values specified in a separate “Comments” column. Also included in the “Comments” column are individual nonlinearity contributions that could be separately determined per constituent material, when stated as such. Lastly, some papers specify the dependence of the NLO parameters on wavelength, waveguide/fiber dimension, doping level, etc. or have notes associated with their measurement/analysis such as the formulas used to calculate conversion efficiencies η. This information is listed within the “Comments” column.

<figure><img src="image_10.png" /><figcaption>**Fig. 8.** SEM images of NLO hybrid fibers and hybrid on-chip waveguides. The labels of the sub-figures correspond to the labels given in Tables 8A and 8B. Panels a and e are reprinted with permission from [Benabid2002, Nielsen2017], AAAS. Panels b, f, and g are reprinted with permission from [Yang2020, Gu2012, Koos2009], Springer Nature. Panels c and h are reprinted with permission from [An2020, Alexander2017], ACS. Panel d is reprinted with permission from [Wang2018], CLP Publishing.</figcaption></figure>

## 3.7.2 Discussion

In view of the relatively recent emergence of low-dimensional materials, their combination with on-chip waveguides and fibers is quite a new development with most progress being made over the last 5 to 10 years. Within this short period of time, a wide variety of 2D materials has been explored for this purpose, as can be seen in Tables 8A and 8B. At the same time, the Tables also show that organic solids, liquids and gasses still are very interesting NLO media for constructing hybrid waveguiding systems.

### 3.7.2.1 Advancement since 2000 and remaining challenges

#### 3.7.2.1.1 Advancement and challenges for hybrid fibers

Table 8A for NLO hybrid fibers shows two distinct novelties as compared to the works before 2000, the first novelty being the emergence of SiO₂ hollow-core photonic-crystal fibers (PCFs) [Russell2014]. In contrast to hollow fiber capillaries which intrinsically are leaky waveguides even when filled with, e.g., gasses, hollow-core PCFs efficiently trap light within their hollow core by exploiting the physics of photonic bandgaps [Russell2014]. As such, various NLO effects have been demonstrated in hollow-core PCFs filled with liquids and gasses [Baghwat2008, Russell2014], exhibiting strong NLO responses as shown by the examples listed in the Table [Benabid2002, Vieweg2010, Renninger2016, Yang2020]. That said, combining ultra-low loss and truly robust single-mode propagation in a hollow-core PCF still
represents a challenge and requires special design approaches [Amrani2021]. It should also be noted that, because of the intrinsic simplicity of fiber capillaries as compared to PCFs, several research groups continued using capillaries after 2000 to realize hybrid fibers for different NLO functionalities [Schneebeli2013, Chemnitz2017].

A second important novelty in NLO hybrid fiber development after 2000 has been the rise of low-dimensional materials [Liu2020]. Both 1D carbon nanotubes and 2D materials such as graphene, graphene oxide, black phosphorus and several types of Transition Metal Dichalcogenides (TMDs) were deposited on side-polished or tapered fibers (see, for example, [Xu2013, Park2015, Lee2015, Martinez2017, Steinberg2018, Wang2018, Chen2019, An2020]). The low-dimensional materials either enhanced the already present NLO response of the bare fibers [Xu2013, An2020] and sometimes even made them tunable by means of electrical gating [An2020], or they enriched the fiber components with additional NLO functionalities such as frequency doubling [Chen2019] and saturable absorption [Park2015, Lee2015, Martinez2017, Steinberg2018, Wang2018]. The latter is particularly useful for the development of pulsed fiber lasers [Liu2020]. Still, transferring low-dimensional materials onto side-polished or tapered fiber surfaces can be quite challenging. Efforts for the direct growth of, e.g., graphene on dielectrics such as glass are currently ongoing [Khan2018] and could circumvent the need for complex material transfer procedures while improving standardization and upscaling of the hybrid fiber fabrication. To assess the quality of fabricated hybrid fibers, careful characterization is required [Vermeulen2022]. Note that particularly for graphene-based devices it is important to know the doping level of the graphene layer as the latter determines whether or not a strong NLO response can be expected from the 2D sheet [CastelloLurbe2020]. Finally, as shown in Table 8A, the strong nonlinear refractive response of fibers enriched with low-dimensional materials is often accompanied by high linear losses [Xu2013, An2020]. At the same time, taking into account that NLO research with low-dimensional materials still is at a relatively early stage, it is remarkable how much progress has been made over the past few years on the implementation of these materials in NLO hybrid fibers.

### 3.7.2.1.2 Advancement and challenges for hybrid on-chip waveguides

A first novelty seen in Table 8B for NLO hybrid on-chip waveguides is that NLO organics are nowadays combined with advanced waveguiding structures such as plasmonic or slot waveguides [Nielsen2017, Koos2009]. This advancement has been enabled by the major progress in photonic chip fabrication technology over the past decades. Both plasmonic and slot waveguides intensify the electromagnetic field in the NLO polymer: the former by focusing the light down to the nanoscale inside a metallic structure close to the polymer [Nielsen2017, Tuniz2021], the latter by having its waveguide mode centered around the polymer located within the slot [Koos2009, Steglich2021]. These two approaches give rise to a strong enhancement of the electromagnetic field strength yielding very high NLO efficiencies, albeit with significant propagation losses in the waveguides. Today's research on hybrid plasmonic or slot waveguides mainly focuses on how to maintain such a strong field enhancement while keeping the losses low [Tuniz2021, Steglich2021].

Hybrid on-chip waveguide development has also greatly benefitted from the emergence of NLO 2D materials [Li2018]. Similarly to the case of hybrid fibers, the combination of on-chip waveguides with graphene, graphene oxide, and TMDs either strengthened the already present NLO response of the waveguides [Gu2012, Vermeulen2016a, Alexander2017, Vermeulen2018, Yang2018, Zhang2020, Qu2020] and sometimes made the response electrically tunable [Alexander2017], or they introduced additional NLO functionalities such as saturable absorption [Demongodin2019], frequency doubling [Chen2017] and difference frequency generation [Yao2018]. Most on-chip waveguides used in these experiments were Si or $Si_3N_4$ waveguides with a standard strip geometry. Transferring 2D materials onto such waveguides embedded within a planar photonic chip poses fewer practical challenges as compared to the transfer onto fiber surfaces. Nevertheless, direct synthesis of the 2D materials on the waveguides would be beneficial from the point of view of standardization and upscaling [Kahn2018]. Just like their fiber-based counterparts, hybrid on-chip waveguides require careful material
characterization of their constituents [Vermeulen2022], and for graphene-based devices this also includes evaluating the doping level of the 2D layer (see Section above). Finally, as shown in Table 8B, the strong nonlinear refractive response of on-chip waveguides covered with 2D materials often comes with high losses (see, for example, [Vermeulen2016a, Alexander2017, Vermeulen2018, Yang2018, Zhang2020]). That said, the obtained results are promising, and further progress can be expected in the coming years as the fabrication of photonic chips and 2D materials will continue to improve and the fundamental understanding of the NLO physics of 2D materials will be further extended.

### 3.7.2.2 Recommendations for future works on hybrid waveguiding systems

Our general recommendation for future works on NLO hybrid waveguides is to report at least those parameters that are included in Tables 8A and 8B, as these provide essential information on the hybrid systems. So, including these parameters in future publications can be considered as a best practice in addition to the more general best practices described in Section 2.

To maximize the impact of future works, it is of particular importance to extract the actual nonlinearity, e.g., $\gamma_{eff}$, of the hybrid system (rather than just specifying the relative NLO enhancement enabled by the added material) and to study the dependence of the nonlinearity on, e.g., wavelength. Ideally, one could also go the extra mile of determining the individual nonlinearities (magnitude and sign) of the fiber/waveguide medium and the added medium along the considerations outlined above. By then comparing the retrieved NLO coefficients with measurements of the same materials in a ‘stand-alone’ configuration, it is possible to evaluate if combining them in a hybrid waveguiding system leaves their individual NLO properties unchanged or not.

We would also like to point out the need for a detailed description of the material properties (including the doping level when working with graphene) and of the linear optical properties of all constituents in the hybrid system. This is not always done in the existing literature, which makes it difficult to properly interpret and reproduce the reported results. For example, in addition to specifying the linear optical loss for the complete hybrid system, one should also quantify the initial optical loss of the bare fiber/waveguide. In a similar way, the material to be added needs to be optically characterized beforehand. To allow for reproducibility, future works should also provide a detailed description of the fabrication procedure for realizing the hybrid system and specify its geometrical outline, both in the cross-sectional plane and along the optical axis. The latter comprises specifying the length over which the added material is present as well as the remaining length of bare fiber/waveguide. This way, the reader can get the full picture of the system under study, which is essential in view of the inherent complexity of hybrid waveguiding structures. As such, the research community will be able to efficiently build upon state-of-the-art results so that the ‘best of both worlds’ idea behind hybrid waveguiding systems can be exploited to its full potential.
### 3.7.3 Data table for hybrid waveguiding systems

<table><caption>**Table 8A.** NLO properties of hybrid fibers from representative works since 2000. Legend for superscripts: see below the table.</caption><thead><tr><th colspan="3">Material Properties</th><th colspan="2">Measurement Details</th><th colspan="4">Nonlinear Properties</th><th rowspan="2">Reference</th></tr><tr><th>Constituent 1<br>Fiber type</th><th>Constituent 2<br>Added material<br>Thickness<br>Fabrication</th><th>Hybrid fiber<br>Length<br>Loss /<br>wavelength<sup>1</sup><br>Effective area<br>SEM image<sup>2</sup></th><th>Method</th><th>Pump<br>Wavelength<br>Peak power<br>Pulse width<br>Rep. rate</th><th>Probe<br>Wavelength<br>Peak power<br>Pulse width</th><th>|Re(γ<sub>eff</sub>)|<br>[m<sup>-1</sup> W<sup>-1</sup>]</th><th>I<sub>sat,eff</sub><br>[W/m<sup>2</sup>]</th><th>Sat.<br>loss<br>[%]</th><th>Additional parameters and comments</th></tr></thead><tbody><tr><td>SiO<sub>2</sub> hollow-core fiber<br>Capillary</td><td>B-carotene<br>-<br>-</td><td>500 [mm]<br>-<br>-<br>-</td><td>IRS Pump-Probe</td><td>1550 [nm]<br>-<br>8 [ps]<br>50 [MHz]</td><td></td><td></td><td></td><td></td><td>g<sub>Raman</sub>: 6.00E-13 [m/W], shift: 1.16E+05 [m<sup>-1</sup>]<br>5 µm core diameter</td><td>Schneebeli2013</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>Capillary</td><td>CCl<sub>4</sub><br>-<br>-</td><td>1 [m]<br>-<br>-<br>-</td><td>IRS Pump-Probe</td><td>1550 [nm]<br>-<br>3 [ps]<br>50 [MHz]</td><td></td><td></td><td></td><td></td><td>g<sub>Raman</sub>: 1.50E-12 [m/W], shift: 4.59E+04 [m<sup>-1</sup>]<br>32 mW average pump power; 10 µm core diameter</td><td>Schneebeli2013</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>Capillary</td><td>CS<sub>2</sub><br>-<br>-</td><td>450 [mm]<br>-<br>-<br>-</td><td>IRS Pump-Probe</td><td>1550 [nm]<br>-<br>8 [ps]<br>50 [MHz]</td><td></td><td></td><td></td><td></td><td>g<sub>Raman</sub>: 3.00E-11 [m/W], shift: 6.6E+04 [m<sup>-1</sup>]<br>1.5 mW average pump power; 2 µm core diameter</td><td>Schneebeli2013</td></tr></tbody></table>

### Hybrid fibers
<table><tr><td>SiO<sub>2</sub> hollow-core fiber<br>Capillary</td><td>CS<sub>2</sub></td><td>140 [mm]<br>-<br>-</td><td>Other</td><td>1950 [nm]<br>-<br>460 [fs]<br>5.6 [MHz]</td><td>2.80E-01</td><td>Supercontinuum generation; 14 nJ pump pulse energy; 4.7 µm core diameter</td><td>Chemnitz2017</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>PCF</td><td>Air<br>-<br>-</td><td>1.61 [m]<br>-<br>-</td><td>Two-color<br>SBS Pump-Probe</td><td>1535 [nm]<br>48 [mW]<br>CW<br>-</td><td>1.60E-04</td><td>Γ<sub>Brillouin</sub>: 9.00E-04 [W<sup>-1</sup>m<sup>-1</sup>], shift: 3.50E+07 [Hz], linewidth: 4.00E+06 [Hz]<br>Forward Brillouin scattering; 1 atm air pressure; 5.65 µm core diameter</td><td>Renninger2016</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>PCF</td><td>CCl<sub>4</sub><br>-<br>-</td><td>190 [mm]<br>-<br>-</td><td>Other</td><td>1030 [nm]-<br>210 [fs]<br>44 [MHz]</td><td>3.70E-01</td><td>Supercontinuum generation; 330 mW average pump power; 2.5 µm core diameter</td><td>Vieweg2010</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>PCF</td><td>CO<sub>2</sub><br>-<br>-</td><td>50 [m]<br>-<br>80 [µm<sup>2</sup>]<br>Fig. 8(b)</td><td>Two-color<br>SBS Pump-Probe</td><td>1550 [nm]<br>6 [mW]<br>CW<br>-</td><td></td><td>Γ<sub>Brillouin</sub>: 1.68 [W<sup>-1</sup>m<sup>-1</sup>], shift: 3.20E+08 [Hz], linewidth: 3.65E+06 [Hz]<br>Backward Brillouin scattering; 41 bar CO<sub>2</sub> pressure; 10 µm core diameter</td><td>Yang2020</td></tr><tr><td>SiO<sub>2</sub> hollow-core fiber<br>PCF</td><td>H<sub>2</sub><br>-<br>-</td><td>320 [mm]<br>-<br>-<br>Fig. 8(a)</td><td>SRS</td><td>532 [nm]<br>-<br>6 [ns]<br>20 [Hz]</td><td></td><td>30% pump-to-Stokes conversion efficiency for 4.5 uJ pump pulse energy; 17 bar H<sub>2</sub> pressure; 15 µm core diameter</td><td>Benabid2002</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Side polished</td><td>Black phosphorus<br>10 [nm]<br>Mechanical exfoliation</td><td>100 [mm]<br>73.6 [%]<br>/1566 [nm]<br>-<br>-</td><td>Nonlinear transm./refl.</td><td>1566 [nm]<br>-<br>1.2 [ps]<br>14.2 [MHz]</td><td>1.25E+11</td><td>3.31</td><td>Park2015</td></tr></table>
<table><tr><td>SiO<sub>2</sub> solid-core fiber<br>Side polished</td><td>Graphene<br>Monolayer<br>CVD</td><td>500 [µm]<br>47 [%]<br>/1561 [nm]<br>40 [µm<sup>2</sup>]<br>Fig. 8(c)</td><td>FWM</td><td>1561 [nm]<br>300 [W]<br>265 [fs]<br>37.8 [MHz]</td><td>1480-1610<br>[nm]<br>10 [mW]<br>CW</td><td>1.14</td><td></td><td>Graphene doping: variable (values here for<br>0.2 eV)<br>|χ<sup>(3)</sup>|: 8E-17 [m<sup>2</sup>/V<sup>2</sup>]<br>Nonlinearity dependence on graphene doping<br>level shown in Fig. 2 of reference paper</td><td>An2020</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Side polished</td><td>Graphene<br>2 layers<br>CVD</td><td>5 [mm]<br>8.7 [%]<br>/1550 [nm]<br>-<br>-</td><td>Nonlinear<br>transm./refl.</td><td>1609 [nm]<br>-<br>423 [fs]<br>30.9 [MHz]</td><td></td><td></td><td>2.57E+12</td><td>1.06</td><td>Graphene doping: variable (values here for<br>-1.2V gate voltage)</td><td>Lee2015</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Side polished</td><td>Graphene oxide<br>100 [nm]<br>Modified<br>Hummers<br>method</td><td>10 [mm]<br>42 [%]<br>/1550 [nm]<br>-<br>-</td><td>Nonlinear<br>transm./refl.</td><td>1550 [nm]<br>19.4 [kW]<br>150 [fs]<br>89 [MHz]</td><td></td><td></td><td></td><td>22</td><td>Saturation fluence: 7E-2 [J/m<sup>2</sup>]</td><td>Steinberg2018</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Tapered</td><td>Carbon<br>nanotubes<br>-<br>High-pressure<br>CO process</td><td>100 [mm]<br>68 [%]<br>/1550 [nm]<br>50 [µm<sup>2</sup>]<br>-</td><td>FWM</td><td>1550 [nm]<br>1.6 [W]<br>CW<br>-</td><td>1552 [nm]<br>-<br>0.1 [ns]</td><td>1.82</td><td></td><td></td><td>Semiconducting nanotubes; 1ppm in PTFEMA<br>Conversion efficiency: 0.2 [%]<br>Conversion efficiency formula:<br>idler_output/signal_input<br>Conversion bandwidth: 9 [nm]</td><td>Xu2013</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Tapered</td><td>Carbon<br>nanotubes<br>-<br>High-pressure<br>CO process</td><td>4 [mm]<br>34 [%]<br>/1550 [nm]<br>-<br>-</td><td>Nonlinear<br>transm./refl.</td><td>1550 [nm]<br>-<br>600 [fs]<br>25 [MHz]</td><td></td><td></td><td></td><td>8.5</td><td>Semiconducting nanotubes; 50% in PTFEMA;<br>variable fiber diameter (values here for 3 µm)<br>Average saturation power: 0.45E-3 [W]<br>Nonlinearity dependence on diameter of<br>tapered fiber shown in Table II of reference<br>paper</td><td>Martinez2017</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Tapered</td><td>MoTe<sub>2</sub><br>11.6 [nm]<br>Magnetron<br>sputtering<br>deposition</td><td>10 [mm]<br>44.6 [%]<br>/1550 [nm]<br>-<br>Fig. 8(d)</td><td>Nonlinear<br>transm./refl.</td><td>1572.4<br>[nm]<br>-<br>642 [fs]<br>50.12<br>[MHz]</td><td></td><td></td><td>9.60E+10</td><td>25.5</td><td></td><td>Wang2018</td></tr><tr><td>SiO<sub>2</sub> solid-core fiber<br>Tapered</td><td>MoTe<sub>2</sub><br>11.6 [nm]<br>Magnetron<br>sputtering<br>deposition</td><td>10 [mm]<br>43.3 [%]<br>/1900 [nm]<br>-<br>Fig. 8(d)</td><td>Nonlinear<br>transm./refl.</td><td>1915.5<br>[nm]<br>-<br>1.25 [ps]<br>18.72<br>[MHz]</td><td></td><td></td><td>1.23E+11</td><td>22.1</td><td></td><td>Wang2018</td></tr></table>
<table><tr><td>SiO<sub>2</sub> solid-core fiber Tapered</td><td>WS<sub>2</sub> Monolayer CVD</td><td>60 [µm]<br/>&lt;25 [%]<br/>/1530-1590 [nm]<br/>-</td><td>SHG</td><td>1550 [nm]<br/>-</td><td>10 [ns]<br/>1 [MHz]</td><td>20-fold enhancement of SHG signal as compared to bare fiber for 50 mW average pump power</td><td>Chen2019</td></tr></table>

<footnote n="1">¹: Wavelength at which the specified loss has been measured</footnote>

<footnote n="2">²: Selected SEM images presented in this work</footnote>
<table><caption>**Table 8B.** NLO properties of hybrid on-chip waveguides from representative works since 2000. Legend for superscripts: see below the table.</caption><thead><tr><th colspan="3">Material Properties</th><th colspan="3">Measurement Details</th><th colspan="3">Nonlinear Properties</th><th rowspan="2">Reference</th></tr><tr><th>Constituent 1<br>Waveguide<br>(wg) medium<br>Lithography<br>(lith.) type</th><th>Constituent 2<br>Added material<br>Thickness<br>Fabrication</th><th>Hybrid wg<br>Length<br>Loss/<br>wavelength<sup>1</sup><br>Effective area<br>SEM image<sup>2</sup></th><th>Method</th><th>Pump<br>Wavelength<br>Peak power<br>Pulse width<br>Rep. rate<br>Polarization</th><th>Probe<br>Wavelength<br>Peak power<br>Pulse width<br>Polarization</th><th>|Re(γ<sub>eff</sub>)|<br>[m<sup>-1</sup> W<sup>-1</sup>]</th><th>η [%]</th><th>Additional parameters and comments</th></tr></thead><tbody><tr><td>Hydex wg<br>Photolith.</td><td>Graphene oxide<br>2 layers<br>Solution-based process</td><td>15 [mm]<br>0.2 [dB/mm]<br>/1550 [nm]</td><td>FWM</td><td>1550 [nm]<br>160 [mW]<br>CW<br>-<br>TE mode</td><td>1551 [nm]<br>160 [mW]<br>CW<br>TE mode</td><td>9.00E-01</td><td>2.00E-03</td><td>Conversion efficiency formula:<br>idler_output/signal_output<br>Conversion bandwidth: 2.50E12 [Hz]<br>Nonlinearity dependence on wavelength<br>detuning and length shown in Fig. 4 of<br>reference paper</td><td>Yang2018</td></tr><tr><td>Plasmonic<br>(gold) wg<br>E-beam lith.</td><td>Polymer MEH-PPV<br>-<br>Spin-coating</td><td>2 [μm]<br>-<br>-<br>Fig. 8(e)</td><td>FWM</td><td>1480 [nm]<br>30 [W]<br>1.04 [ps]<br>-<br>TE mode</td><td>1450 [nm]<br>-<br>1.11 [ps]<br>TE mode</td><td>3.09E+04</td><td>4.60</td><td>Im(γ<sub>eff</sub>): 7.00E2 [m<sup>-1</sup> W<sup>-1</sup>]<br>Conversion efficiency formula:<br>idler_output/signal_input</td><td>Nielsen2017</td></tr><tr><td>Si photonic-crystal wg<br>Photolith.</td><td>Graphene<br>Monolayer<br>CVD</td><td>1.5826 [μm]<br>(cavity length)<br>-<br>-<br>Fig. 8(f)</td><td>FWM</td><td>1562.36<br>[nm]<br>0.6 [mW]<br>CW<br>-<br>TE mode</td><td>1562.09<br>[nm]<br>600 [μW]<br>CW<br>TE mode</td><td></td><td>1.00E-01</td><td>Graphene doping: 5.00E16 [m<sup>-2</sup>]<br>Free-carrier lifetime: 2.00E-10 [s]<br>|n<sub>2,eff</sub>|: 4.80E-17 [m<sup>2</sup>/W]</td><td>Gu2012</td></tr></tbody></table>

Hybrid on-chip waveguides
<table><tr><td>Si slot wg Photolith.</td><td>Organic molecules DDMEBT - Molecular beam deposition</td><td>4 [mm] 1.6 [dB/mm] /1550 [nm] - Fig. 8(g)</td><td>FWM</td><td>1548 [nm] 375 [mW] 3 [ps] 42.7 [GHz] TE mode</td><td>1556 [nm] - 3.00 [ps] TE mode</td><td>1.04E+02</td><td>6.00E-02</td><td>Conversion efficiency formula: idler_output/signal_output</td><td>Koos2009</td></tr><tr><td>Si wg Photolith.</td><td>Graphene Monolayer CVD</td><td>400 [µm] 132 [dB/mm] /1550 [nm] -</td><td>SPM/XPM spectral broadening</td><td>1553 [nm] 1.68 [W] 3 [ps] 80 [MHz] TE mode</td><td></td><td>1.40E+03</td><td></td><td></td><td>Graphene doping: -0.2 [eV] n<sub>2,eff</sub> of graphene only: -1E-13 [m<sup>2</sup>/W]</td><td>Vermeulen2016a</td></tr><tr><td>Si wg E-beam lith.</td><td>MoS<sub>2</sub> 3 [nm] CVD</td><td>1.6 [mm] 5.5 [dB/mm] /1560 [nm] -</td><td>FWM</td><td>1550 [nm] - CW - TM mode</td><td>1550.5 [nm] - CW TM mode</td><td></td><td>6.00E-02</td><td></td><td>Conversion efficiency formula: idler_output/signal_output Nonlinearity dependence on wavelength detuning shown in Fig. 6 of reference paper</td><td>Zhang2020</td></tr><tr><td>Si<sub>3</sub>N<sub>4</sub> wg Photolith.</td><td>Graphene oxide Monolayer Solution-based process</td><td>20 [mm] 0.61 [dB/mm] /1550 [nm] -</td><td>FWM</td><td>1549 [nm] 63 [mW] CW - TE mode</td><td>1550 [nm] 63 [mW] CW TE mode</td><td>1.31E+01</td><td>1.40E-04</td><td></td><td>Conversion efficiency formula: idler_output/signal_input Conversion bandwidth: 1.25E12 [Hz] Nonlinearity dependence on graphene-oxide layer number shown in Fig. 8 of reference paper</td><td>Qu2020</td></tr><tr><td>Si<sub>3</sub>N<sub>4</sub> wg Photolith.</td><td>Graphene Monolayer CVD</td><td>100 [µm] 50 [dB/mm] /1550.18 [nm] - Fig. 8(h)</td><td>FWM</td><td>1550.18 [nm] 11.2 [mW] CW - TM mode</td><td>1550.7 [nm] - CW TM mode</td><td>6.40E+03</td><td>1.00E-03</td><td></td><td>Graphene doping: variable (values here for -0.35 eV, i.e. -0.5 V gate voltage) Conversion efficiency formula: idler_output/signal_output Nonlinearity dependence on graphene doping level and wavelength detuning shown in Fig. 4 of reference paper</td><td>Alexander2017</td></tr></table>
<table><tr><td>Si<sub>3</sub>N<sub>4</sub> wg<br>Photolith.</td><td>Graphene<br>Monolayer<br>-</td><td>3.2 [mm]<br>12.6 [dB/mm]<br>/1540 [nm]<br>0.81 [µm<sup>2</sup>]<br>-</td><td>Nonlinear<br>transm./refl.<br>-</td><td>1547 [nm]<br>-<br>200 [fs]<br>20 [MHz]<br>TE mode</td><td></td><td>Graphene doping: -0.3 [eV]<br>Free-carrier lifetime: 1.50E-13 [s]<br>Saturation density: 1.30E+16 [m<sup>2</sup>]<br>Non-saturable loss: 7.5 [dB/mm]</td><td>Demongodin2019</td></tr><tr><td>Si<sub>3</sub>N<sub>4</sub>/SiO<sub>2</sub><br>wg<br>Photolith.</td><td>Graphene<br>Monolayer<br>CVD</td><td>1.1 [mm]<br>20 [dB/mm]<br>/1563 [nm]<br>-<br>-</td><td>SPM/XPM<br>spectral<br>broadening</td><td>1563 [nm]<br>2.7 [W]<br>3 [ps]<br>80 [MHz]<br>TE mode</td><td>Proportional<br>to free-carrier<br>refraction<br>coefficient</td><td>Graphene doping: 6.50E16 [m<sup>-2</sup>]<br>Free-carrier lifetime: 1.00E-12 [s]<br>Free-carrier refraction coefficient: 1.00E-5 [-]</td><td>Vermeulen2018</td></tr><tr><td>a-Si wg<br>E-beam lith.</td><td>MoSe<sub>2</sub><br>Monolayer<br>Mechanical exfoliation</td><td>22 [µm]<br>-<br>-<br>-</td><td>SHG</td><td>1550 [nm]<br>-<br>82 [fs]<br>TE mode</td><td></td><td>5-fold enhancement of SHG signal as<br>compared to free-space excitation of<br>MoSe<sub>2</sub><br>Pump irradiance: 150 [TW/m<sup>2</sup>]</td><td>Chen2017</td></tr><tr><td>Si<sub>3</sub>N<sub>4</sub> wg<br>Photolith.</td><td>Graphene<br>Monolayer<br>CVD</td><td>80 [µm]<br>100 [dB/mm]<br>/1550 [nm]<br>-<br>-</td><td>DFG</td><td>1531.9 [nm]<br>200 [W]<br>2.2 [ps]<br>TM mode</td><td>1593.2 [nm]<br>1.6 [W]<br>CW<br>TM mode</td><td>4.00E-3<br>[W<sup>-1</sup>]<br><br>Graphene doping: variable (values here for<br>0.05 eV)<br>χ<sup>(2)</sup>: 1.20E-6 [esu]<br>Plasmon-enhanced DFG<br>Conversion efficiency formula:<br>DFG_plasmon_irradiance /<br>(signal_irradiance*pump_irradiance)<br>Nonlinearity dependence on Fermi level<br>shown in Fig. 4 of reference paper</td><td>Yao2018</td></tr></table>

<footnote n="1">¹: Wavelength at which the specified loss has been measured</footnote>

<footnote n="2">²: Selected SEM images presented in this work</footnote>
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## 3.8 THz NLO: data table and discussion

Team: Dmitry Turchinovich

### 3.8.1 Introduction

#### 3.8.1.1 THz NLO mechanisms

As already mentioned in Sections 3.1.1 and 3.3.1, some materials exhibit pronounced nonlinear responses not only at optical wavelengths but also in the THz domain. These THz nonlinearities can originate from various physical processes. The free carrier response typically constitutes the strongest, and most broadband, contribution to the dielectric function of materials at THz frequencies [Dressel and Grüner 2002; Huber 2001; Jepsen 2011; Ulbricht 2011]. Consequently, the strongest NLO response at THz frequencies is also usually dominated by free-carrier effects, and hence is characteristic of conductive materials, or materials that become conductive under intense THz excitation (see, for example, [Blanchard 2011; Cheng 2020; Deinert 2021; Fan 2013; Giorgianni 2016; Grady 2013; Hafez 2018, Hafez 2020; Hirori 2011; Hoffmann 2009b; Hoffmann and Turchinovich 2010; Hohenleutner 2015; Hwang 2013; Jadidi 2016; Junginger 2012; König-Otto 2017; Kovalev 2020, Kovalev 2021; Lee 2020; Liu 2012; Matsunaga 2013; Mayer 2015; Mics 2015; Schubert 2014; Sharma 2010; Shimano 2012; Turchinovich 2012]). Here, the THz electromagnetic field couples to the free carriers via optical conductivity mechanisms, leading to absorption of a part of the electromagnetic energy by the electronic system of the material. This energy transfer from the driving THz field to the free electrons typically results in heating of the electron population of the material, or quasi-coherent ponderomotive acceleration of carriers within the band structure, both leading to the concomitant temporal modification of the THz optical conductivity of the material, and hence to its nonlinear response to the driving THz field.

THz nonlinearities may also arise from THz-driven phase transitions (e.g., [Liu 2012]), or from direct, quasi-resonant excitation of THz intersubband transitions in semiconductor nanostructures such as quantum wells or superlattices (e.g., [Houver 2019; Kuehn 2011; Raab 2019, Raab 2020]). IR-active optical phonons with resonant frequencies in the THz range also contribute to the THz nonlinearity in crystals [Dekorsy 2003; Lu 2021; Mayer and Keilmann 1986a].

The THz nonlinearities resulting from the direct THz excitation of electrons or IR-active phonon modes in crystals are dissipative in nature, as the driving THz field is physically absorbed by the material in the first steps of the light-matter interaction. The materials exhibiting the strongest nonlinearities at THz frequencies are, typically, doped semiconductors, superconductors, and doped quantum materials such as graphene, topological insulators, and 3D Dirac semimetals (see, for example, [Blanchard 2011; Cheng 2020; Deinert 2021; Fan 2013; Giorgianni 2016; Grady 2013; Hafez 2018, Hafez 2020; Hirori 2011; Hoffmann 2009b; Hoffmann and Turchinovich 2010; Hohenleutner 2015; Hwang 2013; Jadidi 2016; Junginger 2012; König-Otto 2017; Kovalev 2020, Kovalev 2021; Lee 2020; Matsunaga 2013; Mayer 2015; Mics 2015; Schubert 2014; Sharma 2010; Shimano 2012; Turchinovich 2012]). Doped graphene was shown to have by far the strongest electronic nonlinearity at THz frequencies [Hafez 2018]. Its nonlinear coefficients at THz frequencies surpass those of all other known materials by many orders of magnitude, and this possibly holds for other spectral ranges as well [Hafez 2018, Hafez 2020].

The non-dissipative THz nonlinearities such as the THz Kerr effect [Hoffmann 2009a; Sajadi 2015] or the THz-driven quantum-confined Stark effect (QCSE) [Hoffmann 2010], also reported in the literature, are usually weaker in strength than the dissipative electronic nonlinearities. These non-dissipative nonlinear effects are typically observed in a THz pump – optical probe arrangement, where the strong THz field modifies the conditions of light-matter interactions for the optical-frequency probe in the material. For THz Kerr measurements, optically transparent solids and liquids are typically used as samples, whereas for the observation of THz QCSE, quantum nanostructures featuring resonant absorption at the optical probe wavelength are used. A somewhat related type of THz NLO experiment is the THz-driven side-band generation on a CW optical carrier wave, which propagates through a THz-
nonlinear material or device [Dhillon2007; Zaks2012]. These experiments are usually performed in a quasi-CW mode, and the THz sideband generation is registered in the spectrum of the carrier probe signal at optical frequencies.

### 3.8.1.2 Brief history of THz NLO research

THz NLO is a relatively young discipline. The key reason for this is the relative difficulty of generating strong fields at THz frequencies, as compared to visible and IR optical signals. In the early works on THz NLO from the 1980-1990s gas lasers and THz free-electron lasers (FELs) were used as sources, and the experiments were performed in a quasi-CW mode (see, for example, [Bewley1993; Van Dantzig and Planken 1999; Dekorsy2003; Ganichev and Prettl 2006; Heyman1994; Markelz1994a; Markelz1994b; Mayer and Keilmann 1986a, Mayer and Keilmann 1986b; Pellemans and Planken 1998; Winnerl2000]). Almost all these experiments were performed on semiconductors, both bulk and nanostructured.

Since the 1990s THz spectroscopy is dominated by the THz time-domain spectroscopy (THz-TDS) method (see, for example, [Grischkowsky1990; Jepsen2011; Tonouchi2007; Ulbricht2011]). In THz-TDS, single-cycle THz pulses are generated from femtosecond laser pulses in photoconductive antennas or via optical rectification in $\chi^{(2)}$-nonlinear crystals. The detection of such THz transients in THz-TDS is enabled via photoconductive detection, or via free-space electrooptic sampling (FEOS) in $\chi^{(2)}$-nonlinear crystals [Gallot and Grischkowsky 1999; Planken2001; Zhang and Turchinovich 2021]. Both of these techniques involve a femtosecond laser pulse as a time gate, providing for the field-resolved detection of THz electromagnetic transients with sub-cycle temporal resolution. Importantly, FEOS also permits the calibrated detection of THz fields, yielding the temporal evolution of the instantaneous electric field strength in absolute units within the detected THz field transient [Gallot and Grischkowsky 1999; Planken2001; Zhang and Turchinovich 2021]. If certain conditions are observed, the recorded FEOS signals can be rigorously reconstructed back to the electric field evolution in the propagating THz signal, and even to the initial polarization or magnetization dynamics in the THz emitter (see, for example, [Hafez2018; Zhang2020; Zhang and Turchinovich 2021]). We note that standard table-top THz generation via photoconductive mechanisms or via optical rectification yields THz fields not exceeding a few kV/cm in strength, thus only providing for spectroscopy in the linear regime.

In 2002 highly efficient strong-field THz generation via optimized optical rectification by tilted pulse front pumping (TPFP) of $\chi^{(2)}$-nonlinear crystals was proposed by Hebling et al. [Hebling2002], which later led to the demonstration of THz pulses with peak field strength reaching 250 kV/cm by Yeh et al. [Yeh2007]. This demonstration paved the way to modern nonlinear THz spectroscopy using table-top sources. Since then, the TPFP of lithium niobate (LN) crystals by mJ-level femtosecond laser pulses dominates the table-top strong-field THz generation, typically yielding single-cycle THz pulses with the spectrum covering the range 0–3 THz, and with (sub-)MV/cm peak electric fields. Amplified Ti:Sapphire femtosecond lasers, delivering mJ-level femtosecond pulses with 800 nm central wavelength remain the most popular pumping sources for strong-field THz generation via the TPFP method in LN, however other types of femtosecond lasers, e.g., Yb-based [Hoffmann2007], can also be used. Other modern methods of table-top strong-field THz generation include air-plasma THz generation [Tani2012], typically yielding ultrabroadband single-cycle THz pulses with the spectrum covering the range of 0–20 THz or even broader, and optical rectification in organic nonlinear crystals [Shalaby and Hauri 2015]. We refer the reader to the following reviews and article collections on modern methods of strong-field THz generation and nonlinear THz spectroscopy [Elsaesser2019; Hafez2016; Hoffmann and Fülöp 2011; Hwang2015; Kampfrath2013; Leitenstorfer2014].

Furthermore, latest-generation relativistic accelerator-based sources, such as those at the TELBE facility at Helmholtz-Zentrum Dresden-Rossendorf [Green2016], are also actively used in modern THz NLO research. They deliver quasi-monochromatic multi-cycle THz pulses with peak fields currently reaching 100s kV/cm (and with the potential to reach MV/cm levels), and are precisely synchronized
to a table-top femtosecond laser [Kovalev2017], thus enabling calibrated FEOS of the THz fields in the experiment [Hafez2018; Kovalev2021].

### 3.8.1.3 Considerations for THz NLO research

Since the strongest THz nonlinearities are based on the physical absorption of the driving THz field by the electrons or polar phonons in the materials, such THz nonlinear effects are therefore dissipative. Furthermore, almost all reported THz nonlinearities are based on field effects. Therefore, the THz field strength is the key parameter characterizing the pump signal in THz NLO, and not the integrated THz pulse energy, power or irradiance, as is common in “traditional” NLO.

Interestingly, almost all reported strong THz nonlinearities are generally non-perturbative, i.e., they do not follow a clear power law scaling over the entire range of applied THz peak fields, and often demonstrate saturation behavior. However, in certain cases it is possible to observe a clear perturbative, power law scaling over a considerable range of pumping THz field strengths, usually around the lower end of pumping strengths. In this case it is possible to use this small-signal nonlinearity regime and establish the effective nonlinear coefficients of the material similar to “traditional” NLO [Cheng2020; Deinert2021; Hafez2018].

Owing to a generally dissipative nature of THz NLO processes, "classical" phase matching between the pump wave and the generated NLO signal is not the dominating concept dictating the conversion efficiency in THz NLO. The conversion efficiency here is rather dependent on the efficiency of energy transfer from the pump signal to the material itself (pump absorption), reabsorption of the generated THz NLO signal within the material, and the outcoupling efficiency of the generated NLO signal.

Finally, and in stark contrast to “traditional” NLO, most of the experiments on THz NLO are performed using strong-field single-cycle THz pulses, usually produced via TPFP in LN. Such single-cycle pulses are per default ultra-broadband, and contain many octaves of frequencies. This often leads to the observation of frequency-dependent effective nonlinearities, sometimes even of different signs (see, e.g., [Turchinovich2012]), co-existing within the ultrabroadband bandwidth of a single-cycle THz pulse. However, observation of THz nonlinearities in a multi-cycle, quasi-monochromatic THz pumping regime is also possible. Such THz pump fields are either generated at large-scale facilities such as FELs (see, e.g., [Jadidi2016; König-Otto2017; Zaks2012]) and TELBE (see, e.g., [Deinert2021; Hafez2018; Kovalev2020, Kovalev2021]), or are produced via the monochromatization of a strong-field LN-generated single-cycle THz pulse using bandwidth filtering [Cheng2020].

### 3.8.1.4 Description of general table outline

Table 9 shows a representative list of THz nonlinearities taken from the literature since 2000. As THz NLO is quite different from “traditional” NLO, Table 9 has been organized in a different way as compared to the previously presented tables and contains the following columns:

- Material type and relevant material information such as doping level (specified either as doping concentration $N_c$ or Fermi energy $E_F$) and type (electrons or holes)

- Additional material information if available

- Material dimensionality (dim.) which can be 0D/1D/2D/3D

- Parameters of THz pump signal used in the measurement: source type, signal type namely single-cycle / few cycle / quasimonochromatic, central frequency (for quasimonochromatic / few cycle pulses) or frequency range (for single-cycle THz pulses), and peak THz field strength

- Physical mechanism of THz nonlinearity

- Nonlinearity type: dissipative (diss.)/non-dissipative (non-diss.)

- Type of observed nonlinear effect, such as nonlinear absorption, high-harmonics generation (HHG) etc.

- Measured nonlinear coefficients. Note that the coefficients $\chi^{(n)}$ are reported for the pumping field range corresponding to the perturbative nonlinearity regime.
Within each column the information is given in the order of the header description, and powers of 10 (e.g., 10<sup>±α</sup>) are written as E±α for compactness.

In Table 9 we only listed papers presenting nonlinear coefficients of various materials, measured and quantified in a fashion compatible with “traditional” NLO (χ<sup>(n)</sup> coefficients, conversion efficiency, saturable absorption parameters, nonlinear refractive index or refractive index modulation etc). We thus did not include papers demonstrating the nonlinear effect, but not quantifying its strength, nor papers that only report a relative transmission change of the THz field through the material.

Furthermore, we only listed papers where the THz pump field was characterized using the calibrated FEOS [Gallot and Grischkowsky 1999; Planken2001; Zhang and Turchinovich 2021] method at the sample position, with one exception of an FEL signal where the THz field at the sample position could be precisely calibrated from the power measurement [König-Otto 2017]. We thus excluded papers reporting only the integrated power/energy/irradiance as the characteristic of the pump THz signal, since it is the THz field that is key for the THz nonlinearity. In addition, we excluded the works reporting on THz Z-scans, especially those using single-cycle ultrabroadband THz pulses. The focal spot for such an ultrabroadband pulse is per definition strongly frequency-dependent, and the measurement of a precise spot shape is highly nontrivial since the THz cameras usually have a rather limited, and strongly frequency-dependent sensitivity. As a result, the precise quantitative determination of NLO coefficients of materials using THz Z-scan is a highly challenging task, prone to too many experimental uncertainties. Finally, we remark that the works included in Table 9 nominally report data obtained at room temperature, unless denoted otherwise.

## 3.8.2 Discussion

Most, and also the strongest nonlinearities presented in Table 9 are dissipative and are based on nonlinear conduction of free electrons in strong THz fields. These nonlinearities are observed directly in the THz field interacting with the material. The non-dissipative nonlinearities such as the THz Kerr effect or THz-driven QCSE, are rather observed in THz pump – optical probe measurements, where the THz field modifies the propagation conditions for the optical-frequency probe in the material, such as its polarization state (Kerr effect) or resonant absorption coefficient (QCSE).

### 3.8.2.1 Advancement and remaining challenges in THz NLO research

The major advancement in THz NLO since the demonstration of table-top generation of strong-field THz pulses via TPFP of LN around 2002-2007 is the ability to implement a nonlinear THz-TDS scheme. Here one is able to “look inside an optical cycle” during the NLO interaction, i.e., to time-resolve the nonlinear propagation of a THz field transient with sub-cycle resolution. Such a type of measurement, giving an unprecedented direct look into the initiation and development of an optical nonlinearity at the level of the light field, is presently unattainable at other frequency ranges featuring much faster oscillating optical fields, that are therefore more problematic to directly sample in the time domain.

The future progress in THz NLO will be most likely driven by the wider availability of strong-field THz sources in combination with highly sensitive field- and time-resolved THz signal detection, the development of novel nonlinear spectroscopy techniques, and the availability of novel nonlinear materials. The following factors will contribute: (i) broader availability of strong-field single-cycle and multi-cycle THz field sources, both table-top and at large-scale facilities; (ii) improved sensitivity in the THz field detection, in particular by increasing the repetition rate of the experiment using high-pulse-energy (multi-mJ level), high-repetition-rate (100 kHz or higher) femtosecond lasers; (iii) development of more advanced data acquisition and analysis protocols; (iv) further development of novel nonlinear THz spectroscopy techniques, e.g., multi-dimensional THz spectroscopy (see, e.g., [Elsaesser2019; Grechko2018; Junginger2012; Kuehn2011; Woerner2013]), and (v) broader
availability of novel nonlinear materials such as quantum materials with higher potential for nonlinear ultrafast electron conduction.

### 3.8.2.2 Recommendations for future works on THz NLO research

In the future works on THz NLO, we do strongly recommend to precisely characterize the THz field at the position of the sample via calibrated FEOS and to provide this calibrated field transient in the publication. Once all the THz signals in the experiment – pump and nonlinear product(s) – are calibrated, it becomes rather straightforward to quantify the parameters of the NLO interaction, such as the effective nonlinear coefficient, in the fashion used in “traditional” NLO.

For example, a broader use of quasi-monochromatic strong-field pumping, either at large-scale facilities such as TELBE (e.g., [Deinert2021; Hafez2018; Kovalev2021]), or via table-top quasi-monochromatic strong-field generation (e.g., [Cheng2020; Lee2020]), combined with fully calibrated FEOS [Gallot and Grischkowsky 1999; Planken2001; Zhang and Turchinovich 2021] detection, should lead to more rigorously calibrated experiments on discrete THz HHG in various materials (see [Deinert2021; Hafez2018; Kovalev2021]). Such measurements allow in particular to extract the THz nonlinear susceptibility coefficients $\chi^{(n)}$ by measuring the conversion efficiency from the pump field to the n-th harmonic, over a wide range of pumping field strengths and frequencies [Hafez2018; Hafez2020; Cheng2020].

Finally, we strongly recommend providing comprehensive information on the nonlinear material used, on the details of the experimental setups, and on the protocols of data processing. This should allow reproducing the THz NLO results by the broader community.
### 3.8.3 Data table for THz NLO

<table><caption>**Table 9:** THz nonlinearities from representative works since 2000. Legend for superscripts: see below the table.</caption><thead><tr><th>Material type; Fabrication; Key material properties;</th><th>Dim.</th><th>THz source; Single-cycle / quasimonochromatic; THz pump frequency range (for single cycle) or central pump frequency (for quasimonochromatic); THz peak field strength range</th><th>Physical mechanism of THz-induced NLO effect</th><th>Diss./ Non-diss.</th><th>Observed nonlinear effects</th><th>Measured THz nonlinear coefficients</th><th>Reference</th></tr></thead><tbody><tr><td>Al<sub>2</sub>O<sub>3</sub></td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 0.7E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 0.6E-6</td><td>[Sajadi 2015]</td></tr><tr><td>Benzene (liquid)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; 30 - 150 kV/cm</td><td>THz-induced Kerr effect in liquid, probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>χ<sup>(3)</sup> = 0.22E-20 m<sup>2</sup>/V<sup>2</sup>; n<sub>2</sub> = 56E-16 cm<sup>2</sup>/W; K = 0.26E-14 m/V<sup>2</sup> (Kerr coefficient*)</td><td>[Hoffmann 2009]</td></tr></tbody></table>
<table><tr><td>CCl<sub>4</sub> (liquid)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; 30 - 150 kV/cm</td><td>THz-induced Kerr effect in liquid, probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>χ<sup>(3)</sup> = 0.10E-20 m<sup>2</sup>/V<sup>2</sup>; n<sub>2</sub> = 27E-16 cm<sup>2</sup>/W; K = 0.12E-14 m/V<sup>2</sup> (Kerr coefficient*)</td><td>[Hoffmann 2009]</td></tr><tr><td>CHCl<sub>3</sub> (liquid)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; 30 - 150 kV/cm</td><td>THz-induced Kerr effect in liquid, probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>χ<sup>(3)</sup> = 0.04E-20 m<sup>2</sup>/V<sup>2</sup>; n<sub>2</sub> = 10E-16 cm<sup>2</sup>/W; K = 0.045E-14 m/V<sup>2</sup> (Kerr coefficient*)</td><td>[Hoffmann 2009]</td></tr><tr><td>CH<sub>2</sub>I<sub>2</sub> (liquid)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; 30 - 150 kV/cm</td><td>THz-induced Kerr effect in liquid, probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>χ<sup>(3)</sup> = 0.70E-20 m<sup>2</sup>/V<sup>2</sup>; n<sub>2</sub> = 140E-16 cm<sup>2</sup>/W; K = 0.75E-14 m/V<sup>2</sup> (Kerr coefficient*)</td><td>[Hoffmann 2009]</td></tr><tr><td>CS<sub>2</sub> (liquid)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; 30 - 150 kV/cm</td><td>THz-induced Kerr effect in liquid, probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>χ<sup>(3)</sup> = 2.08E-20 m<sup>2</sup>/V<sup>2</sup>; n<sub>2</sub> = 440E-16 cm<sup>2</sup>/W; K = 2.4E-14 m/V<sup>2</sup> (Kerr coefficient*)</td><td>[Hoffmann 2009]</td></tr><tr><td>Diamond; polycrystalline</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 3.0E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 1.03E-6</td><td>[Sajadi 2015]</td></tr></table>
<table><tr><td>GaAs; doped, N<sub>c</sub> = 8E15 cm<sup>-3</sup> (electrons), thickness d = 0.4 mm;</td><td>3D</td><td>TPFP of LN; single-cycle; 0.2 - 2.5 THz; 9 - 292 kV/cm</td><td>intervalley transfer and increase of effective mass of THz-driven free electrons</td><td>Diss.</td><td>Frequency-dependent nonlinear conductivity and saturable absorption</td><td>Frequency-dependent index change Δn = -0.13 - 0.08; Reduction of power absorption coefficient by ca. 50% across the whole measurement spectrum</td><td>[Turchinovich 2012]</td></tr><tr><td>GaAs; doped, N<sub>c</sub> = 8E15 cm<sup>-3</sup> (electrons), thickness d = 0.4 mm;</td><td>3D</td><td>TPFP of LN; single-cycle; 0.2 - 2.5 THz; 9 - 292 kV/cm</td><td>intervalley transfer and increase of effective mass of THz-driven free electrons</td><td>Diss.</td><td>Frequency-integrated saturable absorption; nonlinear pulse group delay</td><td>Saturation fluence = 8.2 μJ/cm<sup>2</sup>; Max. nonlinear pulse compression ΔT/T ~ 0.1; Max. group index change Δn<sub>g</sub> ~ 0.1</td><td>[Hoffmann and Turchinovich 2010]</td></tr><tr><td>GaAs in external electric field; intrinsic, photoexcited at 400 nm to create conductivity;</td><td>3D</td><td>Unspecified source; single-cycle, monochromatized using bandpass filter; quasimonochromatic; 0.6 THz; 50 kV/cm;</td><td>THz E-FISH in static applied symmetry-breaking electric bias field up to E<sub>b</sub> = 15 kV/cm</td><td>Diss.</td><td>SHG and THG</td><td>Maximum effective χ<sup>(2)</sup> = χ<sup>(3)</sup>*E<sub>b</sub> = 1.7E-7 m/V, corresponding to natural χ<sup>(3)</sup> = 1E-13 - 1E-14 m<sup>2</sup>/V<sup>2</sup>; Max power conversion efficiency η<sub>SHG</sub> ~ 5E-5; η<sub>THG</sub> ~ 1E-5, at optical pump fluence F<sub>p</sub> = 4 μJ/cm<sup>2</sup></td><td>[Lee 2020]</td></tr></table>
<table><tr><td>GaP; doped, N<sub>c</sub> = 1E16 cm<sup>-3</sup> (electrons), thickness d = 0.3 mm;</td><td>3D</td><td>TPFP of LN; single-cycle; 0.2 - 2.5 THz; 9 - 292 kV/cm</td><td>intervalley transfer and increase of effective mass of THz-driven free electrons</td><td>Diss.</td><td>Frequency-integrated saturable absorption; nonlinear pulse group delay</td><td>Saturation fluence = 20.9 µJ/cm<sup>2</sup>; Max. nonlinear pulse compression ΔT/T ~ 5E-2; Max. group index change Δn<sub>g</sub> ~ 0.05</td><td>[Hoffmann and Turchinovich 2010]</td></tr><tr><td>Ge; doped, N<sub>c</sub> = 1E14 cm<sup>-3</sup> (electrons), thickness d = 6 mm;</td><td>3D</td><td>TPFP of LN; single-cycle; 0.2 - 2.5 THz; 9 - 292 kV/cm</td><td>intervalley transfer and increase of effective mass of THz-driven free electrons</td><td>Diss.</td><td>Frequency-integrated saturable absorption; nonlinear pulse group delay</td><td>Saturation fluence = 3.1 µJ/cm<sup>2</sup>; Max. nonlinear pulse compression ΔT/T ~ 5E-2; Max. group index change Δn<sub>g</sub> ~ 2E-3</td><td>[Hoffmann and Turchinovich 2010]</td></tr><tr><td>Graphene; produced by thermal decomposition of SiC on the C-face of 4H-SiC; n-doped (inhomogeneously within the sample)</td><td>2D</td><td>FEL (FELBE); quasimonochromatic; 19 THz; max. 25 kV/cm</td><td>FWM in Landau-quantized graphene (under 4.5 T magnetic field and 10 K cryogenic temperature)</td><td>Diss.</td><td>Degenerate FWM and transient grating generation</td><td>χ<sup>(3)</sup> ~ 9.2E-20 m<sup>3</sup>/V<sup>2</sup> (bulk susceptibility) Temperature = 10 K</td><td>[König-Otto 2017]</td></tr><tr><td>Graphene; CVD-grown; doped, E<sub>F</sub> = 0.2 eV (holes);</td><td>2D</td><td>air-plasma source; single-cycle; 0 - 15 THz; 100 - 300 kV/cm</td><td>THz-driven impact ionization of carriers</td><td>Diss.</td><td>Modulation of optical density at 800nm probe wavelength</td><td>Optical density change ΔOD ~ 0.01 - 0.1 at 800 nm</td><td>[Tani 2012]</td></tr></table>
<table><tr><td>Graphene; CVD-grown; doped, N<sub>c</sub> = 2.1E12 cm<sup>-2</sup> (holes), E<sub>F</sub> = 0.17 eV</td><td>2D</td><td>TELBE; quasimonochromatic; 0.3 THz, 0.37 THz, 0.68 THz; 0-100 kV/cm</td><td>thermodynamic response of free electrons</td><td>Diss.</td><td>HHG (odd): 3, 5, 7</td><td>χ<sup>(3)</sup> ~ 1E-9 m<sup>2</sup>/V<sup>2</sup>; χ<sup>(5)</sup> ~ 1E-22 m<sup>4</sup>/V<sup>4</sup>; χ<sup>(7)</sup> ~ 1E-38 m<sup>6</sup>/V<sup>6</sup>; Max. field conversion efficiencies: THG: η = 2E-3; 5HG: η = 2.5E-4; 7HG: η = 8E-5</td><td>[Hafez 2018]</td></tr><tr><td>Graphene; CVD-grown; doped, N<sub>c</sub> = 6E11 cm<sup>-2</sup> (holes), E<sub>F</sub> = 0.07 eV</td><td>2D</td><td>TPFP of LN; single-cycle; 0.3 - 2 THz; 2 - 120 kV/cm</td><td>thermodynamic response of free electrons</td><td>Diss.</td><td>Frequency-dependent nonlinear conductivity and saturable absorption</td><td>Power loss = 12.5%, non-saturable power loss = 12.5%</td><td>[Mics 2015]</td></tr><tr><td>Graphene</td><td>2D</td><td>TPFP of LN, monochromatization using bandpass filter; quasimonochromatic; 0.8 THz; 12-31 kV/cm</td><td>thermodynamic response of free electrons</td><td>Diss.</td><td>THG</td><td>χ<sup>(3)</sup> ~ 1E-9 m<sup>2</sup>/V<sup>2</sup>;</td><td>[Cheng 2020]</td></tr><tr><td>InGaAs/GaAs QD-based SESAM for 1040 nm; MBE-grown, 80 QD layers; intrinsic;</td><td>OD</td><td>TPFP of LN; single-cycle; 0.2 - 3.0 THz; 10 - 100 kV/cm</td><td>THz-driven QCSE in QDs</td><td>Non-diss.</td><td>Modulation of absorption at 1040 nm probe wavelength</td><td>Absorption change Δα = 3% at 1040 nm</td><td>[Hoffmann 2010]</td></tr></table>
<table><tr><td>LDPE (low-density polyethylene)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 2.0E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 3.15E-6</td><td>[Sajadi 2015]</td></tr><tr><td>LiNbO<sub>3</sub></td><td>3D</td><td>TPFP of LN; few-cycle; 0.35 THz and 1.1 THz; max. 10 kV/cm</td><td>THz DFG at 0.76 THz</td><td>Diss.</td><td>DFG</td><td>χ<sup>(2)</sup> > 1.5E-6 m/V</td><td>[Lu 2021]</td></tr><tr><td>Metamaterial based on GaAs; metallic (Au/Cr) splitting resonators deposited on n-doped (N<sub>c</sub> = 1E16 cm<sup>-3</sup>) and semi-insulating GaAs.</td><td>3D</td><td>TPFP of LN; single-cycle; 0.2 - 1.2 THz; 24 - 400 kV/cm</td><td>THz-driven intervalley scattering and impact ionization in GaAs</td><td>Diss.</td><td>Modulation of THz relative permittivity ε</td><td>Relative permittivity change at the metamaterial resonant frequency. Doped GaAs: ε = - 2.5 - 1 at f = 0.82 THz, THz field range 24 - 400 kV/cm; Semi-insulating GaAs: ε = - 12.5 - 1 at f = 0.85 THz, THz field range 100 - 400 kV/cm</td><td>[Fan 2013]</td></tr><tr><td>Metamaterial based on golden grating deposited on graphene; CVD-grown graphene; doped, E<sub>F</sub> ~ 0.1 eV (holes)</td><td>2D</td><td>TELBE; quasimonochromatic; 0.7 THz; 5 - 70 kV/cm</td><td>thermodynamic response of free electrons; plasmonic THz field concentration in a grating</td><td>Diss.</td><td>THG</td><td>χ<sup>(3)</sup> ~ 3E-8 m<sup>2</sup>/V<sup>2</sup>; Field conversion efficiency η ~ 1E-2</td><td>[Deinert 2021]</td></tr></table>
<table><tr><td>MgO</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 0.5E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 0.7E-6</td><td>[Sajadi 2015]</td></tr><tr><td>Semiconducting SWCNT; produced by CoMoCAT-process; nominally intrinsic</td><td>1D</td><td>TPFP of LN; single-cycle; 0.2 - 3 THz; 50 - 420 kV/cm</td><td>THz-driven QCSE and interband transitions in CNTs</td><td>Non-diss.</td><td>Modulation of absorption at 1.2-1.25 eV probe energy</td><td>Absorption change Δα = 4% at 1.2-1.25 eV</td><td>[Shimano 2012]</td></tr><tr><td>Si; nominally intrinsic</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 56E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 65E-6</td><td>[Sajadi 2015]</td></tr><tr><td>SiN</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 0.08E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 0.05E-6</td><td>[Sajadi 2015]</td></tr><tr><td>TPX (polymethylpentene)</td><td>3D</td><td>TPFP of LN; single-cycle; 0 - 3 THz; max. 2.1 MV/cm</td><td>THz-induced Kerr effect in bulk material probed by an 800 nm probe pulse</td><td>Non-diss.</td><td>Transient birefringence at optical probe wavelength of 800 nm, measured via optical polarization evolution</td><td>n<sub>2</sub> (1 THz) = 0.3E-16 cm<sup>2</sup>/W; max. Δn (800 nm) = 0.36E-6</td><td>[Sajadi 2015]</td></tr></table>

<footnote>* Kerr coefficient is defined in the paper as K = Δn / λE<sup>2</sup></footnote>
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# 4 Conclusion

We have identified general best practices for performing and reporting NLO measurements regardless of the NLO technique used, and have also highlighted several technique-specific best practices. Furthermore, we have introduced a set of tables with representative NLO data from the literature since 2000 for bulk materials, solvents, 0D-1D-2D materials, metamaterials, fiber waveguiding materials, on-chip waveguiding materials, hybrid waveguiding systems, and materials suitable for THz NLO. The data were selected based on the identified best practices and on special considerations for the different material types. For each of the material categories, we have also discussed the background prior to 2000, highlighted the recent advancements and remaining challenges, and concluded with recommendations for future NLO studies.

As shown in the discussions, the field of NLO has gained considerable momentum over the past two decades thanks to major breakthroughs in material science and technology. This has given rise to an enormous growth in NLO publications. However, many of them were not included in the tables presented here as they provided too limited information to comply with the best practices. The publications that brought most value to the tables are those that provide one or several NLO coefficients – and possibly also conversion efficiencies – for one or several materials, wavelengths, pulse durations, etc., measured and reported along the best practices. The dependence of NLO coefficients on wavelength, pulse duration, etc. is very insightful information not only from a fundamental science perspective (as it allows distinguishing NLO processes while ruling out, e.g., thermal effects) but also from an applications point of view. To assess the practical applicability of a NLO material, it is key that papers also clearly specify the material properties and fabrication details, and provide information on both the nonlinear and linear optical characteristics, such as the linear loss. This will allow evaluating the suitability of the material for specific NLO applications along well-defined figures-of-merit.

We encourage the NLO community to take all these aspects into account and implement the presented best practices in future works. In fact, there is still much to be discovered in NLO research, and in the coming years we intend to update the data tables by considering additional NLO processes and by adding materials that are currently not included or yet to be investigated. Hence, for those future investigations we want to stimulate the use of the listed best practices to allow a more adequate comparison, interpretation and practical implementation of the published parameters and as such further the fundamental understanding of NLO as well as its exploitation in real-life applications.
# 5 Description of author contributions

In the order of the author list:

Nathalie Vermeulen: co-coordinated the data table initiative together with Eric Van Stryland; contributed to the data collection and text of hybrid waveguiding systems (as team leader), on-chip waveguiding materials, and 0D-1D-2D materials; compiled the data table for hybrid waveguiding systems; contributed to the general introduction and conclusion; wrote the best practices Sections 2.2.2.1, 2.2.3, and the introductory part of Section 3; compiled the manuscript and contributed to the final editing of the manuscript

Daniel Espinosa: contributed to the data collection and text of bulk materials, 0D-1D-2D materials, metamaterials, and on-chip waveguiding materials; compiled the data table for on-chip waveguiding materials

Adam Ball: contributed to the data collection and text of bulk materials and metamaterials

John Ballato: contributed to the data collection and text of fiber waveguiding materials (as team leader) and hybrid waveguiding systems; compiled the data table for fiber waveguiding materials; contributed to the final editing of the manuscript

Philippe Boucaud: contributed to the data collection and text of bulk materials and on-chip waveguiding materials

Georges Boudebs: contributed to the data collection and text of bulk materials

Cecília Campos: contributed to the text of 0D-1D-2D materials; compiled the data table for 0D-1D-2D materials

Peter Dragic: contributed to the data collection and text of fiber waveguiding materials and hybrid waveguiding systems

Anderson Gomes: contributed to the data collection and text of 0D-1D-2D materials (as team leader), and bulk materials; contributed to the best practices Section 2.2.2.4

Mikko Huttunen: contributed to the data collection and text of metamaterials (as team leader) and 0D-1D-2D materials; compiled the data table for metamaterials; wrote the best practices Sections 2.2.1.1, 2.2.1.2, 2.2.2.6

Nathaniel Kinsey: contributed to the data collection and text of bulk materials (as team leader) and metamaterials; compiled the data table for bulk materials; contributed to the best practices section 2.2.2.4

Rich Mildren: contributed to the data collection and text of bulk materials and on-chip waveguiding materials; wrote the best practices Section 2.2.2.2

Dragomir Neshev: contributed to the data collection and text of 0D-1D-2D materials and metamaterials

Lázaro Padilha: contributed to the data collection and text of 0D-1D-2D materials
Minhao Pu: contributed to the data collection and text of on-chip waveguiding materials (as team leader) and hybrid waveguiding systems

Ray Secondo: contributed to the data collection and text of bulk materials and metamaterials

Eiji Tokunaga: contributed to the data collection and text of bulk materials and solvents; wrote the best practices Section 2.2.2.5

Dmitry Turchinovich: collected the data and wrote the text for THz NLO (as team leader); compiled the data table for THz NLO

Jingshi Yan: contributed to the data collection and text of 0D-1D-2D materials

Kresten Yvind: contributed to the data collection and text of on-chip waveguiding materials and hybrid waveguiding systems

Ksenia Dolgaleva: contributed to the data collection and text of bulk materials and 0D-1D-2D materials; contributed to the data collection and wrote most of the text of metamaterials and on-chip waveguiding materials

Eric Van Stryland: co-coordinated the data table initiative together with Nathalie Vermeulen; collected the data and wrote the text for fused silica and solvents (as team leader); compiled the data tables for fused silica and solvents; contributed to the data collection and text of bulk materials; contributed to the general introduction, conclusion and the best practices Section 2.2.2.4; wrote the best practices Section 2.2.2.3; contributed to the final editing of the manuscript

## Acknowledgements

NV acknowledges the financial support from Fonds Wetenschappelijk Onderzoek (FWO) under Grants G005420N and G0F6218N (EOS-convention 30467715).

JB acknowledges support from the J. E. Sirrine Foundation.

PB acknowledges the French National Research Agency (Agence Nationale de la Recherche, ANR) - OPOINt project (ANR-19-CE24-0015).

GB acknowledges the support from the University of Angers and from the NNN-TELECOM Program, region des Pays de la Loire, contract n° 2015 09036.

PD acknowledges the U.S. Department of Defense Directed Energy Joint Transition Office (DE JTO) (N00014-17-1-2546) and the Air Force Office of Scientific Research (FA9550-16-1-0383).

ASLG and CLAVC acknowledge support from Brazilian INCT of Photonics (CNPq, CAPES, FACEPE), and Air Force Office of Scientific Research (AFOSR) under grant FA9550-20-1-0381.

MJH acknowledges the support of the Flagship of Photonics Research and Innovation (PREIN) funded by the Academy of Finland (Grant No. 320165).

NK, AB, RS acknowledge support from Air Force Office of Scientific Research (FA9550-18-1-0151) and the National Science Foundation (1808928).

RPM acknowledges funding from AFOSR FA2386-21-1-4030 and Australian Research Council LP200301594.

DN and JY acknowledge the support by the Australian Research Council through the Centres of Excellence program (CE20010001) and NATO SPS program (OPTIMIST).
MP and KY acknowledge the financial support from Danish National Research Foundation through the Research Centre of Excellence, Silicon Photonics for Optical Communications (SPOC) (ref. DNRF123). MP also acknowledges the financial support from the European Union's Horizon 2020 research and innovation programme (grant agreement N° 853522 REFOCUS).

DT acknowledges the financial support from European Union's Horizon 2020 research and innovation programme (grant agreement N° 964735 EXTREME-IR), and from Deutsche Forschungsgemeinschaft (DFG) within the project 468501411-SPP2314 INTEGRATECH under the framework of the priority programme SPP2314 - INTEREST.

KD and DE are grateful to Roberto Morandotti and Sisira Suresh for suggesting relevant references for the tables. KD and DE acknowledge the financial support from Canada Research Chairs program and Natural Science and Engineering Council's Discovery program RGPIN-2020-03989.