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## 3.2 Solvents: data table and discussion

Team: Eric Van Stryland (team leader), Eiji Tokunaga

As mentioned in Section 3.1 on bulk materials, we include here a separate section dedicated specifically to solvents. In Table 3, we show a list of the 3rd-order nonlinearities of a variety of liquids, mainly organic solvents at room temperature, taken from representative experimental works in the literature. These are included primarily because our understanding of their NLO responses has advanced and one of them, carbon disulfide ($CS_2$) is often used as a reference. Besides reporting the values of the effective nonlinear refractive indices, $n_{2,eff}$, where listed in the original works, we report the 2PA coefficient, $\alpha_2$, and the 3PA coefficient, $\alpha_3$. However, in most of the publications the 2PA is not listed since the wavelengths used are with photon energies where such absorption is not energetically possible and 3PA is also extremely small or not energetically allowed.

These solvent nonlinearities are important not only because $CS_2$ is used as a reference material, but also because of the many measurements of organic dye nonlinearities in the literature (not included in the current data tables) as well as for measurements of particle suspensions etc., as they are usually performed for these materials dissolved or dispersed in various solvents. A lack of knowledge of the nonlinearities of the solvents can potentially confuse interpretation of such measurements. We also refer the reader to Section 3.7 on hybrid waveguiding systems that includes, amongst others, hollow fibers filled with solvents.

The primary difficulty in reporting the solvent nonlinearities in Table 3, is that it is now known that there are four separate physical processes leading to the reported NLR, i.e., the bound-electronic nonlinearity, which can be considered instantaneous for any pulse widths reported in this publication, and three nuclear contributions whose temporal characteristics depend on the solvent but are on the order of femtoseconds to picoseconds. These nonlinearities are reorientational, librational, and collisional [McMorrow 1998, Reichert 2014]. For nanosecond and longer pulses, other nonlinearities of electrostriction and thermal origin also become important. We restrict ourselves here to the subnanosecond range. Using single-beam measurements, it is problematic to separate the various nonlinearities. Excite-probe experiments are needed for this. The bound-electronic and collisional nonlinearities can be separated from the rotational and librational nonlinearities by their symmetries given the signals using different relative polarizations of the excitation and probe beams. The nuclear nonlinearities have both rise and fall times and each has an associated response function given, e.g., in [Reichert 2014]. Thus, the response is quite complicated. In rare cases the NLR response is dominated by a single nonlinearity. In fact, this only occurs in the case of extremely short pulses (usually a few femtoseconds) where only the bound-electronic response is activated. Even in 3-dimensionally symmetrical molecules, the nuclear collisional nonlinearity is still present for pulses longer than a few femtoseconds. Thus, the overall NLO response depends on the pulsewidth used for the experiment. This has led to difficulties in reported nonlinearities of multiple materials since $CS_2$, included in this table, has often been used as a reference standard for measurements of other materials. In this regard, we point out [Miguez 2017, Zhao 2018, Reichert 2014] where measurements of $CS_2$ include the nuclear components. The effective nonlinearity can be calculated from the values of the four nonlinear responses along with their temporal responses as in [Zhao 2018, Reichert 2014]. We show results for $CS_2$ in Fig. 1 taken from [Reichert 2014] because of its wide use as a reference material. [Ganeev 2004] shows a dependence consistent with that of [Reichert 2014] for several pulsewidths from 110 fs to 75 ns. The fast and slow nonlinearities referred to in [Miguez 2017] are combinations of the four nonlinearities; thus, attempts to combine information from [Miguez 2017, Zhao 2018, Reichert 2014] which are the only ones in Table 3, reporting the separate nuclear nonlinear responses, into a single table are problematic. Therefore, the reader will have to consult the original papers to get the full details of the NLR [Miguez 2017, Zhao 2018, Reichert 2014]. The two values of $n_2$ reported in Table 3 for [Zhao 2018, Reichert 2014] are the predicted bound-electronic $n_2$ that would be measured using few femtosecond pulses along with an estimate of the effective nonlinear refractive index, $n_{2,eff}$, that would
be measured in a single beam experiment such as Z-scan using pulses of >100 ps, i.e. long compared to the rise and fall times of the nuclear nonlinearities but short compared to times where electrostriction and thermal effects become important. These values are estimated from the temporal dependence plots given in those works [Zhao 2018, Reichert 2014]. These values can be compared to the predictions from [Miguez 2017] for which we report the 2 values of the short and long pulse components as described in [Miguez 2017]. The sum of these components should give the same long pulse limit as the 2nd value given in the table for [Zhao 2018, Reichert 2014].

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Results of beam deflection (BD) experiments for CS₂ yielding the four nonlinearities with their response functions to calculate the predicted n₂,eff in a Z-scan experiment (solid red line), along with Z-scan measurements at various pulse widths and 2 wavelengths namely 700nm (black circles) and 1064nm (green triangles). Reprinted with permission from [Reichert 2014] © 2014 Optica Publishing Group.</figcaption></figure>

<table><caption>**Table 3.** NLO measurement results for solvents. The methods used are Z-scan, Beam Deflection (BD), Ellipse Rotation (ER), and Third-Harmonic Generation (THG). All liquid samples reported in this table were held in either 1- or 2-mm internal thickness cuvettes and were at room temperature. The $\chi^{(3)}$ values specified in the table are in fact $\chi^{(3)}_{eff}$ values containing multiple nonlinearity contributions rather than just the bound-electronic contribution. Legend for superscripts: see below the table. The following abbreviations have been used: DMSO = dimethyl sulfoxide; ODCB = o-dichlorobenzene; DCM = dichloromethane; BS = butyl salicylate; ACN = acetonitrile; NB = nitrobenzene; CB = Chlorobenzene; THF = tetrahydrofuran; CH = Cyclohexane; CCl₄ = carbon tetrachloride; CS₂ = carbon disulfide; D₂O = deuterated water; lin = linear polarization; cir = circular polarization.</caption><thead><tr><th>Material</th><th>Method</th><th>χ<sup>(3)</sup><br>x10<sup>-14</sup><br>(esu)</th><th>χ<sup>(3)</sup><br>x10<sup>-22</sup><br>(m<sup>2</sup>/W)</th><th>n<sub>2,eff</sub><br>x10<sup>-20</sup><br>(m<sup>2</sup>/W)</th><th>2:α<sub>2</sub>(m/W)<br>3:α<sub>3</sub>(m<sup>3</sup>/W<sup>2</sup>)</th><th>λ<sub>pump</sub><br>(nm)</th><th>τ<sub>pump</sub><br>Rep-Rate<br>(Hz)</th><th>λ<sub>probe</sub><br>(nm)</th><th>τ<sub>probe</sub><br>(fs)♦</th><th>Ref</th></tr></thead><tbody><tr><td rowspan="2">CCl<sub>4</sub></td><td>Z-scan</td><td>20.8</td><td>28</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>11.1</td><td>15</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td></tr><tr><td rowspan="2">Chloroform</td><td>Z-scan</td><td>27.8</td><td>38</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>9.77</td><td>13.3</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td></tr><tr><td rowspan="2">DCM</td><td>Z-scan</td><td>26.8</td><td>37</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>8.56</td><td>11.8</td><td></td><td></td><td>1064</td><td>16ps 10 Hz</td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td rowspan="2">CH</td><td>Z-scan</td><td>14.0</td><td>19</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>9.03</td><td>12.2</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">n-Hexane</td><td>Z-scan</td><td>11.2</td><td>17</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>6.17</td><td>9.35</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Acetone</td><td>Z-scan</td><td>14.9</td><td>23</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>6.55</td><td>10.1</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Methanol</td><td>Z-scan</td><td>11.1</td><td>18</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>4.16</td><td>6.74</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Ethanol</td><td>Z-scan</td><td>11.5</td><td>18</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>4.82</td><td>7.57</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">DMF</td><td>Z-scan</td><td>29.8</td><td>42</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>7.78</td><td>11</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Water</td><td>Z-scan</td><td>8.70</td><td>14</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>3.91</td><td>6.28</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">CB</td><td>Z-scan</td><td>95.9</td><td>117</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>14.8</td><td>18.1</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Toluene</td><td>Z-scan</td><td>73.4</td><td>93</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>13.7</td><td>17.4</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">CS<sub>2</sub></td><td>Z-scan</td><td>289</td><td>310</td><td>1064</td><td>16ps 10 Hz</td><td rowspan="2"></td><td rowspan="2">1</td></tr><tr><td>THG</td><td>32.1</td><td>34.5</td><td>1064</td><td>16ps 10 Hz</td></tr><tr><td rowspan="2">Toluene</td><td>Z-scan</td><td></td><td>lin 5.7<br/>cir 3.3</td><td>800</td><td>125fs<br/>1 kHz</td><td rowspan="2"></td><td rowspan="2">2</td></tr><tr><td></td><td></td><td>lin 4.8<br/>cir 2.8</td><td>800</td><td>125fs<br/>1 kHz</td></tr><tr><td rowspan="2">ODCB</td><td>Z-scan</td><td></td><td>lin 4.0<br/>cir 2.2</td><td>800</td><td>125fs<br/>1 kHz</td><td rowspan="2"></td><td rowspan="2">2</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">DMF</td><td>Z-scan</td><td></td><td></td><td>800</td><td>125fs<br/>1 kHz</td><td rowspan="2"></td><td rowspan="2">2</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan="2">Acetone</td><td>Z-scan</td><td>20.5</td><td>43.7</td><td>2: 6.0x10<sup>-13</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>4.37</td><td>9.34</td><td>3: 7.8x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">Chloroform</td><td>Z-scan</td><td>27.6</td><td>51.9</td><td>2: 1.25x10<sup>-12</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>4.96</td><td>9.37</td><td>3: 5.2x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">DMF</td><td>Z-scan</td><td>24.5</td><td>47.0</td><td>2: 1.22x10<sup>-12</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>5.37</td><td>10.4</td><td>3: 9.7x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">THF</td><td>Z-scan</td><td>20.1</td><td>40.1</td><td>2: 5.70x10<sup>-13</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>6.46</td><td>12.9</td><td>3: 150x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">Toluene</td><td>Z-scan</td><td>25.4</td><td>42.7</td><td>2: 3.20x10<sup>-12</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>7.95</td><td>14</td><td>3: 20.5x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">CS<sub>2</sub></td><td>Z-scan</td><td>762</td><td>1130</td><td>2: 2.00x10<sup>-11</sup></td><td>532</td><td>30ps 10 Hz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>12.0</td><td>17.8</td><td>3: 74.7x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">Ethanol</td><td>Z-scan</td><td>5.11</td><td>10.9</td><td>3: 21.0x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>6.31</td><td>11.7</td><td>3: 18.2x10<sup>-23</sup></td><td>800</td><td>100fs 1kHz</td></tr><tr><td rowspan="2">CCl<sub>4</sub></td><td>Z-scan</td><td></td><td></td><td></td><td></td><td></td><td rowspan="2">3</td></tr><tr><td>Z-scan</td><td>87.1</td><td>122</td><td>519</td><td>532</td><td>35ps 10 Hz</td></tr><tr><td rowspan="2">CS<sub>2</sub></td><td>Z-scan</td><td>0.668</td><td>0.932</td><td>3.97</td><td>800</td><td>40fs 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>23.3</td><td>32.6</td><td>153</td><td>532</td><td>35ps 10 Hz</td></tr><tr><td rowspan="2">NB</td><td>Z-scan</td><td>0.151</td><td>0.21</td><td>0.988</td><td>800</td><td>40fs 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>18.6</td><td>26.0</td><td>122</td><td>532</td><td>35ps 10 Hz</td></tr><tr><td rowspan="2">DB</td><td>Z-scan</td><td>0.158</td><td>0.22</td><td>1.03</td><td>800</td><td>40fs 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>17.8</td><td>24.9</td><td>112</td><td>532</td><td>35ps 10 Hz</td></tr><tr><td rowspan="2">Aniline</td><td>Z-scan</td><td>0.165</td><td>0.23</td><td>1.03</td><td>800</td><td>40fs 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td rowspan="2">Toluene</td><td>Z-scan</td><td>15.3</td><td>21.4</td><td>108</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.143</td><td>0.20</td><td>1.01</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">Benzene</td><td>Z-scan</td><td>12.6</td><td>17.5</td><td>88</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.246</td><td>0.343</td><td>1.72</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">Chloroform</td><td>Z-scan</td><td>4.4</td><td>6.1</td><td>33</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.11</td><td>0.153</td><td>0.827</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">DMF</td><td>Z-scan</td><td>4.3</td><td>6.0</td><td>33</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.101</td><td>0.141</td><td>0.781</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">DCM</td><td>Z-scan</td><td>4.1</td><td>5.7</td><td>32</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.100</td><td>0.14</td><td>0.781</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">DMSO</td><td>Z-scan</td><td>3.9</td><td>5.4</td><td>28</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.102</td><td>0.142</td><td>0.735</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">Acetone</td><td>Z-scan</td><td>2.6</td><td>3.6</td><td>22</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.075</td><td>0.105</td><td>0.528</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">ACN</td><td>Z-scan</td><td>2.3</td><td>3.2</td><td>20</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.0474</td><td>0.0661</td><td>0.413</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">n-Hexane</td><td>Z-scan</td><td>2.0</td><td>2.8</td><td>17</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.0496</td><td>0.0692</td><td>0.413</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">THF</td><td>Z-scan</td><td>1.4</td><td>1.9</td><td>11</td><td>532</td><td>35ps 10 Hz</td><td rowspan="2">4</td></tr><tr><td>Z-scan</td><td>0.0721</td><td>0.101</td><td>0.574</td><td>800</td><td>40fs 10 Hz</td></tr><tr><td rowspan="2">CS<sub>2</sub></td><td>BD</td><td></td><td></td><td>15.0;<br/>275°</td><td>800</td><td>50fs 1kHz</td><td>650</td><td>158</td><td>5</td></tr><tr><td>Z-scan</td><td></td><td></td><td>15.0;<br/>300°</td><td>1064</td><td>65fs-25ps<br/>1kHz</td><td></td><td></td><td>5</td></tr><tr><td rowspan="2">CS<sub>2</sub></td><td>Z-scan</td><td></td><td></td><td>15.0;<br/>300°</td><td>700</td><td>32fs-8ps<br/>1kHz</td><td></td><td></td><td>5</td></tr><tr><td>Z-scan</td><td></td><td></td><td>BEn<sub>2</sub>=30<br/>BEn<sub>2</sub>=60</td><td>2: 2.0x10<sup>-11</sup><br/>2: 0.7x10<sup>-11</sup></td><td>420<br/>545</td><td>** 1KHz<br/>** 1kHz</td><td></td><td></td><td>5</td></tr><tr><td>Toluene</td><td>BD</td><td></td><td></td><td>6.0; 49°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>NB</td><td>BD</td><td></td><td></td><td>6.0; 77°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Benzene</td><td>BD</td><td></td><td></td><td>6.0; 44°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>p-Xylene</td><td>BD</td><td></td><td></td><td>6.2; 52°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Pyridine</td><td>BD</td><td></td><td></td><td>6.0; 53°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>ODCB</td><td>BD</td><td></td><td></td><td>6.0; 51°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>DCM</td><td>BD</td><td></td><td></td><td>3.0; 15°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Chloroform</td><td>BD</td><td></td><td></td><td>4.1; 16°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>CCl<sub>4</sub></td><td>BD</td><td></td><td></td><td>4.8; 6.8°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Acetone</td><td>BD</td><td></td><td></td><td>4.0; 12°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>ACN</td><td>BD</td><td></td><td></td><td>3.5; 11°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>DMF</td><td>BD</td><td></td><td></td><td>4.0; 21°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>BS</td><td>BD</td><td></td><td></td><td>3.8; 26°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>THF</td><td>BD</td><td></td><td></td><td>3.2; 6.2°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Hexane</td><td>BD</td><td></td><td></td><td>3.2; 7.0°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>CH</td><td>BD</td><td></td><td></td><td>3.5; 5.9°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Methanol</td><td>BD</td><td></td><td></td><td>3.0; 4.4°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>1-Octanol</td><td>BD</td><td></td><td></td><td>4.0; 4.9°</td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr></tbody></table>
<table><tr><td>1-Butanol</td><td>BD</td><td></td><td>3.3; 4.4<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>Ethanol</td><td>BD</td><td></td><td>3.2; 4.2<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td></td><td></td><td></td><td>15.0;</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>CS<sub>2</sub></td><td>BD</td><td></td><td>275<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>DMSO</td><td>BD</td><td></td><td>4.5; 8.2<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>D<sub>2</sub>O</td><td>BD</td><td></td><td>2.8; 3.7<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td>H<sub>2</sub>O</td><td>BD</td><td></td><td>2.5; 3.2<sup>◊</sup></td><td>800</td><td>150fs 1kHz</td><td>700</td><td>150</td><td>6</td></tr><tr><td></td><td></td><td></td><td>23.9;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>260<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>11.5;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>45.2<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>9.15;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>5.23<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>5.45;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>9.1<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>5.66;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>9.6<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>4.62;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>1.44<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>4.43;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>1.02<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr><tr><td></td><td></td><td></td><td>3.35;</td><td></td><td>Fast ~fs</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>1.06<sup>*</sup></td><td>790</td><td>Slow ~ps</td><td></td><td></td><td>7</td></tr></table>

[1- Rau 2008] [2- Yan 2012] [3- Bala Murali Krishna 2013] [4- Iliopoulos 2015] [5- Reichert 2014] [6- Zhao 2018] [7- Miguez 2017]

* Repetition Rate for the probe is the same as for the pump.

* The 1st entry is the sum of the fast components, and the 2nd entry, which is the sum of the slow entries, is also the predicted n<sub>2,eff</sub> for pulses >100ps as described in [Miguez 2017].

◊ The 2 entries are the predicted bound-electronic n<sub>2</sub>, and the predicted n<sub>2,eff</sub> for pulses >100ps.

◊◊ The 1st entry is the predicted bound-electronic n<sub>2</sub>, while the 2nd is from measured Z-scan data for the longest ps pulses. See Fig. 1.

** The peak 2PA is at 420nm, and the peak bound-electronic n<sub>2</sub> is at 545 nm. Depending on the wavelength, the minimum pulse width varies from 32 to 165 fs.

There is one final set of data that does not fit on this table, and that is the dispersion of the NLO response. The data in the table only covers wavelengths from 532 nm to 1064 nm so little dispersion information is shown. However, the dispersion of the three nuclear contributions to the NLR should be small [Reichert 2014, Alms 1975, Zahedpour 2015]. On the other hand, the bound-electronic nonlinearity should follow similar dispersion to other materials having UV resonances [Hutchings 1992]. For CS<sub>2</sub>, the short pulse limit values for n<sub>2,eff</sub> show essentially no dispersion between 1064 nm and 700 nm (see Fig. 1). However, [Reichert 2014] performed both open and closed aperture Z-scans over a much larger wavelength range from 390 nm to 1550 nm using the shortest pulsewidths available which resulted in signals dominated by bound-electronic n<sub>2</sub> and/or 2PA. Furthermore, the values of the nuclear contributions at these pulsewidths could be confidently subtracted resulting in the plot of n<sub>2</sub> along with α<sub>2</sub> as a function of wavelength as shown in Fig. 2.
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Z-scan measurements of bound-electronic $n_2$ (black circles – labeled as $n_{2,el}$) for femtosecond pulses with non-instantaneous components subtracted, and $\alpha_2$ (blue triangles). Curves represent model fits as described in [Reichert 2014]. Reprinted with permission from [Reichert 2014] erratum © 2016 Optica Publishing Group.</figcaption></figure>

Future work on solvents could include measurements of the 2PA spectra along with the dispersion of the bound-electronic $n_2$ to test theory as done in Fig. 2, as well as measurements verifying the lack of dispersion of the nuclear contributions, and the inclusion of more solvents. Additionally, there are few measurements of solvents in spectral regions where there is significant linear loss [Marble2018, Bautista2021]. Determination of the influence of polar versus nonpolar solvents on the second-order hyperpolarizabilities could also be of interest to spur theoretical calculations for some of these molecules. This could also test local field effects as was done for $CS_2$ in [Reichert 2014].

### References of Section 3.2

[Alms 1975] G. R. Alms, A.K. Burnham, and W. H. Flygare, 1975, Small Nuclear Dispersion References: Measurement of the dispersion in polarizability anisotropies, J. Chem. Phys. 63 (8), 3321-3326, DOI: 10.1063/1.431821

[Bala Murali Krishna 2013] M. Bala Murali Krishna and D. Narayana Rao, 2013, Influence of solvent contribution on nonlinearities of near infra-red absorbing croconate and squaraine dyes with ultrafast laser excitation, J. Appl. Phys. 114, 133103-1-133103-8, DOI: 10.1063/1.4823817

[Bautista2021] J. E. Q. Bautista, M. L. da Silva-Neto, C. L. A. V. Campos, M. Maldonado, C. B. de Araujo, and A. S. L. Gomes, 2021, Thermal and non-thermal intensity dependent optical nonlinearities in ethanol at 800 nm, 1480 nm, and 1560 nm, J. Opt. Soc. Am. B 38, 1104-1111, DOI: 10.1364/JOSAB.418635

[Ganeev2004] R. A. Ganeev, A. I. Ryasnyansky, M. Baba, M. Suzuki, N. Ishizawa, M. Turu, S. Sakakibara, and H. Kuroda, 2004, Nonlinear refraction in $CS_2$, Appl. Phys. B 78, 433-438.

[Hutchings 1992] D.C. Hutchings, M. Sheik-Bahae, D.J. Hagan, and E.W. Van Stryland, 1992, Kramers-Kronig Relations in Nonlinear Optics, Optical and Quantum Electronics, 24, 1-30, DOI: 10.1364/JOSAB.9.002065

[Iliopoulos 2015] K. Iliopoulos, D. Potamianos, E. Kakkava, P. Aloukos, I. Orfanos, and S. Couris, 2015, Ultrafast third order nonlinearities of organic solvents, Opt. Express 23, 24171-24176, DOI:10.1364/OE.23.024171

[Marble2018] C. B. Marble, J. E. Clary, G. D. Noojin, S. P. O'connor, D. T. Nodurft, A. W. Wharmby, B. A. Rockwell, M. O. Scully, and V. V. Yakovlev, 2018, Z-scan measurements of water from 1150 to 1400 nm, Opt. Lett. 43, 4196-4199, DOI: 10.1364/OL.43.004196

[McMorrow1998] D. McMorrow, W. T. Lotshaw, and G. A. Kenney-Wallace, 1988, Femtosecond Optical Kerr Studies on the Origin of the Nonlinear Responses in Simple Liquids, IEEE J. Quantum Electronics 24, 443-454

[Miguez 2017] M. L. Miguez, T. G. B. De Souza, E. C. Barbano, S. C. Zilio, and L. Misoguti, 2017, Measurement of third-order nonlinearities in selected solvents as a function of the pulse width, Opt Exp. 17, 3553-65, DOI: 10.1364/OE.25.003553

[Rau 2008] I. Rau, F. Kajzar, J. Luc, B. Sahraoui, and G. Boudebs, 2008, Comparison of Z-scan and THG derived nonlinear index of refraction in selected organic solvents, J. Opt. Soc. Am. B 25, 1738-1747, DOI: 10.1364/josab.25.001738

[Reichert 2014] M. Reichert, H. Hu, M.R. Ferdinandus, M. Seidel, P. Zhao, T.R. Ensley, D. Peceli, J.M. Reed, D.A. Fishman, S. Webster, D.J. Hagan, & E.W. Van Stryland, 2014, Temporal, spectral, and polarization dependence of the nonlinear optical response of carbon disulfide, Optica 1, 436-445, DOI: 10.1364/OPTICA.1.000436. and see erratum 2016, Optica 3, 657-658, DOI: 10.1364/OPTICA.3.000657

[Yan 2012] X.-Q. Yan, Z.-B. Liu, Y.-S. Chen, and J.-G. Tian, 2012, Polarization characteristics of nonlinear refraction and nonlinear scattering in several solvents, J. Opt. Soc. Am. B 29, 2721-28. DOI: 10.1364/josab.29.002721
[Zahedpour 2015] S. Zahedpour, J. K. Wahlstrand, and H. M. Milchberg, 2015, Measurement of the nonlinear refractive index of air constituents at mid-infrared wavelengths, Opt. Lett. 40, 5794-5797, DOI: 10.1364/OL.40.005794

[Zhao 2018] P. Zhao, M. Reichert, S. Benis, D. J. Hagan, and E. W. Van Stryland, 2018, Temporal and polarization dependence of the nonlinear optical response of solvents, Optica 5(5), 583-594, DOI: 10.1364/OPTICA.99.099999
## 3.3 0D-1D-2D materials: data table and discussion

**Team:** Cecília Campos, Ksenia Dolgaleva, Daniel Espinosa, Anderson Gomes (team leader), Mikko Huttunen, Dragomir Neshev, Lázaro Padilha, Jingshi Yan, Nathalie Vermeulen

### 3.3.1 Introduction

#### 3.3.1.1 0D-1D-2D materials and their NLO applications

Among the ways to categorize nanomaterials, at least two approaches can be followed [Ngo 2014]. In one case, the bulk material is the starting point, and one employs the general definition that a nanomaterial contains at least one dimension of 100 nm or less. Thus, a thin film whose thickness is 100 nm or less is a nanomaterial, such as a nanofilm or a nanoplate. If two dimensions are under 100 nm, then they are named nanowires, nanofibers or nanorods. Finally, under the above definition, if all three dimensions are smaller than 100 nm, it is a nanoparticle. An alternative way becomes relevant if one looks at the material nanostructure. In this case, the dimensionality of the nanoscale component leads to the definition. When the nanostructure has a length larger than 100 nm in one direction only, it is 1D (wire, fiber or rod, for instance). If no dimension is larger than 100 nm, it is 0D (nanoparticle), and a thin film, with two dimensions larger than 100 nm, is a 2D nanostructure (this holds as well for plates and multilayers). A typical representative diagram of the subcategories 0D-1D-2D for nanostructured materials (NSM), as used in this text, is shown in Fig. 3 (adapted from ref. [Sajanlal 2011]). Besides these general definitions, we note that the dimension shorter than 100 nm might confine the movement of free charge carriers and produce a quantum confinement effect [Fox 2001]. In this case, the optical properties of the material would become size-dependent. 0D, 1D and 2D materials presenting quantum confinement effects are called quantum dots (QDs), quantum wires, and quantum wells (QWs), respectively.

<figure><img src="image_5.png" /><figcaption>**Fig. 3** - A typical representative diagram of 0D-1D-2D NSM, as used in this text (figure adapted from [Sajanlal 2011]. Reprinted under CC BY license from [Sajanlal 2011].</figcaption></figure>

There is a myriad of NLO and other applications for 0D-1D-2D NSM, which can be revisited in [Liu 2017; Zhang 2017; Xu 2017; Autere 2018a; Jasim 2019; Eggleton 2019; Liu 2019; Wang 2020; Ahmed 2021]. Based upon the already known and well-established fundamental understanding, and upon the NLO applications in bulk materials, researchers have demonstrated that most NLO effects could also occur at the nanoscale or can be engineered in different ways [Zhang 2021] when they take place in nanoscale dimensions. Although the practical applicability of 0D-1D-2D NSM is at times limited by e.g. fabrication quality issues and/or losses, the NLO properties of 0D-1D-2D NSM have already been exploited in LEDs, nanolasers, nanobiosensors, imaging, as well as in nonlinear microscopy, photoacoustics, photocatalysis, energy harvesting, optical limiting, and saturable-absorber-based mode-locked lasing. Nonlinear nanoplasmonics is another field where NLO in 0D-1D-2D materials has been gaining much attention. The field of nanoplasmonics deals with the study of optical phenomena and applications in the nanoscale neighborhood of dielectric-metal interfaces [Stockman 2011]. It goes
"beyond" plasmonics – with decreasing interaction dimension – aiming at focusing light below the diffraction limit, determined by the Abbe diffraction formula, which can be roughly approximated by $\lambda/2$, where $\lambda$ is the wavelength of light being used in the light-matter interaction. Physically, it is performed by converting photons into localized charge-density oscillations – so-called surface plasmons– on metallic nanostructures. Further reviews on nanoplasmonics and applications can be found at [Barbillon 2017; Panoiu 2018; Krasavin 2019]. Finally, it should be noted that 0D-1D-2D NSM can also feature unusual NLO properties in the THz domain. Whereas this Section focuses on the nonlinearities of 0D-1D-2D NSM at optical wavelengths, further details on the THz nonlinearities are provided in Section 3.8.

### 3.3.1.2 Background prior to 2000

The history of nanostructured materials has been traced back to the 9th century and several other examples of ancient artifacts using nanocomposites emerged throughout the centuries, which of course were only explained after the 19th century with the availability of electron microscopes capable of measuring in the nanoworld [Heiligtag 2013]. But the general nanoscience and nanotechnology scientific history and first publication date back to 1857 with Michael Faraday's work [Faraday 1857]. Then came Feynman's lecture (1959), Taniguchi's "nanotechnology" (1974) and Drexler (1981), as reviewed in [Heiligtag 2013]. Thereafter, the broad field of nanoscience and nanotechnology flourished. Nowadays the material subcategories described here have become a reality, and the study of their NLO properties is a continuously growing research field.

#### 3.3.1.2.1 Background for 0D materials

The first studies about the optical properties of so-called semiconductor microspheres dated back from the early 1980s after the works from Efros and Efros [Efros 1982] and Brus [Brus 1984]. Those nanomaterials were nano-sized semiconducting spheres embedded in a glass matrix. However, they exhibited poor optical quality, mostly due to the difficulties in controlling the particle size distribution and in eliminating surface trap sites. Later, in the early 1990s, colloidal semiconductor quantum dots were first synthesized, narrowing the size distribution and improving the optical properties after surface treatment with organic ligands [Murray 1993]. Major breakthroughs were achieved after the control over the optical properties in II-VI QDs and the field grew rapidly in the late 1990s and early 2000s. During this period, the scientific interest moved from simple core-only nanospheres to more complex nanostructures, including nanorods, nanoplatelets, tetrapods, and core-shell heterostructures, reaching an unprecedented level of control over the optical and electronic properties of those semiconductors by manipulating their sizes, compositions, and shapes. At this stage, the NLO properties of those nanomaterials came to the spotlight as they emerged as promising candidates for a number of applications including bio-labeling, all-optical signal processing, light detectors, and lighting technologies [Kairdolf 2013; García de Arquer 2021].

#### 3.3.1.2.2 Background for 1D materials

Unidimensional structures of different materials were fabricated before 2000. Single-crystalline nanowires made of semiconductors such as GaN, SnO$_2$, ZnO, and Si, as well as carbon nanotubes were particularly attractive for photonics applications. The confinement of light and the cavity-like resonances in the nanowires are interesting features to realize nanolasers, LEDs, frequency converters, solar cells, photodetectors, and sensors [Pauzauskie 2006]. Some linear optical characterizations of the 1D structures were performed before 2000, but there were no extensive experimental measurements of the optical nonlinearities, although some specific works started exploring NLO responses in unidimensional materials in 2000 [Kishida 2000; Ogasawara 2000]. In 2002, the first measurement of
the NLO properties of a single semiconductor nanowire was reported [Johnson 2002]. In the same year, the saturable absorption nonlinearity of single-wall carbon nanotubes (SWCNTs) was demonstrated for the first time [Chen 2002]. Generally speaking, prior to 2002, if the information on the NLO coefficient of a particular 1D material was needed, one usually had to rely on the bulk value.

### 3.3.1.2.3 Background for 2D materials

The potential of semiconductor QW structures for efficient third-order NLO processes has been explored since the early 1980s, especially GaAs-AlGaAs multiple quantum wells (MQWs). Resonant excitonic effects near the band edge region provide the mechanism for NLA and NLR in these materials [Chemla 1985]. By 2000, the importance of semiconductor QWs had already been established, with various applications in optoelectronics, such as laser diodes, LEDs, solar cells, modulators, and detectors [Fox 1996].

Single-layer 2D materials, as we know today, have had very little or almost no scientific impact before 2000. As pointed out in [Miró 2014], according to classical physics, at any finite temperature they would be thermodynamically unstable due to thermal lattice fluctuations (see [Miró 2014] and references therein). The major experimental breakthrough came only in the post-2000 years, after the 2004 publication by Novoselov *et al.* [Novoselov 2004] who first isolated a single-layer 2D material, namely graphene, through the Scotch tape exfoliation of graphite. After 2004, the field of single-layer 2D materials grew rapidly, with many novel materials beyond graphene [Lv 2015; Zhang 2015a], and among the several review articles that have been published, we highlight those related to NLO in 2D materials published in recent years [Liu 2017; Autere 2018a; You 2018; Eggleton 2019; Yamashita 2019; Zhou 2020; Ahmed 2021; Vermeulen 2022]. Several other excellent review articles on different aspects of 2D materials can be found in the literature and have not been indicated here.

### 3.3.1.3 Considerations for 0D-1D-2D materials when performing NLO measurements

Assessing NLO properties in photonic materials, regardless of their dimensions, requires a great amount of scientific care and proper planning regarding the optical source to be employed, the chosen technique, and whether the technique will adequately address the desired information. From the optical material point of view, preferably all the morphological and optical information should be known in order to perform a proper analysis of the results. Regarding 0D-1D-2D materials, further considerations must be taken into account. For 0D materials, size and shape must be properly characterized, and the same holds for the environment in which the 0D material is embedded, e.g., a solvent, in the case of suspensions, or a substrate. It is very important to characterize the NLO properties of the solvent or substrate. This is also valid for 1D and 2D materials. Another aspect to be considered, in the case of 0D-1D-2D materials in suspension, is the scattering, both linear and nonlinear. Therefore, when measuring the linear absorption, the researcher should also insure the knowledge of the extinction coefficient and, in turn, the scattering coefficient of the sample. During the NLO measurement, the influence of nonlinear scattering can contribute to the NLO properties in an undesirable manner. We also point out that for some materials such as graphene, the level of doping strongly influences the NLO response [Jiang 2018] and as such needs to be specified in order to correctly interpret the measurement data. When working with 0D-1D-2D materials in a free-space excitation setup, it is often preferred to use ultrashort excitation pulses (picosecond, femtosecond) to avoid damage to the sample. We point out that the short propagation distance of the excitation beam through nanometer-thick 0D-1D-2D materials typically rules out the need for phase matching in NLO experiments. It is also worth noting that some works on 2D materials report $\chi^{(2)}/\chi^{(3)}$ values either considering sheet susceptibilities or bulk-like susceptibilities, as differentiated in the tables below. The units are different for both cases, as the bulk-like susceptibility is typically assumed to be the sheet susceptibility divided by the monolayer thickness [Autere 2018a]. Finally, we point out that the nonlinearities can also be expressed in terms of
conductivities $\sigma^{(2)}/\sigma^{(3)}$ rather than susceptibilities [Cheng 2014]. In addition to these specific considerations for 0D-1D-2D materials, also the best practices described in Section 2 should be taken into account when performing NLO measurements.

### 3.3.1.4 Description of general table outline

Tables 4A and 4B show a representative list of, respectively, second-order and third-order NLO properties of 0D, 1D and 2D materials taken from the literature since 2000, with the entries arranged in alphabetical order. Based upon the best practices in Section 2, Tables 4A and 4B were put together using inclusion criteria for publications that clearly state the required technical information for an unambiguous and clear identification of the NLO properties being evaluated. In some cases, publications missing only one important parameter could unfortunately not be considered, although they were technically sound. The tables are subdivided into “Material properties,” “Measurement details” and “Nonlinear properties.” The same set of columns was used for 0D, 1D and 2D materials. Within each column the information is given in the order of the header description, and powers of 10 (e.g., $10^{\pm\alpha}$) are written as $E^{\pm\alpha}$ for compactness. The NLO technique used is provided in the “Method” column. Some papers relied on non-standard NLO measurement techniques (e.g., method listed as “Other”) or have notes associated with their measurement/analysis. These values and information are listed within the “Comments” column. The works included in the tables nominally report data obtained at room temperature, unless denoted otherwise in the “Comments” column. For works that report dependences of the nonlinearity on parameters such as wavelength/energy, polarization, sample size, input fluence, doping level, pulse length/peak power, and concentration, these dependencies are denoted by “1234567” respectively, in the “Comments” column. If dispersive values for the NLO parameter were provided, the cited value represents the peak value for the material within the measurement range. The tables are restricted to optical nonlinearities up to the third order, and although higher orders have become of interest [Reyna 2017], they are not covered here.

## 3.3.2 Discussion

### 3.3.2.1 Advancement since 2000 and remaining challenges

#### 3.3.2.1.1 Advancement and challenges for 0D materials

Since 2000, as the field of 0D semiconductor nanomaterials was growing, stable, monodisperse nanoparticles were becoming available, and the interest in investigating their NLO response has increased. This was mainly due to the initial excitement created by the predictions that the NLO response in 0D materials would be highly enhanced as compared to bulk semiconductors [Brunhes 2000a] since the quantum confinement should enhance the oscillator strength as the nanomaterial size was reduced. Thus, several groups started to experimentally study distinct optical properties regarding QDs, including the second-order susceptibility [Brunhes 2000b, Sauvage 2001], third-order susceptibility [Valdueza-Felip 2008], NLR [Wang 2019], and NLA [Pu 2006; Wang 2019; Alo 2020]. The first reports on the magnitude of the 2PA cross-section indicated that, for spherical nanomaterials, the magnitude of this process was linearly dependent on the nanoparticle volume [Pu 2006; Padilha 2007; Makarov 2014]. Strategies to better control the NLO response in 0D materials have involved band-structure optimization, obtained with PbS QDs [Padilha 2011], and shape control [Scott 2015]. In the last decade, a fair amount of effort has been put into gaining further control over the electronic and optical properties of these nanostructures by developing sophisticated heterostructures, and by shape control. Strong enhancement of 2PA cross-sections has been reported in CdSe nanoplatelets, with superlinear dependence on the volume [Scott 2015]. On the other hand, sublinear volume dependence has been recently reported for core/shell heterostructures [Alo 2020]. By shape controlling and
heterostructuring these nanomaterials, it is expected that one can obtain further control over the nanomaterials' NLO response towards the development of tailored nanostructures for on-demand NLO.

### 3.3.2.1.2 Advancement and challenges for 1D materials

Since the year 2000, NLO characterizations of 1D structures in various arrangements and shapes were reported. We give several examples in the following. While many studies focused on experiments involving many nanowires with a distribution of dimensions, a few involve only single nanowires. SHG and SRS measurements were performed on individual GaP nanowires [Wu 2009; Sanatinia 2014]. SHG was also evaluated in a single GaAs nanoneedle [Chen 2010] or even in areas within a single GaAs nanowire with different crystal phases [Timofeeva 2016]. Often, the NLO technique is applied to a collection of 1D structures, either random or spatially ordered. For example, SHG was measured in an array of GaP nanowires embedded in a polydimethylsiloxane matrix [Fedorov 2020] or grown on a GaP substrate [Sanatinia 2012]. Also, Z-scan experiments were performed in single-walled carbon nanotubes (SWCNTs) deposited on a glass substrate as a thin-film [Seo 2006] or dispersed in a colloidal suspension [Shi 2019]. In the latter case, the authors reported that a semiconductor-SWCNTs colloid exhibits a lower saturation irradiance and a lower 2PA coefficient as compared to a mixture of metallic-SWCNTs and semiconducting-SWCNTs [Shi 2019].

One exciting feature of 1D materials is the optical nonlinearity dependence on the structure dimensions due to resonances or surface effects. For example, in GaP nanowires (with a diameter of 210 nm), the SHG efficiency increases with increasing equivalent thickness (total volume of nanowires per unit area) [Fedorov 2020], while the SRS irradiance decreases with increasing length [Wu 2009]. Furthermore, the SHG irradiance in GaP nanopillars (with diameters from 100 nm to 250 nm) presents a strong dependence on the pillar diameter [Sanatinia 2012]. Moreover, polarization-dependent measurements can be used to distinguish between the second-harmonic light generated in bulk or on the surface [Johnson 2002; Sanatinia 2012; Sanatinia 2014; Sanatinia 2015]. Indeed, by using polarization-dependent measurements, the nonlinear coefficient at the surface of a GaP nanopillar was observed to be approximately 15 times higher than that from the bulk [Sanatinia 2015]. Conversely, the SHG light from a wurtzite (WZ) GaAs single nanoneedle was shown to be primarily originating from the bulk [Chen 2010].

Advances in SHG polarimetry also allow studying the crystal structure of a nonlinear material because the SHG is sensitive to crystallographic symmetry. A per-pixel analysis of SHG images was used to distinguish between GaAs nanowires in the pure WZ, pure zinc blende (ZB), WZ-ZB mixed phases, or with ZB rotational twins [Timofeeva 2016].

Despite the advancements reported on SHG measurements, this material category is far from being thoroughly characterized. There is still room to use other second- and third-order NLO techniques to exploit the large surface-to-volume ratio and strong Mie and Fabry-Perot resonances of the 1D structures.

### 3.3.2.1.3 Advancement and challenges for 2D materials

The study of NLO in quantum-well structures post-2000 emphasizes the effects of intersubband transitions (ISBT) rather than excitonic effects. GaN-AlN and GaN-AlGaN MQW structures have been shown to be particularly suitable for both second-order and third-order processes resonant at the ISBT frequency [Rapaport 2003, Nevou 2006].

The advancements in single-layer 2D materials, which can be considered a novel field of research with less than two decades of R&D, have been outstanding both from the point of view of scientific understanding and applications. Both refractive and absorptive nonlinearities have been investigated. Particularly, graphene has been shown to exhibit extremely strong third-order nonlinearities and a low saturable absorption threshold [Chen 2015; Miao 2015; Dremetsika 2017; Jiang 2018; Soavi 2018; Thakur 2019], and there are several other 2D materials and particularly Transition Metal
Dichalcogenides (TMDs) such as MoS₂ that feature a pronounced second-order NLO response as well (see, e.g., [Woodward 2016, Autere 2018b]). Nevertheless, in view of the large family of 2D materials, further characterization will be required to get the full picture of their NLO behavior, including the contributions from exciton resonances, free carriers, etc., in order to fully exploit their application potential in, e.g., optoelectronics [Sundaram 2018]. The progress made so far has been promoted by advances in the 2D materials synthesis and proper morphological characterization. Also, the tunability of the NLO response using electrical gating [Jiang 2018, Soavi 2018] is an important advantage for the practical use of both the refractive and absorptive nonlinearities in, e.g., wavelength converters and saturable-absorption-based modelocked lasers, respectively [Vermeulen 2022]. Nevertheless, further fabrication improvement is necessary to achieve large-area films on a wide range of substrates, easy placement of contact points, proper etching, and advanced metrology control. The NLO characterization might also provide metrology methods to control parameters of interest for practical NLO applications of 2D materials.

### 3.3.2.2 Recommendations for future works on 0D-1D-2D materials

The future of NLO in 0D-1D-2D materials relies on a more in-depth NLO characterization of the already existing materials and on the development of novel materials. A multidisciplinary approach will also be required to further exploit their optical nonlinearities. Among the novel 0D-1D-2D materials, we highlight nanodiamonds, nanoporous materials, core-shell nanoparticles, perovskite nanostructures, silicene, antimonene, MXenes, 2D metal-organic framework nanosheets, boron nitride nanosheets, and metal-based nanomaterials [Baig 2021]. Metal-based nanomaterials are at the basis of nanoplasmonics, and the field of nonlinear nanoplasmonics is also a flourishing and promising area, opening new avenues for future works [Panoiu 2018; Krasavin 2019]. Further NLO research on 0D materials could benefit, amongst others, biosensing applications [Wang 2020]. Regarding 1D materials, assemblies of 1D nanomaterials have been recently introduced, and their NLO properties need to be further understood [Chen 2019]. 2D materials also have significant potential for future study and applications. For instance, 2D rare-earth based materials are opening new avenues [Chen 2021], since rare-earth materials are quite well studied and have already found a great deal of applications. Also, various methods to enhance the NLO response, including plasmonics, gating, and functionalization [Wei 2019] can be further explored. That said, to enable further progress in the overall understanding of 0D-1D-2D materials, it will be important that future works report in detail both the fabrication aspects, the material properties and the NLO experiments carried out with the materials. Parameters such as the linear absorption loss (which is typically quite high in 0D-1D-2D materials) and the doping level are often overlooked, yet essential to properly evaluate the observed NLO behavior [Vermeulen 2022]. In addition, great care is needed to adequately extract and describe the actual NLO coefficients that underpin the measurement data, especially when the dependence of the NLO coefficients on, e.g., wavelength is also being studied. Only such a systematic approach allows building up an extensive and in-depth understanding of the NLO physics of this promising category of materials and the potential they hold for NLO applications.
### 3.3.3 Data table for 0D-1D-2D materials

<table><caption>**Table 4A.** Second-order NLO properties of 0D-1D-2D materials from representative works since 2000. Legend for superscripts: see below the table.</caption><thead><tr><th colspan="11">Second-Order Nonlinearities</th></tr><tr><th rowspan="2">Material</th><th rowspan="2">Method</th><th colspan="3">Material Properties</th><th rowspan="2">Measurement Details</th><th colspan="2">Nonlinear Properties</th></tr><tr><th>Fabrication<br>Substrate</th><th>Width<br>Number of<br>layers</th><th>Crystallinity<br>Bandgap<br>Doping Level</th><th>Pump<br>Wavelength<br>Peak Power<br>Peak Irradiance<br>Beam spot size<br>Pulse Width<br>Rep. Rate</th><th>χ<sup>(2)</sup><br>deff</th><th>Reference<br>Additional parameters and comments</th></tr></thead><tbody><tr><th colspan="9">0D Materials</th></tr><tr><td>InAs/GaAs QD</td><td>SHG</td><td>MBE<br>GaAs</td><td>-<br>40</td><td>-<br>-<br>-</td><td>7381 nm<br>-<br>2.00E+12 W/m<sup>2</sup><br>-<br>-<br>-</td><td>1.20E-9 m/V</td><td>[Brunhes 2000]<br>Nonlinearity dependence<sup>1,2</sup> shown in Fig. 2 of reference paper<br>InAs QD height/diameter = 0.2 (diameter = 2.3E-8 m)<br>χ<sup>(2)</sup> :<br>1.2E-9 m/V (sample)<br>2E-7 m/V (1 QD layer)<br>Pump polarization:<br>TE (in the layer plane)</td></tr></tbody></table>
<table><tr><td>InAs/GaAs QD</td><td>SHG</td><td>MBE<br>(001)-GaAs</td><td>-<br>30</td><td>-<br>-<br>-<br>-</td><td>20000 nm<br>-<br>1.00E+12 W/m²<br>1.50E-4 m<br>-<br>-</td><td>2.50E-6 m/V</td><td>[Sauvage 2001]<br>χ<sup>(2)</sup> for 1 QD layer<br>1 layer:<br>InAs QD height of 3E-9 m<br>InAs QD base length 1.5E-8 m<br>GaAs barrier thickness of 5E-8 m<br><br>Pump polarization:<br>50% in the layer plane along [110] direction,<br>50% along z-growth axis<br><br>Temperature: 10K</td></tr><tr><td colspan="8">1D Materials</td></tr><tr><td>GaP nanopillars</td><td>SHG</td><td>Nanosphere<br>lithography /<br>RIE /<br>CAIBE<br>GaP (100)</td><td>1.50E-7 m<br>-</td><td>Monocrystalline<br>2.26 eV</td><td>840 nm<br>-<br>1.00E-6 m<br>1.00E-13 s<br>8.20E+7 Hz</td><td>-<br>-<br>-<br>2.00E-7</td><td>[Sanatinia 2012]<br>Nonlinearity dependence³ shown in Fig. 2b of reference paper<br>The efficiency is for the SHG light generated by the array of<br>nanopillars. The authors did not take into account the light<br>collection efficiency of the microscope objective</td></tr><tr><td>GaP nanowires</td><td>SHG</td><td>MBE /<br>G-coating<br>Silicon (111)</td><td>1.50E-7 m<br>-</td><td>Monocrystalline<br>-<br>-</td><td>1048 nm<br>-<br>5.00E-6 m<br>1.50E-13 s<br>8.00E+7 Hz</td><td>-<br>-<br>-<br>1.00E-2</td><td>[Fedorov 2020]<br>Nonlinearity dependence¹ shown in Fig. 2c of the reference paper</td></tr></table>
2D Materials

<table><tr><td>Al<sub>0.08</sub>Ga<sub>0.92</sub>N/<br>GaN QW</td><td>SHG</td><td>MOCVD<br>GaN-on-(0001)-<br>csapphire</td><td>Monocrystalline<br>-<br>10</td><td>1064 nm<br>-<br>-<br>5.00E-9 s<br>1.40E+1 Hz</td><td>d<sub>31</sub> = 2.2E-12 m/V<br>d<sub>33</sub> = -4.80E-12 m/V</td><td>[Passeri 2004]<br>Thickness for 1 layer Al<sub>0.08</sub>Ga<sub>0.92</sub>N/GaN:<br>1.4E-8 m/2.4E-9 m<br>The authors also provided information for<br>Al<sub>0.15</sub>Ga<sub>0.85</sub>N/GaN QW and<br>Al0.08Ga<sub>0.92</sub>N/GaN with different thickness<br>Pump: p and s-polarization</td></tr><tr><td>GaN-AlN QW</td><td>SHG</td><td>MBE<br>AlN-on-csapphire</td><td>-<br>200</td><td>1980 nm<br>2.50E+3 W<br>-<br>1.50E-4 m<br>6.00E-9 s<br>3.00E+1 Hz</td><td>1.14E-10 m/V</td><td>[Nevou 2006]<br>Thickness for 1 layer GaN-AlN:<br>2.593E-9 m/3E-9 m<br>Nonlinearity dependence¹ shown in Fig. 3 of reference paper<br>The nonlinear conversion efficiency in MQW sample was 16 times<br>higher compared to that in a reference bulk GaN sample<br>Pump: p-polarization</td></tr><tr><td>Graphene</td><td>DFG</td><td>CVD<br>Quartz</td><td>Monolayer<br>-<br>0.3 eV</td><td>547 nm<br>-<br>3.00E-4 m<br>1.00E-13 s<br>1.00E+3 Hz</td><td>3.00E-7 m/V</td><td>[Constant 2016]<br>DFG enhanced through plasmonic excitation<br>Probe parameters:<br>λ = 615 nm<br>Intensity: 1.5E+11 W/m²<br>Polarization: p-polarization (transverse magnetic)<br>Pump: p-polarization</td></tr></table>
<table><tr><td>MoSe<sub>2</sub></td><td>SHG</td><td>Micromechanical exfoliation<br>Si/SiO<sub>2</sub> (285 nm)</td><td>~ 0.65E-9 m<br>Monolayer</td><td>-<br>-<br>-</td><td>1560 nm<br>2.70E+3 W<br>-<br>-<br>1.50E-13 s<br>5.00E+7 Hz</td><td>3.70E-11 m/V<br>-</td><td>[Autere 2018b]<br>Nonlinearity dependence<sup>2</sup> in Fig. 5 of the reference paper</td></tr><tr><td>WS<sub>2</sub></td><td>SHG</td><td>Micromechanical exfoliation<br>Si/SiO<sub>2</sub> (285 nm)</td><td>~ 0.65E-9 m<br>Monolayer</td><td>-<br>-<br>-</td><td>1560 nm<br>2.70E+3 W<br>-<br>-<br>1.50E-13 s<br>5.00E+7 Hz</td><td>1.62E-11 m/V<br>-</td><td>[Autere 2018b]<br>Nonlinearity dependence<sup>2</sup> in Fig. 5 of the reference paper</td></tr><tr><td>WS<sub>2</sub></td><td>SHG</td><td>CVD<br>SiO<sub>2</sub>/Si</td><td>6.50E-10 m<br>Monolayer</td><td>-<br>-<br>-</td><td>832 nm<br>-<br>-<br>1.80E-6 m<br>1.06E-13 s<br>8.80E+7 Hz</td><td>-<br>4.50E-9 m/V</td><td>[Janisch 2014]<br>-</td></tr><tr><td>WS<sub>2</sub>-MS</td><td>SHG</td><td>CVD<br>Sapphire</td><td>-<br>Monolayer</td><td>-<br>-<br>-</td><td>804 nm<br>-<br>-<br>4.00E-6 m<br>8.00E-15 s<br>8.00E+7 Hz</td><td>-<br>-</td><td>[Li 2019]<br>The authors reported a 20-fold enhancement of the optical SHG from WS<sub>2</sub> monolayers by cooperating with SiO<sub>2</sub> dielectric microspheres (MSs)</td></tr><tr><td>WS<sub>2</sub></td><td>SHG</td><td>Micromechanical exfoliation<br>Si/SiO<sub>2</sub> (285 nm)</td><td>~ 0.65E-9 m<br>Monolayer</td><td>-<br>-<br>-</td><td>1560 nm<br>2.70E+3 W<br>-<br>-<br>1.50E-13 s<br>5.00E+7 Hz</td><td>1.65E-11 m/V<br>-</td><td>[Autere 2018b]<br>Nonlinearity dependence<sup>2</sup> in Fig. 5 of the reference paper</td></tr></table>
<table><tr><td>WSe<sub>2</sub></td><td>SHG</td><td>CVT<br>SiO<sub>2</sub>/Si (300 nm)</td><td>7.00E-10 m<br>Monolayer</td><td>-<br>-<br>-</td><td>816 nm<br>-<br>-<br>1.07E-13 s<br>8.80E+7 Hz</td><td>5.00E-9 m/V</td><td>[Ribeiro-Soares 2015]<br>Nonlinearity dependence<sup>2</sup> in Fig. 3 of the reference paper</td></tr><tr><td>WSe<sub>2</sub></td><td>SHG</td><td>Micromechanical<br>exfoliation<br>Fused silica</td><td>1.17E-9 m<br>Monolayer</td><td>-<br>1.65 eV</td><td>1546 nm<br>-<br>(0.8 ± 0.1)E-6 m<br>2.00E-13 s<br>8.00E+7 Hz</td><td>(0.7 ± 0.09)E-19 m<sup>2</sup>/V</td><td>[Rosa 2018]<br>Nonlinearity dependence<sup>2,3</sup> in Fig. 2 and Fig. 4 of the reference paper<br>The authors also provided information for 4, 5, 6 and 9 layers at 1546 nm<br>Specified second-order nonlinearity is the sheet susceptibility</td></tr></table>

Superscripts indicate the work reports the nonlinearity dependence on ¹wavelength/energy, ²polarization, ³sample size (width, diameter, thickness, number of layers), ⁴input fluence, ⁵doping level, ⁶pulse length/peak power, ⁷concentration.
<table><tr><td>CdSe QDs</td><td>Other (add in comments)</td><td>Organic Chemical Route<br/>Toluene</td><td>4.80E-9 m</td><td>-</td><td>-</td><td>800 nm<br/>-<br/>-<br/>-<br/>1.00E-13 s<br/>8.20E+7 Hz</td><td>-<br/>-<br/>-<br/>1.03E+4 m/W</td><td>[Pu 2006]<br/>The authors also provided NLO parameters for different diameters (2.4E-9 m to 4.6E-9 m)<br/>Nonlinearity dependence³ shown in Fig. 1 of the reference paper<br/>Two-photon excited photoluminescence</td></tr><tr><td>CdSe/CdZnS QDs</td><td>Other (add in comments)</td><td>Organic Chemical Route<br/>Toluene</td><td>1.24E-8 m</td><td>-</td><td>-</td><td>800 nm<br/>-<br/>-<br/>-<br/>8.00E-14 s<br/>1.00E+3 Hz</td><td>-<br/>-<br/>2.70E+4 m/W</td><td>[Alo 2020]<br/>Nuclei diameter: 4.0 nm<br/>Shell thickness: 4.2 nm<br/>The authors also provided NLO parameters for shell thicknesses of 1.2 nm, 2.5 nm, and 3.0 nm<br/>Nonlinearity dependence¹ shown in Fig. 4 of the reference paper<br/>The technique is called MPAPS. The bandgap defined as the excitonic peak position is shown in Fig. S1 of the reference paper</td></tr><tr><td>CdTe QDs</td><td>Other (add in comments)</td><td>Organic Chemical Route<br/>Toluene</td><td>5.40E-9 m</td><td>-</td><td>-</td><td>840 nm<br/>-<br/>-<br/>-<br/>1.00E-13 s<br/>8.20E+7 Hz</td><td>-<br/>-<br/>7.96E+3 m/W</td><td>[Pu 2006]<br/>The authors also provided NLO parameters for different diameters (4.4E-9 m to 5.2E-9 m) and doping levels<br/>Nonlinearity dependence³ shown in Fig. 1 of the reference paper<br/>Two-photon excited photoluminescence</td></tr></table>
<table><tr><td>GaN-AlN QD</td><td>FWM</td><td>MBE</td><td>-</td><td>-</td><td>-</td><td>1500 nm</td><td>1.3E-6 esu</td><td>[Valdueza-Felip 2008]<br/>The authors also provided NLO parameter for 200 layers GaN-AlN QD</td></tr><tr><td></td><td></td><td>AlN-on-sapphire</td><td>20</td><td>-</td><td>-</td><td>1.00E-13 s<br/>1.00E+3 Hz</td><td>-</td><td>For 1 layer:<br/>GaN QD height 1.1E-9 m<br/>AlN barrier thickness 3E-9 m</td></tr><tr><td>Graphene quantum dots</td><td>Z-scan</td><td>-</td><td>&lt; 5E-9 m</td><td>-</td><td>-</td><td>355 nm<br/>2.6E+14 W/m²<br/>2.30E+1 m<br/>1.00E-11 s<br/>1.00E+1 Hz</td><td>(5.7 ± 1.2)E-19 m²/W<br/>-1.40E-11 m/W</td><td>[Wang 2019]<br/>Details regarding fabrication:<br/>https://www.strem.com/catalog/v/06-0334/44/nanomaterials_1034343-98-0<br/>Peak irradiance: 2.5E+13 W/m² to 26E+13 W/m²<br/>No NL response is presented at 532 nm and infrared regions. At 355 nm, authors have brought out a clear saturable absorption effect</td></tr></table>
1D Materials

<table><tr><td>GaP nanowires</td><td>SRS Threshold</td><td>Pulsed Laser Vaporization (PLV)<br/>Silicon</td><td>2.10E-7 m</td><td>-<br/>-<br/>-</td><td>Monocrystalline</td><td>514.5 nm<br/>-<br/>-<br/>4.00E-7 m<br/>CW<br/>-</td><td>-<br/>-<br/>-<br/>-</td><td>[Wu 2009]<br/>Authors studied stimulated Raman scattering from GaP NWs as a function of their length<br/>Raman shift (1/m): 3.62E+4 (TO) / 3.98E+4 (LO)<br/>The quality factor for the nanowire segment was measured as Q ~ 18000 by assuming gRaman = 10 cm/GW (the gain coefficient for GaP bulk material from Rhee B. K., Bron W. E., Kuhl J., 1984, Phys. Rev. B 30, 7358-7361)</td></tr><tr><td>s-SWCNT</td><td>Z-scan</td><td>CoMoCAT catalytic CVD process (SigmaAldrich)<br/>Petroleum ether</td><td>0.6E-9 m to 1.1E-9 m</td><td>-<br/>2.26E-1 dB/m<br/>1064 nm</td><td>-<br/>-<br/>-</td><td>1064 nm<br/>-<br/>3.20E+12 W/m²<br/>-<br/>4.00E-9 s<br/>1.00E+1 Hz</td><td>-<br/>-<br/>-<br/>2.90E-12 m/W</td><td>[Shi 2019]<br/>Saturation irradiance: 2.13E+12 W/m²<br/>14% non-saturable absorption</td></tr><tr><td>SWCNT</td><td>Z-scan</td><td>HIPCO SWNT<br/>Glass</td><td>1.00E-9 m</td><td>-<br/>8.40E+4 1/m<br/>532 nm</td><td>-<br/>-<br/>-</td><td>532 nm<br/>-<br/>1.60E+10 W/m²<br/>1.20E-5 m<br/>8.00E-9 s<br/>1.00E+1 Hz</td><td>(-1.4E-15 + i 4.3E-16) m²/V²<br/>-</td><td>[Seo 2006]<br/>-</td></tr><tr><td>SWCNT</td><td>Z-scan</td><td>CoMoCAT catalytic CVD process (SigmaAldrich)<br/>Petroleum ether</td><td>-</td><td>-<br/>1.72E-1 dB/m<br/>1064 nm</td><td>-<br/>-<br/>-</td><td>1064 nm<br/>-<br/>3.20E+12 W/m²<br/>-<br/>4.00E-9 s<br/>1.00E+1 Hz</td><td>-<br/>-<br/>-<br/>5.20E-12 m/W</td><td>[Shi 2019]<br/>Saturation irradiance: 2.95E+12 W/m²<br/>12% non-saturable absorption</td></tr></table>
2D Materials

<table><tr><td>AlGaN-GaN QW</td><td>Nonlinear transm./refl.</td><td>MBE</td><td>-</td><td>-</td><td>-</td><td>1500 nm</td><td>5.00E-19 m²/V²</td><td>[Rapaport 2003]<br/>NLO parameters are for GaN/Al<sub>0.65</sub>Ga<sub>0.35</sub>N/GaN<br/>The technique used to measure n<sub>2</sub> was cross-phase modulation α<sub>2</sub> due to SA<br/>The saturation irradiance was measured using many different pump wavelengths (see Fig. 3 of the reference paper)</td></tr><tr><td></td><td></td><td>c-Sapphire</td><td>100</td><td>-</td><td>-</td><td>7.00E+10 W/m²<br/>1.00E-13 s<br/>1.00E+3 Hz</td><td>2E-17 m²/W<br/>4.00E-10 m/W</td><td>Beam radius of 8.5E-5 m for the saturation irradiance measurement<br/>Pump and probe polarization: p-polarized<br/>Thickness for 1 layer of GaN/Al<sub>0.65</sub>Ga<sub>0.35</sub>N/GaN<br/>1.2E-9 m/1.5E-9 m/7.8E-10 m [(x 3 periods) barrier]<br/>Information also for GaN/Al<sub>0.15</sub>Ga<sub>0.85</sub>N/GaN</td></tr><tr><td>Black Phosphorus</td><td>THG</td><td>Exfoliated<br/>Glass + AlOx encapsulation</td><td>9.50E-9 m<br/>20</td><td>-<br/>7.00E+7 1/m<br/>520 nm</td><td>-<br/>-<br/>-</td><td>1560 nm<br/>5.58E+15 W/m²<br/>2.00E-6 m<br/>1.00E-13 s<br/>8.00E+6 Hz</td><td>1.64E-19 m²/V²<br/>-</td><td>[Autere 2017]<br/>Nonlinearity dependence³ in Fig. 4(b) of the reference paper</td></tr></table>
<table><tr><td>Black Phosphorus</td><td>Z-scan</td><td>Solvent exfoliation + gradient centrifugation<br>NMP</td><td>-<br>13 to 15</td><td>-<br>3.86E+2 1/m<br>800 nm</td><td>-<br>-<br>-</td><td>800 nm<br>-<br>3.54E+15 W/m²<br>-<br>1.00E-13 s<br>1.00E+3 Hz</td><td>-3.01E-14 esu<br>-2.07E-20 m²/W</td><td>[Xu 2017b]<br>Saturation irradiance: 6E+13 W/m²<br>Nonlinearity dependence³ in Table 1 of the reference paper<br>Real part of χ<sup>(3)</sup>, 28.5% non-saturable absorption</td></tr><tr><td>Black Phosphorus</td><td>Z-scan</td><td>Nanoplatelets obtained by grinding bulk material and then dispersed<br>NMP/PVA</td><td>30E-9 m to 60E-9 m<br>50 to 100</td><td>-<br>around 7 %<br>500 nm to 2000 nm</td><td>-<br>-<br>-</td><td>800 nm<br>-<br>1.30E+16 W/m²<br>4.00E-5 m<br>1.00E-13 s<br>1.00E+3 Hz</td><td>-<br>1.20E-13 m²/W<br>4.50E-10 m/W</td><td>[Zheng 2015a]<br>-</td></tr><tr><td>Electrochemical graphene oxide</td><td>Z-scan</td><td>Electrochemical method<br>Vacuum filtration<br>-</td><td>3.00E-7 m<br>Thin film</td><td>> 2.0 above 350 nm<br>-<br>300 nm to 1800 nm</td><td>-<br>0.88 eV<br>-</td><td>800 nm<br>-<br>-<br>1.00E-5 m<br>8.50E-14 s<br>1.00E+4 Hz</td><td>-<br>3.63E-13 m²/W<br>7.00E-11 m/W</td><td>[Ren 2016]<br>Nonlinearity dependence⁴ in Fig. 3 and Fig. 4 of the reference paper<br>n<sub>2</sub> is 3.63E-13 m²/W at 8E+02 J/m²<br>n<sub>2</sub> is 2.82E-13 m²/W at 1E+03 J/m²<br>n<sub>2</sub> is 1.91E-13 m²/W at 2E+03 J/m²<br>n<sub>2</sub> is 0.57E-13 m²/W at 4E+03 J/m²<br>Nonlinear absorption coefficient is 7E-11 m/W at 4E+03 J/m²</td></tr></table>
<table><tr><td>Graphene</td><td>Other (add in comments)</td><td>CVD</td><td>-</td><td>-</td><td>-</td><td>1600 nm<br/>-<br/>5.00E+12 W/m²<br/>2.00E-5 m<br/>1.80E-13 s<br/>8.20E+7 Hz</td><td>(-6 - i 9.6)E-16 m²/V²<br/>(-1 - i 1.6)E-13 m²/W</td><td>[Dremetsika 2017]<br/>Technique: optically-heterodyne-detected optical Kerr effect<br/>Probe parameters:<br/>λ = 1600 nm;<br/>Irradiance = 3E+11W/m²;<br/>beam radius = 1.5E-05 m<br/>Specified values of pump and probe polarization are for χ<sub>xyxy</sub>+χ<sub>xyyx</sub></td></tr><tr><td>Graphene</td><td>FWM</td><td>CVD<br/>Fused silica</td><td>-<br/>Monolayer</td><td>-<br/>-<br/>0 eV</td><td>-<br/>-<br/>0 eV</td><td>1040 nm<br/>-<br/>2.00E-13 s<br/>8.00E+7 Hz</td><td>3.00E-17 m²/V²</td><td>[Jiang 2018]<br/>Nonlinearity dependence⁵ in Fig. 5d of the reference paper</td></tr><tr><td>Graphene</td><td>Z-scan</td><td>CVD<br/>Quartz</td><td>-<br/>6 to 8</td><td>1.49E-1 dB/m<br/>1000 nm to 2500 nm</td><td>-<br/>Approx. 0</td><td>1930 nm<br/>-<br/>5.75E+11 W/m²<br/>3.50E-5 m<br/>2.80E-12 s<br/>3.23E+7 Hz</td><td>4.58E-11 m²/W</td><td>[Miao 2015]<br/>Saturation irradiance: 1E+10 W/m²<br/>Nonlinearity also at 1562 nm specified in the reference paper</td></tr><tr><td>Graphene</td><td>THG</td><td>CVD<br/>Sapphire</td><td>-<br/>Monolayer</td><td>2.30E-2 dB/m<br/>1250 nm</td><td>-<br/>0.25 eV</td><td>3100 nm<br/>-<br/>2.40E+12 W/m²<br/>4.70E-6 m<br/>3.00E-13 s<br/>8.00E+7 Hz</td><td>8.00E-17 m²/V²</td><td>[Soavi 2018]<br/>Nonlinearity dependence¹,⁵ in Fig. 3b and Fig. 3d of reference paper<br/>η: 1E-11 %</td></tr></table>
<table><tr><td>Graphene</td><td>Z-scan</td><td>CVD<br>Quartz</td><td>Monolayer</td><td>-<br>-<br>-</td><td>-<br>-<br>Approx. 0</td><td>900 nm<br>-<br>4.59E+13 W/m²<br>-<br>1.00E-13 s<br>8.00E+7 Hz</td><td>-<br>1.08E-12 m²/W</td><td>[Thakur 2019]<br>Nonlinearity dependence¹,⁶ in Fig. 2b and Fig. 5b of the reference paper</td></tr><tr><td>Graphene<br>Oxide (GO)</td><td>Z-scan</td><td>Modified<br>Hummers<br>method<br>Self-assembly<br>Vacuum filtration<br>process<br>H₂SO₄<br>/H₂O₂/deionized<br>water/methanol<br>mixture</td><td>1.00E-6 m<br>Thin film</td><td>-<br>-</td><td>-<br>-</td><td>1560 nm<br>-<br>3.80E+12 W/m²<br>-<br>6.70E-14 s<br>2.00E+7 Hz</td><td>-<br>4.50E-14 m²/W</td><td>[Xu 2017a]<br>Nonlinearity dependence⁶ in Fig. 4(a) of the reference paper</td></tr><tr><td>Graphene<br>Oxide (GO)</td><td>Z-scan</td><td>Self-assembly<br>H₂SO₄<br>/H₂O₂/deionized<br>water/methanol<br>mixture</td><td>8E-7 m to<br>2E-6 m<br>Thin film</td><td>-<br>-<br>-</td><td>-<br>-<br>-</td><td>800 nm<br>-<br>-<br>2.50E-6 m<br>1.00E-13 s<br>-</td><td>-<br>-<br>4.00E-7 m/W</td><td>[Zheng 2014]<br>α₂ is 4.00E-7 m/W at 32 μJ/cm²<br>Nonlinearity dependence⁴ in Fig. 4(a) of the reference paper<br>Information also for reduced Graphene Oxide (rGO)</td></tr></table>
<table><tr><td>Graphene Oxide (GO) with gold nanoparticles</td><td>Z-scan</td><td>Vacuum filtration process<br>Water<br>1.00E-6 m<br>Thin film</td><td>-<br>&lt; 20%<br>800 nm</td><td>-<br>-<br>-</td><td>800 nm<br>-<br>-<br>-<br>-</td><td>-<br>-<br>2.40E-16 m²/W<br>-<br>1.00E-9 m/W</td><td>[Fraser 2015]<br>Nonlinearity dependence⁷ in Fig. 4 of the reference paper<br>n₂ of reference GO film is 4E-17 m²/W<br>α₂ of reference GO film is 1.7E-10 m/W<br>Input fluence laser is 1.4E+02 J/m²</td></tr><tr><td>Mo₀.₅W₀.₅S₂</td><td>Z-scan</td><td>CVT<br>Glass<br>2.20E-5 m<br>Multilayer</td><td>-<br>6.22E-2 dB/m<br>1064 nm</td><td>Polycrystalline<br>-<br>-</td><td>1064 nm<br>-<br>8.50E-16 W/m²<br>-<br>2.50E-11 s<br>2.00E+1 Hz</td><td>[-(4.18 ± 2.32)E-8 + i(5.73 ± 1.12)E-11] esu<br>-(8.73 ± 1.47)E-15 m²/W<br>(1.91 ± 0.78)E-10 m/W</td><td>[Bikorimana 2016]<br>α₂ due to TPA</td></tr><tr><td>MoS₂</td><td>THG</td><td>Micromechanical exfoliation<br>Si/SiO₂ (285 nm)</td><td>~ 0.65E-9 m<br>Monolayer</td><td>-<br>-<br>-</td><td>1560 nm<br>2.70E+3 W<br>-<br>1.50E-13 s<br>5.00E+7 Hz</td><td>3.60E-19 m²/V²</td><td>[Autere 2018b]<br>Nonlinearity dependence² in Fig. 5 of the reference paper</td></tr><tr><td>MoS₂</td><td>THG</td><td>Micromechanical cleavage<br>Si/SiO₂</td><td>-<br>Monolayer</td><td>Close to 0<br>1560 nm</td><td>1560 nm<br>8.00E+3 W<br>-<br>9.30E-7 m<br>1.50E-13 s<br>5.00E+7 Hz</td><td>-<br>-<br>-<br>-<br>-</td><td>[Säynätjoki 2017]<br>Nonlinearity dependence²,³,⁶ in Fig. 4 and Fig. 5 of reference paper<br>η: 4.76E-8 %</td></tr></table>
<table><tr><td>MoSe<sub>2</sub></td><td>THG</td><td>Micromechanical exfoliation<br>Si/SiO<sub>2</sub> (285 nm)</td><td>~ 0.65E-9 m<br>Monolayer</td><td>-<br>-<br>-</td><td>-<br>-<br>-</td><td>1560 nm<br>2.70E+3 W<br>-<br>1.50E-13 s<br>5.00E+7 Hz</td><td>2.20E-19 m<sup>2</sup>/V<sup>2</sup><br>-<br>-</td><td>[Autere 2018b]<br>Nonlinearity dependence<sup>2</sup> in Fig. 5 of the reference paper</td></tr><tr><td>MoSe<sub>2</sub></td><td>Z-scan</td><td>Liquid-phase exfoliation technique<br>Cyclohexyl pyrrolidinone</td><td>Mono or triple</td><td>7.93E+2 1/m<br>800 nm</td><td>indirect: 1.1 eV/<br>direct: 1.5 eV<br>-</td><td>800 nm<br>-<br>-<br>1.00E-13 s<br>1.00E+3 Hz</td><td>-i(1.45 ± 0.34)E-15 esu<br>-<br>-(2.54 ± 0.60)E-14<br>m/W</td><td>[Wang 2014]<br>Saturation irradiance: (590 ± 225)E+13 W/m<sup>2</sup><br>The authors also provided NLO parameters for measurements at 515 nm, 532 nm, 1030 nm and 1064 nm</td></tr><tr><td>MoTe<sub>2</sub></td><td>Z-scan</td><td>Liquid-phase exfoliation technique<br>Cyclohexyl pyrrolidinone</td><td>Mono or triple</td><td>1.07E+2 1/m<br>1064 nm</td><td>indirect: 1.0 eV/<br>direct: 1.0 eV<br>-</td><td>1064 nm<br>-<br>-<br>1.00E-10 s<br>1.00E+4 Hz</td><td>[-(0.92 ± 0.15)E-11<br>- i(2.27 ± 0.39)E-12]<br>esu<br>-(0.160 ± 0.027)E-16<br>m<sup>2</sup>/W<br>-(2.99 ± 0.52)E-11<br>m/W</td><td>[Wang 2014]<br>Saturation irradiance: (0.19 ± 0.04)E+13 W/m<sup>2</sup><br>The authors also provided NLO parameters for measurements at 515 nm, 532 nm, 800 nm and 1030 nm</td></tr><tr><td>Reduced GO (rGO)</td><td>Z-scan</td><td>Self-assembly<br>H<sub>2</sub>SO<sub>4</sub>/H<sub>2</sub>O<sub>2</sub>/deionized water/methanol mixture</td><td>8E-7 m to 2E-6 m<br>Thin film</td><td>-<br>-<br>-</td><td>-<br>-<br>-</td><td>800 nm<br>-<br>-<br>2.50E-6 m<br>1.00E-13 s<br>-</td><td>-<br>-1.00E-13 m<sup>2</sup>/W</td><td>[Zheng 2014]<br>Fully rGO with no oxygen-containing group for optical fluence > 50 µJ/cm<sup>2</sup><br>Nonlinearity dependence<sup>4</sup> in Fig. 4(a) of reference paper<br>Information also for Graphene Oxide (GO)</td></tr></table>
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**3.4 Metamaterials: data table and discussion**

**Team:** Adam Ball, Ksenia Dolgaleva, Daniel Espinosa, Nathaniel Kinsey, Mikko Huttunen (team leader), Dragomir Neshev, Ray Secondo

**3.4.1 Introduction**

**3.4.1.1 Metamaterials definitions**

Efficient NLO interactions are essential for many applications in modern optics. However, they typically require high-irradiance laser sources or long interaction lengths. These requirements cannot be satisfied in the small-footprint optical devices used in many applications. With the growing importance of photonic integrated circuits (PICs) and ultracompact nanostructured optical devices, enabling efficient NLO interactions at the nanoscale is essential. This may be achieved in nanostructured optical devices with engineered NLO responses. Along these lines, metamaterials offer several advantages such as the ability to confine light for enhanced nonlinearities as well as to shape and control scattered light in new ways [Chen2018, Minovich2015, Husu2012, Litchinitser2018, BinHasan2014, Kauranen2012, Smirnova2016, Chang2018, Butet2015, Huang2020, Gigli2019a, Reshef2019b, DeAngelis2020]. It is also worth noting that in ultra-thin metasurfaces the phase-matching condition is largely relaxed. It is expressed in the form of transverse phase matching, together with mode-matching that replaces the conventional longitudinal phase matching for propagating waves in bulk materials. Recent work also demonstrates that stacked metasurface structures can be longitudinally phasematched by utilizing phase-engineered metasurfaces [Stolt2021].

In this Section, we focus on highlighting recent achievements in the broader field of NLO metamaterials and nanostructured materials. Metamaterials are artificial nanostructures comprised of building blocks called meta-atoms, which serve as their structural units. The constituent materials can be metallic [Kauranen2012, Husu2012, BinHasan2014, Litchinitser2018], semiconducting [Vabishchevich2018, Shcherbakov2017] or dielectric [Aain2019, Gigli2019a, Yan2020]. Moreover, one can choose the arrangement geometry of the building blocks in such a way as to change the enhancement mechanism. The structural units can be positioned in a regular or irregular 2D array to form a 2D metamaterial or a metasurface [Kauranen2012, Saad-Bin-Alam2021a]. Alternatively, they can comprise 3D metamaterials through structures such as extruded 2D arrays, multi-layer films [Stolt2021, Suresh2021], or fully 3D nanostructure arrays [Kadic2019]. The latter requires far more sophisticated fabrication approaches and is not discussed in this work. We focus instead on the NLO performance of more practical metasurfaces and planar nanostructures made of metal, semiconductor and dielectric materials. Since metamaterials are manmade artificial materials different from all other material types considered in this article, a short tutorial on metamaterials is provided below.

**3.4.1.2 Mechanisms of nonlinearity enhancement**

The power of artificial nanostructures resides in their ability to manipulate the flow of light in a manner not accessible in bulk materials by engineering optical confinement, scattering and interference on a subwavelength scale. An artificial nanostructure thus represents a new material comprised of the structural elements (meta-atoms), which has an optical response that can be very different from that of its constituent materials. There are several mechanisms responsible for the enhancement, suppression and/or sign reversal of the NLO effects attainable in nanostructures. Although most of the mechanisms are based on the resonances associated with the meta-atoms, the origin and type of resonance differ depending on the type of metasurface, as discussed below. Moreover, there are some additional mechanisms of enhancement one can explore, which are briefly described as well.

*Metal-dielectric nanostructures* with meta-atoms built of noble metals, such as silver and gold, exhibit surface plasmon polaritons and localized surface plasmon resonances (LSPRs) associated with the meta-atoms (often referred to as antennae) [Chen2018, Gomes2020, Lis2014, Wurtz2008,
Minovich2015, Krasavin2018, Husu2012, Litchinitser2018, BinHasan2014, Deka2017, Kauranen2012, Smirnova2016, Chang2018, Butet2015, Huang2020, Gwo2016]. LSPRs are responsible for the localization, and subsequent increase, of the electric field strength in the vicinity of the meta-atoms, which contributes to the enhancement of the NLO interactions. While the metallic meta-atoms can themselves exhibit nonlinearities due to the strong fields outside of the meta-atom, plasmonic structures are typically employed to enhance the electric field within another nonlinear medium, such as lithium niobate or a nonlinear polymer. This allows for the separation and optimization of the confinement and nonlinear material. Yet, the quality (Q-) factors of LSPRs are generally very low (< 10) due to high absorption losses associated with metals. High ohmic losses and Joule heating limit the applicability of the noble metal metasurfaces in practical NLO devices unless other mechanisms of enhancement are explored in addition to LSPRs [Reshef2019a, Saad-Bin-Alam2021a, Huttunen2019, Khurgin2013]. For example, in regular metal-dielectric metasurfaces, there exist collective resonances called surface lattice resonances (SLRs) [Michaeli2017], which exhibit much higher Q-factors on the order of $10^3$ [Saad-Bin-Alam2021a]. SLRs arise from the coherent scattering of light where the periodic structure introduces interference to amplify the resonance and its associated Q-factor. In this way, the collective scattering of the entire lattice, as opposed to the enhancement provided by a single meta-atom, contributes to the NLO processes much more efficiently.

*Dielectric metasurfaces*, including those made of semiconductors, have appeared in recent years as a solution to overcome limitations associated with the high losses of metals [Smirnova2016, Gigli2019a, Yan2020, Hopkins2015, Shcherbakov2017, Sain2019, Vabishchevich2018]. Dielectric metasurfaces are typically divided by the values of their refractive indices into high-index ($n > 3.5$), such as Si, GaAs, Ge, mid-index ($2 < n < 3.5$), such as TiO$_2$, Se, SiC, Si$_3$N$_4$, and low-index ($n < 2$), such as silica- and polymer-based metasurfaces, which are not commonly used in NLO due to their poor confinement or weak NLO characteristics [Yan2020]. The primary mechanisms of enhancement of the NLO interactions in dielectric metasurfaces are electric and magnetic Mie resonances associated with their individual building blocks. These resonances result in moderate Q-factors of 10-100, depending upon the index [Rybin2017], and fields localized within the antenna, enabling enhanced effective NLO responses. In this sense, their nonlinearity is derived from the nanoantenna's constituent material, coupling the linear index and nonlinearity. While this coupling generates some restrictions in the case of second-order nonlinearities, for third-order effects, higher index (needed for high-Q) materials also generally exhibit larger nonlinearities [Sheik-Bahae1990]. High-quality-factor phase-gradient dielectric metasurfaces with a Q-factor around 2,500 have recently been demonstrated and hold the potential for NLO applications and on-chip lasing [Lawrence2020]. Moreover, the NLO performance of dielectric metasurfaces can be boosted by exploiting multimode interference, resulting in Fano resonances [Hopkins2015] or non-radiative anapole modes [Yang2018]. Dielectric metasurfaces hold promise for infrared imaging [Camacho-Morales2021], harmonic generation and other areas of application of NLO processes [Yan2020].

*Photonic bound states in the continuum (BICs)* structures create localized eigenstates with an infinite lifetime coexisting with the continuous spectrum of radiative modes [Bykov2020] and complement the more general photonic topological insulating structures [Khanikaev2013]. Localized types of practical BIC nanostructures are a subgroup of dielectric metamaterials where a special type of resonance, having an ultra-high Q-factor on the order of $10^7$ and higher, is employed [Koshelev2019a, Wang2020, Koshelev2019b, Koshelev2020, Bernhard2020, Liu2019, Azzam2021]. They represent localized states with energies embedded in the continuum of the radiating states. The basic idea behind BICs is the lack of coupling between the resonant mode and all the radiation channels of the surrounding space. Different BICs can be categorized based on the radiation suppression mechanisms. In practice, infinitely high Q-factors of BICs are limited by finite sample sizes, material absorption, symmetry breaking and fabrication imperfections. These limitations result in quasi-BICs with large but finite Q-factors, rendering the opportunity for radiation collection, important for practical nonlinear photonic devices. Dynamical nonlinear image tuning via polarization and wavelength tuning in BIC nanostructures has
been experimentally demonstrated [Xu2019], opening the door for application of such nanostructures in tunable displays, nonlinear holograms and other areas.

*Epsilon-near-zero* (ENZ) nanostructures [Reshef2019b, Neira2015, Alam2018, Suresh2021, Wen2018, Deng2020, Yang2019] represent a new group of metamaterials designed in such a way that the real part of the effective dielectric permittivity is vanishingly small at a certain wavelength called zero-permittivity wavelength $\lambda_{ENZ}$. In contrast, *near-zero-index* (NZI) metamaterials have the effective refractive index $n_{eff}$ vanishing at a specific wavelength called zero-index wavelength $\lambda_{ZI}$. Such materials are generally achieved through the combination of metal and dielectric constituents or through the use of resonant structures [Kinsey2019]. There are several mechanisms of enhancement of the NLO interactions acting as the basis of ENZ and NZI metamaterials. First, near the ENZ/NZI wavelength, the strong dispersion of the index gives rise to slow-light propagation, which results in temporal compression and enhancement of the electric field [Khurgin2010]. Second, the small magnitude of the permittivity in the ENZ region gives rise to additional electric field-enhancement, through the continuity of the normal component of the displacement field $D_{\perp}$. Lastly, ENZ and NZI provide reduced phase advance within the bulk and facilitate easier phase matching. However, it should be noted that these properties result primarily from slow light effects and are not present in NZI schemes which maintain a finite group velocity [Khurgin2019, Khurgin2020].

### 3.4.1.3 Historical development

The precursors to the rise and development of the field of metal-dielectric NLO metasurfaces were the earlier demonstrations in the 1980s of the benefit of surface roughness in surface-emitted second- and third-order NLO processes [Chen1981, Shalaev1998] exhibiting several orders-of-magnitude enhancement in comparison to harmonic generation from a smooth surface. Experiments on hyper-Rayleigh scattering (HRS), reported at the end of the 1990s, have served as the first formal demonstrations of incoherent SHG from spherical gold and silver nanoparticles [Dadap1999, Vance1998]. Noble metal nanoparticles of different dimensions and shapes have been explored in later HRS studies [Hao2002, Nappa2006, Butet2010a, Butet2010b], performed in the 2000s, with an emphasis on the interplay between multipoles [Nappa2006, Butet2010a]. SHG from non-spherical plasmonic objects [Bouhelier2003, Nahata2003, Danckwerts2007, Hanke2012] (before 2012) and more advanced 3D plasmonic nanostructures [Zhang2011, Cai2011, Pu2010, Park2011] (after 2010) have also been investigated.

Structured plasmonic surfaces for efficient NLO interactions were proposed [Wokaun1981, Reinisch1983] and later experimentally demonstrated [Wokaun1981, Coutaz1985] in the 1980s by SHG generation off an extended metal grating into the first order of diffraction. The first experimental demonstration of the NLO effects in a plasmonic array of meta-atoms was SHG in an array of L-shaped gold nanoparticles [Lamprecht1997, Tuovinen2002, Canfield2004] (late 1990s – early 2000s), followed by a split-ring resonator (SRR) metasurface with magnetic resonances [Linden2012, Klein2006, Feth2008] (2006 – 2012). These works laid the foundation to the rapidly evolving field of plasmonic metasurfaces for efficient NLO interactions.

The study of resonant dielectric nanostructures is a relatively new research direction in the field of nanophotonics that started a decade ago [Zhao2009, Schuller2007]. Employing sub-wavelength dielectric nanoparticles with Mie resonances for engineering optical metasurfaces with strong NLO responses resulted in the demonstration of enhanced SHG, THG, FWM and other NLO interactions without paying the cost of high losses associated with metal-dielectric nanostructures [Minovich2015, Smirnova2016, Liu2018a, Liu2018b, Kivshar2018, Grinblat2017a, Grinblat2017b]. Mie-resonant silicon and AlGaAs nanoparticles have received considerable attention for nonlinear frequency generation [Liu2018a, Liu2018b].

Spurred by developments in solid-state physics, work in dielectric metasurfaces has more recently broadened to encompass the physical effects of optical topology, enabling robust unidirectional propagation and control over scattering [Khanikaev2013]. Such effects have been explored in the
nonlinear regime to realize devices such as lasers [Bandres2018] and switches [Shalaev2019] for integrated photonics (see on-chip waveguiding materials category in Section 3.6). Among the many physical effects being explored, BIC structures represent a special case for NLO due to their ability to achieve high Q-factors [Koshelev2019a, Wang2020, Koshelev2019b, Koshelev2020, Bernhard2020, Liu2019, Azzam2021] and are being actively explored. Although BICs were mathematically predicted a long time ago and were investigated in other fields of physics in the 1960s–1970s [Fonda1963, Cumpsty1971], the first experimental demonstrations of BIC-based photonic devices were performed only in 2016 [Hsu2016].

ENZ and NZI metamaterials represent other emerging classes of nanostructured materials for efficient NLO interactions. The research in the field started in the 2010s with a plethora of theoretical works [Ciattoni2010a, Ciattoni2010b, Vincenti2011, Ciattoni2012] predicting enhanced NLO interactions in media with vanishing permittivity, which preceded the first experimental demonstrations both in homogeneous thin films and effective ENZ metastructures [Alam2016, Kinsey2015, Luk2015, Capretti2015, Caspani2016]. Following these works, ENZ nanostructures have been proposed, offering tunability and unprecedented strength and tailorability of NLO interactions [Neira2015, Alam2018, Suresh2021, Wen2018].

### 3.4.1.4 Metamaterial-related best practices and considerations when performing NLO measurements

Among the best experimental practices for a proper assessment of the NLO performances of a metamaterial, the following rules-of-thumb can be recommended. If possible, one needs to provide information about the estimated values of the nonlinearities per meta-atom. Since some of the described systems are fragile and can melt under even moderate levels of optical power in a free-space excitation setup, one must follow precautions to avoid optical damage, and to ensure reproducibility of the obtained NLO results. Additionally, as efforts to realize higher Q-factor structures emerge, analysis should also account for the strong dispersion near these resonances. For example, not all of the incident excitation power may be captured by the resonator [Shcherbakov2019] and the effect of pulse chirp near the resonance will begin to play a role in the analysis and extraction of effective NLO parameters. In the end, the caveats and best practices are the same as those listed in Section 2, with enhanced focus on damage, thermal nonlinearities, etc. due to the relatively strong absorption of plasmonic structures.

### 3.4.1.5 Description of general table outline

Tables 5A and 5B show a representative list of, respectively, second-order and third-order NLO properties of metamaterials taken from the literature since 2000, with the entries arranged in alphabetical order. Our aim has been to compile the most relevant advances in NLO metamaterials since 2000. Because numerous studies have been performed during the last two decades, not all of them could be tabulated and/or referenced here. Instead, the studies highlighted here are only representative works, where extra care has been taken in the characterization of the investigated metamaterial following the recommendations in the section above and the best practices in Section 2. Furthermore, our focus is on the studies that report NLO coefficients (e.g., $\chi^{(2)}$, $\chi^{(3)}$, $n_{2(\text{eff})}$, $\alpha_2$, etc.). Many studies exist where the investigated metamaterial has been shown to result in enhancement of NLO signals when compared to some reference material/system. In order to restrict our scope, we chose not to tabulate works providing only enhancement factors, but instead focused on investigations also reporting NLO coefficients and/or conversion efficiencies. An interested reader is referred to Refs. [Chen2018, Minovich2015, Krasavin2018, Husu2012, Litchinitser2018, BinHasan2014, Kauranen2012, Smirnova2016, Chang2018, Butet2015, Huang2020, Gigli2019a, Reshef2019b, DeAngelis2020] for a broader perspective of the topic.
Table 5A and 5B are subdivided into “Material properties,” “Measurement details” and “Nonlinear properties.” The main material properties that we have tabulated are the material(s), fabrication technique, sample thickness, material crystallinity, substrate and the reference to the sub-figure in Figs. 4-5 showing the fabricated device. All reported experiments have nominally been performed at room temperature. Fig. 4 shows SEM pictures of devices with second-order NLO responses, while Fig. 5 shows SEM pictures of devices with third-order NLO responses. The main pump parameters have also been tabulated, along with the enhancement mechanism utilized in the sample. Within each column the information is given in the order of the header description, and the NLO technique used is provided in the “Method” column. In Table 5A, we list the overall conversion efficiency η and/or absolute value of the effective susceptibility as the NLO parameters of key importance. In Table 5B, in addition to the above parameters, the effective $n_2$ and $\alpha_2$ values have also been listed where relevant.

<figure><img src="image_6.png" /><figcaption>**Fig. 4** SEM images of metamaterial devices with second-order nonlinear properties. The labels of the sub-figures correspond to the labels given in Table 5A. Panels A–D, G, K, L, P, and R–T are reprinted with permission from [Liu2016, Xu2020, Sautter2019, Marino2019, Camacho-Moralez2016, Cambiasso2017, Anthur2020, Czaplicki2018, Wen2018, Bernhardt2020, Semmlinger2018], ACS. Panels M and O are reprinted with permission from [Lee2014, Celebrano2015], Springer Nature. Panels I and N are reprinted under CC BY license from [Liu2018, Wolf2015]. Panels E, F, J and Q are reprinted with permission from [Gili2016, Gigli2019, Wang2020, Klein2007], ©2007,2016,2019,2020 Optica Publishing Group. Panel H is reprinted with permission from [Koshelev2020], AAAS.</figcaption></figure>
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<figure><img src="image_7.png" /><figcaption>**Fig. 5** SEM images of metamaterial devices with third-order nonlinear properties. The labels of the sub-figures correspond to the labels given in Table 5B. Panels A, B, E, G, H, and I are reprinted with permission from [Zubyuk2019, Suresh2021, Grinblat2017, Yang2015, Koshelev2019, Shcherbakov2014], ACS. Panel C is reprinted with permission from [Melentiev2013], ©2013 Optica Publishing Group. Panel D is reprinted with permission from [Lee2014, Celebrano2015], Springer Nature. Panel F is reprinted under CC BY license from [Liu2019].</figcaption></figure>

**3.4.2 Discussion**

**3.4.2.1 Advancements and remaining challenges for metamaterials**

Because most of the early works were focused on metallic nanoparticles, *de facto* metamaterials of that
era, a major advancement in the field of metamaterials can be seen to be the gradual shift from metals
to the lower-loss semiconductors and dielectric nanomaterials [Grinblat2017a, Grinblat2017b,
Semmlinger2018, Koshelev2019b]. New mechanisms of enhancement, associated with these new kinds
of metamaterials, including ENZ, BIC and Fano resonances, have clearly diversified from the early
approaches that were almost solely based on material resonances [Czaplicki2018, Koshelev2020].
Apart from the fundamental studies of NLO phenomena, dielectric metasurfaces have enabled the
observation of such challenging effects as spontaneous parametric down-conversion (SPDC). Recently,
for the first time, the generation of photon pairs via SPDC in lithium niobate quantum optical
metasurfaces [Santiago-Cruz2021] and in a single AlGaAs nanocylinder [Marino2019b] has been
demonstrated. By engineering the quantum optical metasurface, the authors demonstrated tailoring of
the photon-pair spectrum in a controlled way [Santiago-Cruz2021]. These achievements lay the
foundation for the application of dielectric metasurfaces in quantum light generation.

Furthermore, a generally progressing trend is visible in the achieved NLO parameters where newer
NLO metamaterials seem to (clearly) outperform their earlier counterparts. As an example, many of the
recent investigations of dielectric and semiconducting metasurfaces demonstrate conversion
efficiencies on the order of ~10⁻³ for SHG [Sautter2019, Xu2020, Gili2016] and on the order of ~10⁻⁴
for THG [Lee2014, Liu2018a, Koshelev2019b]. In contrast, conversion efficiencies achieved in earlier
works have been around 2×10⁻¹¹ for SHG and 3×10⁻¹² for THG [Klein2008].

NLO metasurfaces have recently started gaining popularity as efficient sources of THz radiation, as this
frequency range is not easily attainable by conventional methods of light generation. There is a growing
number of works where THz generation by DFG off a metasurface is reported [Luo2014, Keren-
Zur2019, Polyushkin2011, Tal2020, McDonnell2021]. These results compare favorably with the more
standard method of THz generation by DFG in bulk crystals. Specifically, the measured sheet NLO
susceptibility ~10⁻¹⁶ m² V⁻¹ far exceeds that of thin films and bulk non-centrosymmetric materials
[Luo2014]. On the other hand, the extraction of the associated NLO coefficients has only been
performed in [Luo2014], leaving this a development area for future studies. There is very little
knowledge about NLO parameters of metamaterials at THz frequencies, and it is highly advisable for
research works to report such values for this frequency window.
On the design side, significant improvements have been made in the development of new analytical [Saad-Bin-Alam2021b] and numerical [Butet2016, Blechman2019, Noor2020] methods for designing NLO metasurfaces optimized for some specific NLO interactions.

Further work is in progress to make NLO metamaterials functional in real-life applications. A partly connected challenge is to develop nanofabrication techniques to the level where fabrication of 3D NLO metamaterials becomes a routine task [Stolt2021]. When these achievements can be combined with recent advances in phase-engineered metasurfaces [Chen2018], one could envisage 3D phase-matched metamaterials that could potentially boost efficiencies up to the levels adequate for practical applications. We note that at that stage, linear absorption/scattering losses would start playing a role [Stolt2021], and actions to mitigate their detrimental effects should be taken.

### 3.4.2.2 Recommendations for future works on metamaterials

It would be extremely beneficial if future publications would in general report more extensive details of the experimental parameters alongside the NLO parameters of the studied metamaterials (see our earlier recommendations for “best practices”). For example, oftentimes publications report only the relative NLO enhancement factors of the metamaterials, making it difficult to estimate the potential importance of the studies. Therefore, we recommend that, in future works, at least the parameters listed in the tables reported here be properly quantified and reported.

In many cases, it would also be important to quantitatively characterize individual meta-atoms acting as the building blocks in the metamaterials (see, for example, [Saad-Bin-Alam2021b]). If such information becomes more abundantly available, designing next-generation NLO metamaterials would become an easier task. Furthermore, such data would improve the transparency of the work because it would facilitate estimating the success of the demonstrated enhancement mechanism and the potential importance of the performed study.

Finally, we recommend the community to continue their effort to estimate the damage thresholds of various metamaterial platforms and to study their possible damage mechanisms. We consider such efforts particularly useful when NLO metamaterials reach the level of maturity where applications start to emerge. At that stage, the relevant parameters should not be limited only to conversion efficiency/susceptibility/$n_2$ values, but can include the measured power of frequency-converted light.

In this case, such information becomes critical for an application-oriented researcher to be able to estimate whether the studied metamaterial system could be scaled up in area to be used with high-power pump lasers.
### 3.4.3 Data table for metamaterials

<table><caption>**Table 5A.** Second-order NLO properties of metamaterials 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</th><th colspan="4">NONLINEAR PROPERTIES</th></tr><tr><th>Material<br>Substrate<br>Fabrication<br>Thickness<br>Crystallinity</th><th>SEM<br>image</th><th>Method</th><th>Pump<br>wavelength<br>Peak irradiance<br>Beam waist<br>Pulse width<br>Rep. rate</th><th>η (%)*</th><th>|χ<sup>(2)</sup>|<br>(m/V)*</th><th>Enhancement<br>mechanism</th><th>Additional<br>parameters and<br>Comments</th><th>Reference</th></tr></thead><tbody><tr><td>GaAs / AlGaO<br>GaAs<br>MBE/e-beam lithography<br>0.45 [μm]<br>Monocrystalline / monocrystalline</td><td>4(a)</td><td>SHG</td><td>1020 [nm]<br>3.4×10<sup>7</sup> [MW/m<sup>2</sup>]<br>3 [μm]<br>0.012 [ps]<br>80000 [kHz]</td><td>2×10<sup>-3</sup> % / 1.5 ×10<sup>-8</sup><br>W<sup>-1</sup></td><td>-</td><td>Magnetic dipole<br>resonance</td><td>-</td><td>[Liu2016]</td></tr><tr><td>(110)-GaAs<br>Fused silica<br>MOCVD/e-beam lithography<br>0.4 [μm]<br>Monocrystalline</td><td>4(b)</td><td>SHG</td><td>1450 [nm]<br>1.2×10<sup>7</sup> [MW/m<sup>2</sup>]<br>2 [μm]<br>0.0255 [ps]<br>80000 [kHz]</td><td>3.00×10<sup>-3</sup></td><td>-</td><td>-</td><td>-</td><td>[Xu2020]</td></tr><tr><td>(111)-GaAs<br>Fused silica<br>MOCVD/e-beam lithography<br>0.4 [μm]<br>Monocrystalline</td><td>4(c)</td><td>SHG</td><td>1556 [nm]<br>1.0×10<sup>7</sup> [MW/m<sup>2</sup>]<br>-<br>0.01 [ps]<br>80000 [kHz]</td><td>4.80×10<sup>-3</sup></td><td>-</td><td>-</td><td>-</td><td>[Sautter2019]</td></tr><tr><td>Al<sub>0.18</sub>Ga<sub>0.82</sub>As / AlOx<br>GaAs<br>MBE/e-beam lithography<br>0.4 / 1.0 [μm]<br>Monocrystalline / amorphous</td><td>-</td><td>SHG</td><td>1550 [nm]<br>1.6×10<sup>7</sup> [MW/m<sup>2</sup>]<br>-<br>0.015 [ps]<br>-</td><td>1×10<sup>-3</sup> % / 1.5 ×10<sup>-7</sup><br>W<sup>-1</sup></td><td>-</td><td>-</td><td>-</td><td>[Ghirardini2017]</td></tr></tbody></table>
<table><tr><td>Al<sub>0.18</sub>Ga<sub>0.82</sub>As / AlOx<br>GaAs<br>MBE/e-beam lithography<br>0.4 / 1.0 [µm]<br>Monocrystalline / amorphous</td><td>4(d)</td><td>SHG</td><td>1550 [nm]<br>5×10<sup>7</sup> [MW/m<sup>2</sup>]<br>25 [µm]<br>0.016 [ps]<br>1000 [kHz]</td><td>2×10<sup>-6</sup> W<sup>-1</sup></td><td>-</td><td>-</td><td>-</td><td>[Marino2019a]</td></tr><tr><td>Al<sub>0.18</sub>Ga<sub>0.82</sub>As / AlOx<br>GaAs<br>MBE/e-beam lithography<br>0.4 / 1.0 [µm]<br>Monocrystalline / amorphous</td><td>4(e)</td><td>SHG</td><td>1554 [nm]<br>1.6×10<sup>7</sup> [MW/m<sup>2</sup>]<br>-<br>0.015 [ps]<br>-</td><td>1.10×10<sup>-3</sup></td><td>-</td><td>-</td><td>-</td><td>[Gili2016]</td></tr><tr><td>Al<sub>0.18</sub>Ga<sub>0.82</sub>As / AlOx<br>GaAs<br>MOCVD/e-beam lithography<br>0.35 / 1.0 [µm]<br>Monocrystalline / amorphous</td><td>4(f)</td><td>SHG</td><td>1550 [nm]<br>-<br>2.36 [µm]<br>0.016 [ps]<br>1000 [kHz]</td><td>6.5×10<sup>-6</sup> W<sup>-1</sup></td><td>-</td><td>-</td><td>-</td><td>[Gigli2019b]</td></tr><tr><td>Al<sub>0.2</sub>Ga<sub>0.8</sub>As<br>Fused silica<br>MOCVD/e-beam lithography<br>0.3 [µm]<br>Monocrystalline</td><td>4(g)</td><td>SHG</td><td>1556 [nm]<br>7.0×10<sup>7</sup> [MW/m<sup>2</sup>]<br>1.1 [µm]<br>0.05 [ps]<br>5000 [kHz]</td><td>8.50×10<sup>-3</sup></td><td>-</td><td>-</td><td>-</td><td>[Camacho-Morales2016]</td></tr><tr><td>AlGaAs<br>SiO<sub>2</sub>/ITO/SiO<sub>2</sub><br>e-beam lithography<br>0.635 [µm]<br>Monocrystalline</td><td>4(h)</td><td>SHG</td><td>1570 [nm]<br>-<br>1.8 [µm]<br>0.2 [ps]<br>5144 [kHz]</td><td>1.3×10<sup>-6</sup> W<sup>-1</sup></td><td>-</td><td>Quasi-BIC, Mie resonance</td><td>-</td><td>[Koshelev2020]</td></tr></table>
<table><tr><td>GaAs / AlGaO<br>GaAs<br>MBE/e-beam lithography<br>0.45 [µm]<br>Monocrystalline / monocrystalline</td><td>4(i)</td><td>SHG</td><td>1570 [nm]<br>-<br>1.5 [µm]<br>0.004 [ps]<br>1 [kHz]</td><td>2.30×10<sup>-4</sup></td><td>-</td><td>-</td><td>-</td><td>[Liu2018a]</td></tr><tr><td>GaN<br>Silicon<br>MOCVD, PECVD, e-beam<br>lithography<br>0.2 [µm]<br>Monocrystalline</td><td>4(j)</td><td>SHG</td><td>1543.55 [nm]<br>-<br>-<br>CW<br>-</td><td>2.4×10<sup>-2</sup> W<sup>-1</sup></td><td>-</td><td>-</td><td>-</td><td>[Wang2020]</td></tr><tr><td>GaP nanoantennas<br>GaP<br>e-beam lithography<br>0.2 [µm]<br>Monocrystalline</td><td>4(k)</td><td>SHG</td><td>910 [nm]<br>2×10<sup>9</sup> [MW/m<sup>2</sup>]<br>-<br>0.018 [ps]<br>100 [kHz]</td><td>2×10<sup>-4</sup> % / 4×10<sup>-9</sup> W<sup>-1</sup></td><td>-</td><td>Scattering<br>resonance and<br>surface effect</td><td>-</td><td>[Cambiasso<br>2017]</td></tr><tr><td>GaP nanodimers<br>Sapphire<br>MOCVD on GaAs then transfer-<br>bonded to Sapphire, e-beam<br>lithography<br>0.15 [µm]<br>Monocrystalline</td><td>4(l)</td><td>SHG</td><td>1190–1220 [nm]<br>100 [MW/m<sup>2</sup>]<br>50 [µm]<br>CW<br>-</td><td>2.00×10<sup>-7</sup></td><td>-</td><td>Quasi-BIC in<br>arrays with slight<br>asymmetry in<br>nanoparticles</td><td>-</td><td>[Anthur2020]</td></tr><tr><td>GaP nanodimers<br>Sapphire<br>MOCVD on GaAs then transfer-<br>bonded to Sapphire, e-beam<br>lithography<br>0.15 [µm]<br>Monocrystalline</td><td>4(l)</td><td>SHG</td><td>1190–1220 [nm]<br>1.0×10<sup>5</sup> [MW/m<sup>2</sup>]<br>50 [µm]<br>0.02 [ps]<br>-</td><td>4.00×10<sup>-5</sup></td><td>-</td><td>Quasi-BIC in<br>arrays with slight<br>asymmetry in<br>nanoparticles</td><td>-</td><td>[Anthur2020]</td></tr></table>
<table><tr><td>Gold / InGaAs/AlInAs MQW<br>InP<br>MBE<br>0.1 / 0.03 / 0.656 [µm]<br>Amorphous / monocrystalline /<br>monocrystalline</td><td>4(m)</td><td>SHG</td><td>8000 [nm]<br>150 [MW/m²]<br>17.5 [µm]<br>40000 [ps]<br>250 [kHz]</td><td>2.00×10<sup>-4</sup></td><td>55×10<sup>-8</sup> (yyy)</td><td>MQW<br>enhancement</td><td>-</td><td>[Lee2014]</td></tr><tr><td>Gold / InGaAs/InAlAs MQW<br>InP<br>e-beam lithography<br>1 [µm]<br>Amorphous / monocrystalline /<br>monocrystalline</td><td>4(n)</td><td>SHG</td><td>10100 [nm]<br>-<br>50 [µm]<br>1200 [ps]<br>1 [kHz]</td><td>2.3 mW<sup>-2</sup></td><td>-</td><td>Resonators on<br>MQWs</td><td>-</td><td>[Wolf2015]</td></tr><tr><td>Gold nanoantenna<br>Fused silica<br>Focused ion beam lithography<br>0.04 [µm]<br>Monocrystalline</td><td>4(o)</td><td>SHG<br>imaging</td><td>1560 [nm]<br>1.69×10<sup>7</sup><br>[MW/m²]<br>0.7 [µm]<br>0.012 [ps]<br>80000 [kHz]</td><td>5×10<sup>-10</sup> W<sup>-1</sup></td><td>-</td><td>Multiresonant<br>antenna</td><td>-</td><td>[Celebrano2015]</td></tr><tr><td>Gold nanoparticle arrays<br>Fused silica<br>e-beam lithography<br>0.02 [µm]<br>Amorphous</td><td>4(p)</td><td>SHG</td><td>1150 [nm]<br>1.2×10<sup>5</sup> [MW/m²]<br>-<br>0.02 [ps]<br>80000 [kHz]</td><td>2.50×10<sup>-11</sup></td><td>-</td><td>Collective lattice<br>effect</td><td>-</td><td>[Czaplicki2018]</td></tr><tr><td>Gold nanoparticles<br>Embedded in gelatin matrix<br>e-beam lithography<br>0.15 [µm]<br>-</td><td>-</td><td>SHG<br>imaging</td><td>794 [nm]<br>-<br>-<br>0.018 [ps]<br>76000 [kHz]</td><td>-</td><td>1×10<sup>-23</sup> (esu)</td><td>-</td><td>-</td><td>[Butet2010b]</td></tr><tr><td>Gold split-ring resonators<br>Fused silica<br>e-beam lithography<br>0.025 [µm]<br>Amorphous</td><td>4(q)</td><td>SHG</td><td>1500 [nm]<br>0.0077 [MW/m²]<br>-<br>0.017 [ps]<br>81000 [kHz]</td><td>2.00×10<sup>-11</sup></td><td>-</td><td>-</td><td>-</td><td>[Klein2007]</td></tr></table>
<table><tr><td>TiN<br>Sapphire<br>Pulsed laser deposition/e-beam<br>lithography<br>0.05 [µm]<br>Monocrystalline</td><td>4(r)</td><td>SHG</td><td>1040 [nm]<br>1.2×10<sup>8</sup> [MW/m<sup>2</sup>]<br>2 [µm]<br>0.02 [ps]<br>80000 [kHz]</td><td>-</td><td>-</td><td>LSPR at the<br>fundamental<br>frequency, ENZ at<br>the SH frequency</td><td>-</td><td>[Wen2018]</td></tr><tr><td>WS<sub>2</sub> / Silicon<br>BK-7<br>PECVD, e-beam lithography<br>- / 0.16 [µm]<br>Monocrystalline</td><td>4(s)</td><td>SHG</td><td>832 [nm]<br>6 [µm]<br>0.008 [ps]<br>80000 [kHz]</td><td>-</td><td>-</td><td>Quasi-BIC</td><td>-</td><td>[Bernhardt2020]</td></tr><tr><td>ZnO nanodisks<br>Soda lime glass<br>Sputtering/focused ion beam<br>lithography<br>0.15 [µm]<br>Polycrystalline</td><td>4(t)</td><td>SHG</td><td>394 [nm]<br>1.68×10<sup>7</sup> [MW/m<sup>2</sup>]<br>8.5 [µm]<br>0.0205 [ps]<br>250 [kHz]</td><td>7.00×10<sup>-9</sup></td><td>9.6×10<sup>-13</sup> (d-<br>coefficient)</td><td>Magnetic dipole<br>resonance</td><td>-</td><td>[Semmlinger<br>2018]</td></tr></table>

*Units as illustrated unless otherwise indicated in the table
<table><caption>**Table 5B.** Third-order NLO properties of metamaterials 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</th><th colspan="3">NONLINEAR PROPERTIES</th></tr><tr><th>Material<br>Substrate<br>Fabrication<br>Thickness<br>Crystallinity</th><th>SEM<br>image</th><th>Method</th><th>Pump<br>wavelength<br>Peak<br>irradiance<br>Beam waist<br>Pulse width<br>Rep. rate</th><th>η (%)*<br>n<sub>2</sub><br>(m<sup>2</sup>/W)<br>α<sub>2</sub> (m/W)</th><th>χ<sup>(3)</sup><br>(m<sup>2</sup>/V<sup>2</sup>)*</th><th>Enhancement mechanism</th><th>Additional<br>parameters<br>and<br>Comments</th><th>Reference</th></tr></thead><tbody><tr><td>(100)-GaAs / AlGaO<br>GaAs<br>MBE/e-beam lithography<br>0.3 [μm]<br>Monocrystalline</td><td>5(a)</td><td>I-scan</td><td>830 [nm]<br>2.5×10<sup>7</sup><br>[MW/m<sup>2</sup>]<br>10 [μm]<br>0.06 [ps]<br>80000 [kHz]</td><td>-<br>-<br>-</td><td>-</td><td>Higher free-carrier generation rate</td><td>-</td><td>[Zubyuk2019]</td></tr><tr><td>Ag nanoantenna / SiO<sub>2</sub><br>Fused silica<br>e-beam lithography<br>0.016 / 0.065 [μm]<br>Amorphous</td><td>5(b)</td><td>Z-scan</td><td>410–560 [nm]<br>-<br>-<br>28 [ps]<br>0.05 [kHz]</td><td>-<br>1.2×10<sup>-12</sup><br>-1.5×10<sup>-5</sup></td><td>-</td><td>ENZ</td><td>-</td><td>[Suresh2021]</td></tr><tr><td>Al split hole resonator / -<br>40 nm of fused silica<br>e-beam lithography/FIB<br>0.2 [μm]<br>Amorphous</td><td>5(c)</td><td>THG</td><td>1560 [nm]<br>10 [MW/m<sup>2</sup>]<br>2.15 [μm]<br>0.2 [ps]<br>70000 [kHz]</td><td>1.0×10<sup>-5</sup><br>-<br>-</td><td>-<br>-<br>-</td><td>Plasmon resonance and lightning-rod effect</td><td>-</td><td>[Melentiev2013]</td></tr><tr><td>Au nanoantenna / ITO<br>Fused silica<br>e-beam lithography<br>0.05 [μm]<br>Amorphous</td><td>5(d)</td><td>Z-scan</td><td>1240 [nm]<br>1.5×10<sup>6</sup><br>[MW/m<sup>2</sup>]<br>-<br>0.14 [ps]<br>-</td><td>-<br>-3.7×10<sup>-13</sup><br>-2.4×10<sup>-7</sup></td><td>-</td><td>ENZ + optimized field coupling and enhancement by antennae</td><td>-</td><td>[Alam2018]</td></tr></tbody></table>
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<table><tr><td>Au nanoparticle<br>Fused silica<br>electrodeposition<br>0.15 [µm]<br>Amorphous</td><td>-</td><td>Z-scan</td><td>600 [nm]<br>8.0×10<sup>8</sup><br>[MW/m<sup>2</sup>]<br>1.5 [µm]<br>0.05 [ps]<br>-</td><td>-<br>-2.4×10<sup>-15</sup><br>-9.97×10<sup>-8</sup></td><td>(0.28 - 1.48i)×10<sup>-16</sup></td><td>ENZ + Au interband transition resonance</td><td>χ<sup>(3)</sup> for 600 nm at 60° and (0.18 - 1.48i)×10<sup>-17</sup> at 550 nm and 20° incidence</td><td>[Neira2015]</td></tr><tr><td>Ge nanodisk<br>Borosilicate glass<br>e-beam lithography<br>0.2 [µm]<br>Amorphous</td><td>5(e)</td><td>THG</td><td>1650 [nm]<br>0.8 [MW/m<sup>2</sup>]<br>-<br>0.18 [ps]<br>100 [kHz]</td><td>1.0×10<sup>-5</sup><br>-<br>-</td><td>2.8×10<sup>-16</sup></td><td>Higher-order (anapole) modes</td><td>-</td><td>[Grinblat2017b]</td></tr><tr><td>Si nanoblock<br>Quartz<br>e-beam lithography<br>0.5 [µm]<br>Monocrystalline</td><td>5(f)</td><td>THG</td><td>1587 [nm]<br>-<br>-<br>5 [ps]<br>-</td><td>1.4×10<sup>-8</sup><br>W<sup>-2</sup><br>-<br>-</td><td>-</td><td>Quasi-BIC in arrays with slight asymmetry in nanoparticles</td><td>-</td><td>[Liu2019]</td></tr><tr><td>Si nanoantenna<br>Quartz<br>LPCVD/e-beam lithography<br>0.12 [µm]<br>Polycrystalline</td><td>5(g)</td><td>THG</td><td>1350 [nm]<br>0.32 [MW/m<sup>2</sup>]<br>-<br>0.25 [ps]<br>1 [kHz]</td><td>1.2×10<sup>-6</sup><br>-<br>-</td><td>-</td><td>High-Q Fano resonance</td><td>-</td><td>[Yang2015]</td></tr><tr><td>Si nanoantenna<br>Fused silica<br>PECVD, e-beam lithography<br>0.538 [µm]<br>Amorphous</td><td>5(h)</td><td>THG</td><td>1425 [nm]<br>-<br>-<br>0.2 [ps]<br>80000 [kHz]</td><td>1.0×10<sup>-4</sup><br>-<br>-</td><td>-</td><td>Quasi-BIC in arrays with broken symmetry</td><td>-</td><td>[Koshelev2019]</td></tr><tr><td>Si nanodisk<br>SOI<br>e-beam lithography<br>0.26 [µm]<br>Monocrystalline</td><td>5(i)</td><td>THG</td><td>1260 [nm]<br>0.5 [MW/m<sup>2</sup>]<br>-<br>0.2 [ps]<br>80000 [kHz]</td><td>1.0×10<sup>-7</sup><br>-<br>-</td><td>-</td><td>Magnetic response</td><td>-</td><td>[Shcherbakov2014]</td></tr></table>

<footnote>*Units as illustrated unless otherwise indicated in the table</footnote>
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**3.5 Fiber waveguiding materials: data table and discussion**

Team: John Ballato (team leader), Peter Dragic

**3.5.1 Introduction**

**3.5.1.1 Background information**

Our understanding and use of NLO are very much intertwined with the history and development of fiber optics. This is because optical fibers are unique tools for studying and utilizing NLO phenomena due to their combination of low loss, long lengths, small core sizes and resulting small mode diameters. Indeed, the first observations of nonlinearities [Stolen1972, Ippen1972, Stolen1974, Hill1974] in fibers are contemporaneous with the first low loss fibers themselves [Keck1972a, Keck1972b]. Furthermore, phase matching of NLO processes in optical fibers can be achieved in more than one way [Stolen1974, Stolen1981]. First, in the case of a multimode fiber, different optical modes across a wide span of wavelengths, for example the signal and idler in FWM, may be found which possess the same propagation constant. Alternatively, the design of the fiber can be suitably tailored to control (e.g., "flatten") the dispersion curve of a given mode (such as the fundamental mode) to compensate for bulk chromatic dispersion. Such fibers can take the form of both solid multilayered conventional and microstructured waveguides.

From a practical perspective, a key driver of nonlinear fiber optics was the concurrent development of commercial communications networks. As noted by Stolen, a pioneer in nonlinear fiber optics, “*Fiber nonlinear optics has grown from a novel medium for the study of nonlinear optical effects, through a period where these effects appeared as system impairments, to the present day where optical nonlinearities are an integral part of high-capacity optical systems*” [Stolen2008]. This sentiment is well-reflected in the history of nonlinearities and communication systems [Smith1972, Chraplyvy1990, Zhang1994, Li2001, Hasegawa2017, Winzer2018, Essiambre2021].

Other applications of modern consequence where fiber nonlinear optics is critical include fiber sensors and high power / high energy fiber lasers. Fiber lasers, of considerable interest for defense, security, remote sensing, and manufacturing uses, generally demand reduced nonlinearities, particularly those that are considered parasitic to power-scaling and beam quality [Richardson2010, Zervas2014, Dawson2008], such as stimulated Brillouin, Raman, and thermal Rayleigh scattering (which manifests as transverse mode instabilities) [Dong2013, Smith2011, Jauregui2020, Zervas2019], as well as thermal lensing [Dong2016]. Fiber-based sensors, on the other hand, including distributed systems [Lu2019] such as those designed for structural health [Barrias2016] and down-hole and geological [Schenato2017] temperature and pressure monitoring, tend to favor higher nonlinearities for enhanced signal-to-noise ratio and therefore higher measurement sensitivity [Bao2021]. ‘Linear’ systems, namely those based on fiber Bragg grating technology and Rayleigh scattering, have found wide commercial success, as have those based on the Raman scattering nonlinearity. The latter systems work on the principle that the ratio of the anti-Stokes to Stokes Rayleigh scattered signals is a strong function of temperature, and therefore Raman scattering primarily is used for distributed temperature sensing. Both temperature and strain sensing are possible with Brillouin scattering, since the scattering frequency is a strong function of both the fiber thermal and mechanical environments. Unfortunately, Brillouin-based systems have not found wide commercial success due to their complexity and cost [Dragic2018].

**3.5.1.2 Considerations for fibers when performing NLO measurements**

All this said, care must be taken in assessing nonlinearities in optical fiber. This is because the measured nonlinearity is contributed to by both the materials comprising the fiber and the design of the fiber itself. Being a waveguide, the latter is important because fiber design influences the spatial (modal) and spectral (dispersion) properties of the propagating light. This, in turn, influences the relative contributions of the core and clad materials to the nonlinearities through their respective NLO
coefficients. Representative examples of such fiber design influences on, for example, the nonlinear refractive index, $n_2$, can be found in Refs. [Kato1995, Boskovic1996]. To this point, even the $n_2$ value of fused silica, the most canonical and well-studied fiber optic material, is not as precisely known as one would expect given the half-century since the first low loss fibers were reported (see Section 1.1). Another case where considerable care is required when applied to waveguides is Brillouin scattering. Namely, the Brillouin gain coefficient $g_{\text{Brillouin}}$ has both material and waveguide influences. Depending on the waveguide design, one may dominate the other, or they may play a more cooperative role, often making it difficult to distinguish between the two. Materially, $g_{\text{Brillouin}}$ depends on refractive index, transverse photoelasticity, isothermal compressibility, Brillouin linewidth, and acoustic velocity such that each of these values should be carefully measured and reported for true completeness. More commonly, publications report the Brillouin shift and linewidth / lineshape but these can strongly be impacted by the fiber itself. These include factors such as the number of acoustic modes present and their relative confinement, whether the waveguide is acoustically anti-guiding or guiding, fiber quality (loss) and length, and birefringence, to name just a few. All waveguide effects broaden the spectrum and reduce the observed Brillouin gain relative to the bulk material. In addition to these specific considerations for fibers, also the best practices described in Section 2 should be taken into account when performing NLO measurements.

### 3.5.1.3 Description of general table outline

Table 6 provides a representative list of NLO properties of fiber waveguiding materials taken from the literature since 2000. Our goal has been to compile and highlight advancements in nonlinear materials in fiber form since 2000 with the above-mentioned caveats and the best practices of Section 2 in mind. The publications included in Table 6 were selected based on their representing measurements on fibers across the spectrum of reasonably common materials and respective nonlinearities. The included works nominally report data obtained at room temperature. The Table is subdivided into “Fiber properties,” “Measurement details” and “Nonlinear properties.” Within each column the information is given in the order of the header description, and the NLO technique used is provided in the “Method” column. The Table compiles representative values for the following fiber material sub-categories: telecommunications-grade silica [Oguama2005, Oguama2005b, Deroh2020, Evert2012], silicate [Tuggle2017, Dragic2014, Lee2005, Cavillon2018], non-silica oxide (e.g., phosphate [Lee2007] and tellurite [Deroh2020]), and non-oxide (e.g., chalcogenides [Deroh2020, Florea2006, Tuniz2006, Fortin2008] and fluoride [Fortin2011, Lambin-Lezzi2013, Deroh2020]) optical fibers. For the more conventional silica and silica-based glass fibers, trends in Brillouin, Raman, and $n_2$ for a wide variety of dopants are provided in Ref. [Cavillon2018a] and references therein. Also noted are NLO properties of representative specialty optical fibers including polymer [Mizuno2010] and crystalline core optical fibers such as those made from sapphire [Harrington2014, Yin2008, Kim2008] and semiconductors [Ren2019, Shen2020]. Since second-order nonlinearities are generally precluded by symmetry in glass and cubic crystalline core fibers, the focus here is on third order $\chi^{(3)}$ wave-mixing as well as nonlinear scattering (e.g., Raman and Brillouin) phenomena. For completeness, second order nonlinearities in poled glass optical fiber also are noted [Canagasabey2009].

Two last points are noteworthy. First, the focus here is on the materials and material nonlinearities in conventional core-cladding configurations and not specifically on nonlinearity contributions from advanced waveguide design (e.g., microstructured or photonic-crystal fibers (PCFs), see, for example [Cordeiro2005, Dudley2009, Hu2019, Nizar2021]) as the range of those enabled influences is virtually endless. Indeed, one can make a very strong case that understanding / quantifying the relevant properties of a material should precede any attempts to design a waveguide from it. Second, nonlinearities can be materially enhanced or reduced (even negated), depending on the application. As such, both are generally treated here by way of discussing trends and ranges. The more fundamental materials science of optical nonlinearities in fibers has been recently reviewed [Ballato2018, Cavillon2018a].