal and their properties^{23,51}. For example, some fabrication and deposition methods leave more residues (e.g., polymer, water), defects in the monolayer crystal, and wrinkles of the flake^{87}. For instance, a CVDgrown sample (e.g. CVD grown graphene prompting $ {\chi }^{\left(3\right)}=~510\times {10}^{19}\;{\rm m}^{2}{\rm V}^{2} $ from THG measurements with a pump at 1566 nm^{88}) presents different performance than an exfoliated one (e.g., $ {\chi }^{\left(3\right)}=~4.2\times {10}^{15}\;{\rm m}^{2}{\rm V}^{2} $ for exfoliated graphene measured from THG measurements with a pump at 1550 nm^{89}). Doping can also generate variation in the measured nonlinear response^{90–92}. Moreover, the environment surrounding the 2D materials (e.g., the substrate^{93}), and induced strain^{94–97} also affect results. The characterization method itself is an important aspect of this matter. An inplane electromagnetic wave excitation produces different results than an outofplane one^{23,98–100}. In connection with this question of the optical excitation mode of 2D sheets, it was shown that for graphene and GO a quasiTE polarization in the waveguide triggered higher responses than the quasiTM polarization^{99101}. Some preliminary studies suggested this might also be true for TMDs, such as MoS_{2}^{92}.
The study and deep understanding of the nonlinear properties of 2D materials remain challenging. Most previous studies are based on experimental measurements without any strong link with the theoretical origin of the observed properties. In this view, the exception is again graphene, its Diraccone shape band structure allowing for analytical investigation of the thirdorder conductivity, i.Fe., equivalent to the thirdorder susceptibility in a bulk material as for its linear properties. Thus, the analytical solution for $ {\sigma }^{\left(3\right)} $, a surface equivalent of the susceptibility tensor $ {\chi }^{\left(3\right)} $, can be explicitly expressed^{98}. For other 2D materials, understanding the mechanisms requires iterative comparisons between experimental results and numerical modelling and analysis, raising at the same time a difficulty but also opening a very interesting field of investigation to explore to evaluate their full potential. Undoubtedly the theoretical understanding of the optical properties of 2D materials will continue to grow in the coming years, as well as the hybrid integration with waveguides simultaneously. However, for the time being, progress and research must mainly rely on experimental results, particularly in the case of newly discovered materials.

Integrating 2D materials on photonic chips is beneficial in various ways^{102}. We have witnessed that hybrid integration of 2D materials not only enhances the nonlinear optical performance of waveguides, but also introduces new nonlinear optical properties that are impossible with the waveguide before the integration. This section focuses on second and thirdorder parametric and a few nonparametric optical processes^{62}.

Finding reliable materials to boost the effective thirdorder nonlinear response of a waveguide while remaining TPAfree is critical for siliconbased waveguides. Whereas some 2D materials present photon absorption around 1.5 µm wavelength (such as graphene, due to its gapless nature^{103}), other families of 2D materials, such as TMDs, have a gap large enough to limit photon absorption in the wavelength range of interest^{102–104}. Related to this, experimental data collected from integrated optics and free space measurements are summarized in Table 1. Although results can vary significantly from one characterization method to another, performances are generally promising with high reported Kerr refractive indices compared to standard materials of the CMOS platform (see $ {n}_{2} $ comparison for different materials in Table 1). Here we summarize the hybrid integration results of using thirdorder optical nonlinearities.

SPM may be one of the easiest methods to measure thirdorder nonlinearities in a waveguide. Indeed, this phenomenon relies on the Kerr refractive indexinduced phase shift during light propagation throughout a waveguide. Thus, this occurs in a typical regime of the waveguide’s chromatic dispersion for the guided mode and can be measured by pulse broadening. Fig. 3 presents several cases of different 2D materials. By adding a highly nonlinear material to the waveguide, a higher effective nonlinear refractive index (n_{2}) is expected, thus causing a more substantial broadening of the output spectrum after a given constant propagation distance (Fig.1). Therefore, measurements of the output spectra can convey information regarding the effective properties of the hybrid integrated waveguide structure. Usually, simulation data calculated by integrating the GNLSE are matched with measurements. For all cases presented in Fig. 3, a more significant broadening effect is obtained from the waveguide with a 2D material on top due to the enhanced SPM of the 2D materials^{26,63,107}.
Fig. 3 SPM enhancement results with 2D materials covered waveguides. a SPM measurements in graphenecovered silicon waveguides. Measurements were done for several lengths (L_{Gr}) of graphene sheetcovered waveguides in contrast with a pure silicon waveguide (SWG) using a pulsed pump at 1560 nm wavelength with a 250 MHz repetition rate, 80 fs pulse duration, and an input pulse energy of 120 pJ ^{32}. b SPM measurements in a planarized buried strip silicon on insulator (SOI) waveguide enhanced with MoS_{2}. The pump is a pulsed laser with a 1 MHz repetition rate and a pulse width of 56 ps at 1554.5 nm^{26}. c SPM measurements in a SiN waveguide integrated with WS_{2}. The pump is a pulsed lasercentered around 800 nm with an 85 MHz repetition rate and a TEduration pulse of 100 fs^{22}.
Results highlighted in Fig.3 show significant broadening spectra for the hybrid waveguides compared to the reference (bare waveguides). Fig. 3a presents larger spectral broadening effects for different lengths of graphene. The full potential of various 2D materials in a guided configuration for Kerr effect enhancement of waveguides was thus successfully demonstrated and quantified using methods discussed in Section 2. An effective nonlinear refractive index of ~ 4.1 × 10^{−17} m^{2}/W was retrieved for graphene^{32} (Fig. 3a). Other similar work in graphene also confirms ~ 2 × 10^{−17} m^{2}/W^{34}. SPM also has been demonstrated with planarized buried strip SOI waveguides integrated with MoS_{2} (Fig. 3b), giving an effective nonlinear refractive index for MoS_{2} of ~ 1.1 × 10^{−16} m^{2}/W^{26}. Ref. 22 reported a n_{2} of ~ 2.18 × 10^{−15} m^{2}/W for WS_{2} (Fig. 3c). Note that those reported n_{2} values are consistent with the results measured with other characterization methods such as FWM (see Table 1). To offer more complex optical functions (e.g., broadband, and ultrashort light sources), future developments are expected, such as using longer spiral waveguides, integrating 2D materials inside the waveguides, using heterostructures or alternative layers^{60} with enhanced nonlinearities. Beyond the spectral applications to the Kerr effect, Kerr modulation can also be introduced and enhanced using 2D materials, as in the example of a fiberbased alloptical Kerr phase shift modulator^{108}. Undoubtedly, similar usage of 2D materials will be developed for the fiber integrated platform as well.

FWM is one of the most widely used thirdorder nonlinear optical functions for applications, such as alloptical parametric amplifiers and photon pairs^{109,110}. FWM is often found in its degenerate form, where $ 2{\omega }_{P}={\omega }_{s}+{\omega }_{c} $, according to the conservation of energy and $ 2{k}_{P}={k}_{s}+{k}_{c} $ according to the conservation of momentum. Here, degenerate FWM involves 3 frequencies ($ {\omega }_{p} $, $ {\omega }_{s} $ and $ {\omega }_{c} $) and their wavenumbers ($ {k}_{P} $, $ {k}_{s} $, and $ {k}_{c} $) for the pump, the signal, and the converted frequency (idler), respectively. FWM is most effective when the frequencies are phasematched together. In the absence of TPA, the efficiency of the FWM process can be characterized in terms of conversion efficiency^{111–114}:
$$ {\eta }_{straight}=\frac{{P}_{C}\left(L\right)}{{P}_{S}\left(0\right)}={\left\gamma {P}_{0}{L}_{eff}\right}^{2}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha L) $$ (12) with $ {P}_{C}\left(L\right) $ the power of the converted frequency after propagation in a waveguide of L length, $ {P}_{S}\left(0\right) $ the input signal frequency power, $ {P}_{0} $ the input pump power, α the propagation loss, and the effective propagation length L_{eff}:
$$ {L}_{eff}=\frac{1{e}^{(\alpha +j\kappa )L}}{\alpha +j\kappa } $$ (13) where κ is the phase mismatch. FWM conversion efficiency can be further enhanced if we use a resonator. However, the integration of 2D materials with a microresonator (e.g., a micro ring resonator) must be carefully considered because of the influence of the thin layer material on the cavity’s passive properties^{59,115,116}. The new equation for the conversion efficiency becomes^{112,113,117–119}:
$$ {\eta }_{resonator}=\frac{{P}_{C}\left(L\right)}{{P}_{S}\left(0\right)}={\left\gamma {P}_{0}{L}_{eff}\right}^{2}{FE}_{p}^{4}{FE}_{c}^{2}{FE}_{s}^{2}\mathrm{e}\mathrm{x}\mathrm{p}(\alpha L) $$ (14) where FE_{i} is the field enhancement factor and can be expressed as follows in an allpass ring:
$$ {FE}_{i}=\left\frac{k}{1r{e}^{\frac{\alpha L}{2}+j\frac{{\omega }_{i}L}{c}}}\right $$ (15) The field enhancement factor is responsible for improving the FWM performances in a ring compared to a straight waveguide. Reciprocally, FWM frequency conversion can also be used as a useful means of characterization of hybrid optical waveguides. By measuring the gain on the converted frequency, one can retrieve the waveguide’s effective nonlinearities. This method was used in numerous examples for different materials due to its reliability and relatively easy implementation. Fig. 4 shows examples of FWM studies carried out for various hybrid waveguides embedding 2D materials. Those studies were either performed in microring resonators (Fig. 4a−f) or regular waveguides^{25}
Fig. 4 FWM results of hybrid integration with 2D materials. a−c FWM experiment conducted on GO coated Siring resonators ^{120}. a Doped silica/silica microring resonator coated with 150 layers of GO. In the zoomedin inset: chemical representation of GO and its crystalline configuration. b FWM spectrum for 05 layers coating of GO. c FWM spectrum for 050 layers coating of GO. d−f FWM measurements in a graphenecoated Si ringresonator^{31}. d Cross section of a Siridge ring resonator. e Comparison between the FWM spectrum in a Si waveguide (black) and a Sigraphene waveguide (red). f Enhancement in the idler gain shows the conversion efficiency versus the pump power for both waveguides. gh Gated grapheneSiN waveguide for tunable FWM, ^{122}. g Gated planarized buried strip with graphene. h Results of conversion efficiency (FWM) versus the gate voltage (V_{GS}) and the wavelength detuning between the pump and the signal.
Using the same reasoning as for SPM, it is then possible, knowing the overlap between the 2D material and the transverse propagating mode, to calculate the effective Kerr refractive index of the 2D material alone. A first example of the FWM approach to the use of hybrid waveguides is highlighted with a ring resonator coated with GO (see Fig. 4a)^{120}. The Kerr refractive index of GO was extracted as ~ 1.5 × 10^{−14} m^{2}/W. The injected pump was a 160mW continuous wave source centered at ~ 1550 nm wavelength. As shown in Fig. 4b, c), the experiment was attempted for several thicknesses of the GO covering layer. The thicker the layer, the higher the nonlinear effect seems to be due to a higher interaction between the mode and the GO. For graphene (Fig. 4d−f), the nonlinear refractive index has been measured at ~1.5 × 10^{−17} m^{2}/W^{31} with a pump and the signal set at resonances of the ring (1557 nm and 1567 nm, respectively). The gain on the idler is ~6.8 dB for a continuous power of 8 mW for the pump and 1 mW for the signal (Fig. 4e). The study also reports a typical behavior of the conversion efficiency versus the pump power (Fig. 4f), confirming the nonlinear origins of this increased performance in the hybrid waveguide. Similar experiments with MoS_{2} on a straight Si waveguide have also been reported (Ref. 25) with a Kerr refractive index of ~ 2.7 × 10^{−16} m^{2}/W^{25}. FWM was measured by injecting a 0.1 mW pump at ~ 1550 nm with a signal at ~ 1550.4 nm. The results between the reference waveguide and the MoS_{2}Si waveguide are compared to assess the effect of MoS_{2} on the waveguide’s properties. FWM results presented here are promising. Depending on the structure and nonlinear 2D material, a gain of up to 10.3 dB (Fig. 4c) on the idler frequency, the lowest being 4 dB for a straight waveguide in SOI (Ref. 25). This also shows the advantages of combining interesting integrated photonics structures with 2D materials. Additionally, 2D materials, such as graphene, can be combined with electrodes, introducing the tunability of the FWM. Indeed, inducing a biased current allows tuning the 2D sheet surface conductivity as a tuning of the Fermi level^{121}. Fig. 4g, h presents an example of tunable effective nonlinear coefficient γ using gated graphene (with polymer electrolyte) combined with electrodes (see Fig. 4g)^{122}. The applied bias voltage effectively induces a shift in the Fermi level, which translates into effective tunable properties of the waveguide (Fig. 4h). This can, of course, be expanded to wider and very promising nonlinear optical functions as the FWM process is involved in the generation of multiple nonlinear integrated functions (e.g., supercontinuum sources, frequency combs). Using the approaches illustrated in Fig. 4g, h, one can improve and diversify the possibilities of hybrid integration by introducing the device’s tunability. This is one more step toward fully integrated onchip applications and other applications such as miniaturized spectroscopy^{123}.

Frequency combs are broadband light sources of multiple equally spaced narrow frequency lines^{124}. They can be formed under specific phase and pump power conditions inside a ring resonator^{3}. To meet these phasematching requirements, the ring waveguide GVD is required to be in the anomalous regime, thus requiring waveguide engineering^{63}. The wider the anomalous window, the broader the frequency comb tends to be^{125}. This phenomenon was established in the literature and can be understood thanks to the socalled Cherenkov radiations^{2,125}. Kerr frequency comb generation is typically strongly relying on cascaded FWM. Hence, good performance in FWM is key to the development of efficient generation processes of frequency combs. Due to the recent advances in boosting FWM using hybrid integration schemes, the demonstration of frequency combs using hybrid devices with 2D materials is likely to be widely addressed in the next few years. Moreover, a report of tunable frequency combs enhanced with graphene has been reported. Using the FermiDirac tunability of a gated graphene sheet on top of a microring, the tunability of the GVD profile brought a tunability of the comb itself^{126} as well as a reported slight enhancement of the nonlinearity introduced by the graphene. In this work, a tunable laser (between 1480 nm and 1640 nm) was driven to modelock Kerr dissipative soliton pulse inside the cavity together with the control of the cavity chromatic dispersion thanks to the gated sheet of graphene. A bias current was applied to the graphene sheet covering the cavity’s waveguide. The proper flow of electrons was further guaranteed at the surface of the monolayer sheet using an ion gel. With different bias voltages applied, various Fermi levels for the graphene sheet can be electrically shifted, thus resulting in various frequency combs due to the shifting dispersion profile. Graphene is not the only material that could benefit this range of applications. Other 2D materials recently showed a lot of potential regarding the tunability of the micro ring cavity using gated TMDs^{61}. The variations of the resonance induced by different biased currents can be explained by an effective index change of the propagative waveguide mode induced by the modified conductivity of the 2D material. From the tunability of the effective refractive index in Ref. 126, it was inferred that such tunable frequency combs could also be achieved in the future using other families of materials (such as WS_{2}^{62} or WSe_{2}^{127}). Beyond the benefit of the dispersion tunability, materials such as graphene can also weight on the gainloss balance and thus further expand tunability. The optoelectronic feedback stabilization^{128} and chemical sensitization^{121} are also huge benefits of that approach for soliton combs. Supercontinuum generation can benefit from similar designs. Although, the complexity brought by cavities and modelocking is removed as regular waveguides are generally used for that purpose, GVD engineering and optimization of the effective nonlinear properties are still required^{65}.

Secondorder nonlinear phenomena involve $ {\chi }^{\left(2\right)} $and are subject to crystal symmetry^{62}. In a centrosymmetric medium, material polarization P is an odd function (P(E) = −P(E)) E being the electrical field. Thus, to satisfy Eq. (4), $ {\chi }^{\left(2\right)} $ vanishes to 0. Hence, typical materials compatible with CMOS processes, such as Si or SiN do not present secondorder nonlinear effects due to their centrosymmetric structures. Yet, secondorder nonlinearities could be beneficial for ultrafast modulation of optical signals, which is required for signal processing. The freecarrier dispersion of pn and pin junctions typically used in the SOI technology for modulation is too slow^{130132}. Therefore, the ultrafast Pockels effect would be preferable, although it is challenging to engineer due to the nature of available materials. The introduction of such an effect relying on the hybrid integration of III/V semiconductors on Si was proven efficient for that purpose^{131}. 2D materials could also be a promising solution to that end. However, materials presenting centrosymmetric crystal layouts (such as graphene, BP, or even numbers of TMD layers) do not provide secondorder nonlinearity^{75,133}. However, some methods exist to introduce SHG to centrosymmetric 2D materials. For example, introducing external electrical fields (in graphene^{134,135} or TMDs^{135137}) or doping the material^{91,138} showed promising results in that regard. On the other hand, other 2D materials (e.g., 3R phase BN and TMDs) can present secondorder nonlinearity due to their noncentrosymmetric nature. As a pilot demonstration scheme of $ {\chi }^{\left(2\right)} $, SHG was particularly mainly studied. As such, promising SHG performance in terms of conversion efficiency was reported in free space^{27,29,51,86,139–141}, but also in a few cases in integrated optics, as discussed in this section. SHG phenomena were investigated in various waveguide structures (see Fig. 5) regarding the intensity of the generated frequency (i.e., second harmonic: SH). For structures coated with a 2D material sheet, the evanescent field of the exciting wave (also referenced as a fundamental wave: FW) overlaps with the 2D material and could thus take advantage of the sheet's secondorder nonlinearity. The SH frequency wave could then propagate in the integrated structures and be collected at the output. Similarly, to what has been established with other frequency conversion mechanisms (see the section relating to FWM), SHG can benefit from the boosting of a cavity^{57,142} (e.g., photonic crystal cavity^{142}). It can be treated mathematically similarly to FWM (see Ref. 112 for SHG and Ref. 112,118 for FWM). The cases discussed here show larger enhancement with the hybrid cavity than in a more classic propagation scheme. In the reported experiments, samples were pumped with a 1550 nm wavelength laser for a generated signal at around λ = 630 nm. Thus far, the 2D materials considered in these experiments typically are WSe_{2}, MoSe_{2,} and WS_{2} for Refs.142,143, and 144 respectively. In the case of Ref. 142, the addition of the 2D materials induces a twofold increase in the conversion efficiency, reaching an impressive ~ 10^{−2} conversion efficiency. The dependence of the results on the pump power is a characteristic of a nonlinear phenomenon, proving the nonlinear origins of the results, thus confirming SHG. Accounting for the cavity, boosting this enhancement, the improvement was up to 200 times more than bare Si. Fig. 5a−c reports a 5fold efficiency enhancement after waveguide integration. This example showed that inplane, longitudinal and extended interaction between the FW frequency and the 2D material was beneficial in terms of conversion efficiency in the MoSe_{2}/Si waveguide compared to MoSe_{2} layers deposited on a homogeneous Si substrate. This demonstrated that an integrated edgebased scheme provided a reinforced lightmatter interaction compared to free space settings, as shown in Fig. 5b, with a prominent SHG peak compared to the free space measurements. Moreover, the nonlinear nature of this intensity increase was indeed demonstrated (Fig. 5c). Other works in optical fibers using various 2D materials coatings were also reported. The example in Fig. 5d, e is a microfiber coated with WS_{2} in which a factor of ~ 20 on the second harmonic intensity was reached^{143}. As in previous examples, the nonlinear origin of this SHG intensity increase was confirmed by ramping the pump power (Fig. 5e). Additionally, work in hybrid photonic crystal fibers (PCF) has also been recently reported. In Ref. 144, PCFs coated with MoS_{2} and measured conversion efficiency of 10^{−4} to 10^{−3}, an enhancement of 3 orders of magnitudes compared to planar 2D MoS_{2}. Additionally, other results with various 2D materials combined with different integrated structures have also shown great results (e.g., monolayer GaSe on a photonic crystal (PhC) cavity^{145}, GaSe^{146} or InSe^{147} on microfibers). All these results and trends collected on the enhancement of secondorder nonlinear effect by the integration of 2D materials on waveguide photonic structures have opened promising application opportunities as a reliable approach to induce effective secondorder nonlinearities in Si photonics that is difficult to induce by other approaches and by relying on classical Si compatible materials (e.g., Si, SiO_{2}, SiN). Enabling SHG for these platforms will extend the range of applications and possibilities. For example, the realization of onchip optical parametric oscillators, as presented in Ref. 148 (in aluminum nitride) or the generation of entangled photons sources with the pair photon generation enabled by SHG for quantum applications (as depicted in Ref. 149 in lithiumniobate) could become a reality for CMOS compatible platforms, using 2D materials as the second order nonlinear medium.
Fig. 5 Examples of SHG enhancement in various hybrid structures. a−c SHG enhancement with a MoSe_{2} integrated planar Si waveguide^{143}. d Schematics of the device. A grating coupler is designed for the pump at the input and the generated frequency at the output. e Measured spectrum with a significant increase in the SHG emission for the hybrid structure. Peak R1 corresponds to SHG, and peak R2 to PL. f SHG intensity versus the input power. d and e SHG enhancement in a WS_{2} covered optical fiber^{144}. d Optical fiber coated with the 2D material and occurring phenomena during propagation. e SHG intensity versus the input power. The WS_{2}opticalfibernanowire (WOFN) is here compared to the pure opticalfibernanowire (OFN).

Raman scattering originates from the coupling between the incident light and the vibrational frequency of the medium (atoms or molecules)^{62}. Materials thus show different responses due to their different eigenvibrational states. In fact, this is largely used to identify 2D materials based on their Raman signatures^{150,151}. Although this effect is widely studied and exploited in free space, very few examples of Raman applications in the 2D material hybrid platforms exist. For example, some studies have shown a reinforcement in the Raman response of graphene by integrating it on top of a 2D slab of a PhC cavity^{152,153}. Fig 6 shows two examples of enhanced Raman response in graphene using this integration method. PhC cavities were designed here using a threehole defect (L3) cavity (Fig. 6a, d)^{154}, which allowed for direct normal coupling inside the cavity. This cavity was designed for the cavity modes to correspond to the proper excitation wavelength of graphene for an optimal Raman response at 532 nm. The Raman mapping of the cavity (intensity of peak G and 2D as depicted by Fig. 6b) shows a large increase in the cavity (Fig. 6c). In a similar fashion, different measurements performed at key locations (locations are represented in Fig. 6d) show a large increase in PL and Raman signature for measurements performed on the cavity (red curves for Fig. 6e). As depicted in both examples presented in Fig 6, the combination of a cavity and a 2D material strengthens the Raman response of the 2D materials. Fig. 6c, e shows a significant increase with a 3.5 and a 6.1fold enhancement in the Raman emission, respectively. Moreover, using these L3 PhC cavities allows normal, direct injection into the cavity, making the characterization process easy. As pointed out in Ref. 152, other types of photoemissions (PL for instance) from other 2D materials could also be enhanced using this principle^{155}. This is a promising step for characterization purposes.This enhanced interaction between 2D materials, and the probing light might resolve limitations in the current characterization methods by improving the magnitude of the obtained signal. Other 2D materials, such as TMDs or hBN, will also provide interesting alternatives to graphene for this kind of application, thanks to the nature of their bandgap offering other possibilities. Photoluminescence in 2D semiconductors (e.g., TMD), or harmonic generation (SHG, THG, higher harmonic generation (HHG)) might also be investigated using PhC devices. Furthermore, Raman effect could be leveraged in the generation of broadband light sources (as demonstrated in Ref. 156), thus expanding the range of possibilities in achieving efficient onchip nonlinear optical functions.
Fig. 6 Raman response enhancement using grapheneintegrated PhC cavities. a−c Enhancement of the Raman response in graphene deposited on a SiL3 cavity^{153}. a Inplane (top) and crosssection (bottom) energy distribution of the fundamental mode of the PhC L3 cavity (simulation). b Raman response for excitation at 637 nm wavelength. The resulting spectrum shows two characteristic peaks (G at 1578 cm^{−1} and 2D at 2640 cm^{−1}). The cavity resonance, set at 636 nm, is displayed in the inset. c Normalized emission intensity for the two peaks G and 2D, showing the spatial distribution. The Scanning electron microscopy (SEM) top image shows the corresponding localization. d−e Enhancement of the Raman response on graphene deposited on a SiL3 cavity^{152}. d SEM image of the PhC L3 cavity. Scale bars are 1 µm. The inset is a zoomedin image of the cavity. e Raman spectrum of graphene for excitation at 532 nm. The spectrum is taken at two locations on the device: red for the cavity and blue for the SOI plane substrate. In the inset: zoomedin on the normalized 2D peak for spectral comparison.

Until now, we considered absorption to be avoided due to its attenuation nature. However, absorption can also be an ally for applications due to its nonlinear properties. For several applications, TPA, a nonlinear conducting/semiconductor materials media property, can be exploited to trigger new effects. For example, because of the absorption of propagating photons through nonlinear processes such as TPA, free carriers are generated^{157} and affect the propagating waves. In a bulk medium, this can be accounted for by calculating the dynamic of the free carrier’s population using the imaginary part of the effective susceptibility tensor $ \Gamma \text{'}\text{'} $^{64}.
$$ \frac{\partial N}{\partial t}=\frac{N}{{t}_{c}}+\frac{3{\beta }_{1}^{2}{P}_{0}^{2}\mathrm{\Gamma }\text{'}\text{'}}{4{\epsilon }_{0}\hslash {A}_{0}^{2}}{\leftu\right}^{4} $$ (16) with u the normalized field amplitude and t_{c} the carrier lifetime.
In the context of a 2D sheet of graphene, it was established that due to Pauli blocking, interband transitions from the valence band to the conduction band become less likely as the electronic states of the conduction band are filled^{158}. Hence the absorption becomes less probable, effectively translating into a saturation of the absorption process. In this context, it is thus more accurate to use an alternative expression describing the carrier population dynamic accounting for this saturation effect^{159}:
$$ \frac{\partial N}{\partial t}=\frac{N}{{t}_{c}}+\frac{{\alpha }_{sat}}{\hslash \omega }\left(1\frac{N}{{N}_{sat}}\right)\frac{{\leftu\right}^{2}}{w} $$ (17) The overall absorption in the 2D materialcovered waveguide can then be written as such:
$$ \alpha \left(N\right)={\alpha }_{NS}+{\alpha }_{sat}\left(1\frac{N}{{N}_{sat}}\right) $$ (18) where $ {\alpha }_{NS} $ is the nonsaturable component of the absorption, N_{sat} is the saturated carrier density, and w accounts for the interaction length between graphene and the propagating mode within the transverse crosssection and corresponds to the waveguide width. Here, we present several examples of devices based on the saturable absorption properties of graphene (Fig. 7). Adding graphene on top of SiN/SiO_{2} (Fig. 7a, b) or SOI (Fig. 7c, d) waveguides (strip or slot waveguides as depicted in Ref. 161) allows for the introduction of saturable absorption to the silicon platform. The graphene sheet modifies the effective properties of the waveguide by interacting with the propagating mode overlapping with the 2D layer, as shown in Fig. 7a. Accordingly, transmission measurements show a saturation of the absorption as the injected power is increased (see Fig. 7b, d) even more if the interaction length between graphene and the mode is increased (see Fig. 7b, d). Adding saturable absorption properties allows for the design of new optical functions such as modelocking, modulation^{160,161}, and optical switching^{162}. Whereas graphene has been the most studied, other materials are also available, operating at various wavelengths. For instance, MoS_{2} shows promising results both at 1 and 1.5 μm^{162,163}. More generally, many different materials have been investigated for their saturable absorption properties, characterized in different guided settings (PCF, tapered fibers, or microfibers), showing the promises in terms of integrated optics (because of the similarity in the propagating evanescent field interacting with the material of interest)^{159,164,165}. The variety of materials and fabrication/integration methods also allow for expanding the spectral range. In a 1 μm regime, work in a PCF was reported with a coating of the topological insulator Bi_{2}Te_{3}^{166}. Other materials operating at this wavelength could also be investigated as they have shown promising results in unguided settings (graphene^{167}, WS_{2}^{168}, MoS_{2}^{169}, BP^{170}, etc.). Similarly, for the 1.5 μm regime, several results in guided settings have been reported using different 2D materials such as Bi_{2}Se_{3} in PCF^{171}, Bi_{2}Te_{3} on a microfiber^{172}, Heterostructures may also be investigated, MoS_{2}/graphene, WSe_{2}/MoS_{2}, hBN/graphene and other nanocomposite films have shown interesting results^{173176}. The extensive work conducted on saturable absorption has enabled the development of various functions such as Qswitching of modelocking for short pulse duration laser with excellent performances down to the femtosecond regime^{176,177}.
Fig. 7 Examples of graphene saturable absorption in CMOScompatible waveguide structures. a and b Saturable absorption in a graphenecovered SiN waveguide^{159}. a Transverse mode distribution of a graphenecovered planarized SiN buried strip. On the right, a slice of the cross section shows the density of the mode around the graphene layer. b Transmission in this waveguide measured for several lengths of graphene covered waveguides. c and d Saturable absorption in a grapheneSOI strip waveguide^{160}. c Strip grapheneSi waveguide. d transmission measurements for different graphene lengths on top of the waveguides.
Hybrid integration of 2D materials for onchip nonlinear photonics
 Light: Advanced Manufacturing , Article number: 14 (2023)
 Received: 18 January 2023
 Revised: 09 May 2023
 Accepted: 09 May 2023 Published online: 01 August 2023
doi: https://doi.org/10.37188/lam.2023.014
Abstract:
Interests surrounding the development of onchip nonlinear optical devices have been consistently growing in the past decades due to the tremendous applications, such as quantum photonics, alloptical communications, optical computing, onchip metrology, and sensing. Developing efficient onchip nonlinear optical devices to meet the requirements of those applications brings the need for new directions to improve the existing photonic approaches. Recent research has directed the field of onchip nonlinear optics toward the hybrid integration of twodimensional layered materials (such as graphene, transition metal dichalcogenides, and black phosphorous) with various integrated platforms. The combination of wellknown photonic chip design platforms (e.g., silicon, silicon nitride) and different twodimensional layered materials has opened the road for more versatile and efficient structures and devices, which has the great potential to unlock numerous new possibilities. This review discusses the modeling and characterization of different hybrid photonic integration structures with twodimensional materials, highlights the current state of the art examples, and presents an outlook for future prospects.
Research Summary
Hybrid integration of 2D materials for onchip nonlinear photonics
In the past decade, the field of micro and nanotechnologies has undergone significant changes due to the emerging demand for highperformance onchip integrated devices and the advent of quantum computing. Although onchip optical devices have been identified as a promising platform, several limitations still need to be addressed to facilitate integrated photonics' next technological evolution. The integration of disruptive new nanomaterials, particularly 2D materials, with the integrated optics approach, is a promising solution to unlock the full potential of the next generation of onchip devices. The unique electronic, optical, and mechanical properties of 2D materials can enable the development of ultracompact and efficient photonic components, such as modulators and detectors, and new functionalities. The integration of 2D materials with integrated optics is a promising avenue for revolutionizing the field of onchip devices, paving the way for significant technological advances.
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