Post-2000 nonlinear optical materials and measurements: data tables and best practices

Second-order wave mixing processes, mainly SHG, SFG and DFG [Shen1989, Shen2002, Sutherland2003, Wang2009, Liang2017, Prylepa2018], are often utilized in characterization of bulk materials, 0D–1D–2D materials, metamaterials, fiber waveguiding materials, on-chip waveguiding materials, and hybrid waveguiding systems. The developed techniques allow one to extract the material's χ⁽²⁾ and d_eff coefficients, and the conversion efficiency of the associated process. In some cases, the techniques also allow characterization of the relevant components of the χ⁽²⁾ tensor, or the conversion bandwidth of the process. Second-order wave mixing is also at the heart of OPO, OPG, and OPA [Shen2002, Sutherland2003]. We note that EFISH is in fact a third-order wave mixing process that in the presence of an external static electric field gives rise to SHG.

Besides the general best practices in section 2.1, it is important to verify successful phase matching of the NLO interaction in the material and to determine the parameters on which the considered phase matching approach relies (e.g., dispersion, poling parameters in some cases of QPM, etc). Furthermore, when reporting the conversion efficiency and conversion bandwidth of the material, the used parameter definitions should be clearly stated. Finally, when performing SHG experiments, it is important to verify that the SHG response is not masked by two-photon-excited fluorescence effects.

Second-order nonlinear imaging techniques, most notably SHG imaging [Gannaway1978, Kumar2013, Yin2014, Woodward2016], can be meaningfully applied to material categories exhibiting spatially varying second-order nonlinear parameters (0D–1D–2D materials, metamaterials, on-chip waveguiding materials, and hybrid waveguiding systems). Second-order nonlinear imaging allows one to characterize the magnitude of χ⁽²⁾ along with the conversion efficiency associated with the material and the performed experiments.

While the general best practices listed in section 2.1 also apply for second-order nonlinear imaging, we further stress the importance of carefully characterizing the NLO excitation parameters (the peak power/irradiance/energy, beam size, pulse width and polarization) inside the material. This is particularly important, because optical components such as microscope objective lenses may considerably modify the properties of incident pulses. Furthermore, when using SHG imaging, the occurrence of two-photon-excited fluorescence effects should be avoided.

Third-order parametric wave mixing refers to techniques such as FWM [Jain1979, Friberg1987, Agrawal2001], THG [Ward1969, Soon2005], and SPM/XPM [Alfano1986, Agrawal2001], and can be applied to all material categories considered in this work. Third-order parametric wave mixing allows measuring the magnitude of Re(χ⁽³⁾), n_2,eff (which is usually assumed to be real), and Re(γ_eff). Specifically in the case of SPM/XPM one can also extract the sign of these nonlinearities [Vermeulen2016a]. When using THG one most often quantifies a Re(χ⁽³⁾) value that is specifically associated with bound-electronic nonlinear transitions. Note also that FWM may be more complex than THG and SPM/XPM as often the imaginary part of the third-order nonlinearity cannot be neglected. To separate the real and imaginary parts of the nonlinearity in a FWM signal, special experimental modifications and careful analysis involving also 2PA and Raman contributions, which will be described in the following sections, are generally required [Burris1985].

In FWM and THG experiments one most often measures the power/energy of the new signal generated by these wave mixing interactions, whereas SPM/XPM experiments are focused on measuring the phase modulation effects (e.g., spectral broadening, SCG [Dudley2010], ...) that these processes induce. Besides the general best practices listed in section 2.1, FWM and THG also require that their phase matching conditions are addressed. When the conversion efficiency of FWM or THG is measured, attention needs to be paid as to how the conversion efficiency and bandwidth are defined. For SPM experiments the spectral shape and phase information, i.e., chirp, of the input pulses also need to be known to allow for a correct measurement of the NLO coefficients [Vermeulen2016a]. However, in the case of wideband SCG, the input chirp typically has a negligible impact on the output spectrum and hence does not necessarily have to be known to allow for a correct extraction of the NLO coefficients. Finally, it is important to keep in mind that, besides the material's bound-electronic (Kerr) nonlinearity, other effects can also contribute to its third-order parametric wave mixing response (e.g., free-carrier effects contributing to SPM- and XPM-induced spectral broadening [Vermeulen2018, Zhang2016]).

The material intrinsic NLO parameter that most often characterizes stimulated scattering for the Raman/Brillouin processes is the gain coefficient g_{Raman/Brillouin} (m W⁻¹). In the context of waveguides, the parameter can be expressed as a gain factor γ_{Raman/Brillouin} (m⁻¹ W⁻¹). Two classes of measurement methods are distinguishable: (a) direct methods—which rely on the amplification of a Stokes field, and (b) indirect methods—which rely on the determination of the spontaneous scattering cross-section and the dephasing time. Both methods can in principle be applied to all material categories considered in this work. Although most of the literature has focused on the Raman gain coefficient, the considerations often apply analogously to Brillouin scattering. Besides the general best practices listed in section 2.1, we have identified the following best practices specifically for Raman and Brillouin gain measurements:

• Direct methods are based on measuring the threshold for SRS/SBS [Ippen1972, Zverev1999], the threshold of laser action [McQuillan1970] and probing gain in an amplifier [Stappaerts1980, Niklès1997]. In the tables presented further on, these are indicated as 'SRS/SBS threshold', 'Raman/Brillouin laser threshold', and 'SRS/SBS pump-probe', respectively. Major contributors to the measurement uncertainty include the precision of beam sizes and shapes, and the specification of the relative polarization of pump and Stokes (in relation to crystal axes in the case of crystals) and the pulse shape. In cases where the Rayleigh range of the beams are comparable to or shorter than the material length, the variation in beam waist upon propagation through the medium should be factored-in [Boyd1969]. The gain is a function of the pump and Stokes linewidths, decreasing markedly as the widths approach that of the phonon resonance [Georges1991, Agrawal2001, Bonner2014] or, equivalently, as pulse durations approach the T₂ phonon relaxation time [Basiev1999b]. For methods involving co-propagating multi-longitudinal mode beams, correlations between the pump and Stokes irradiances also need to be considered [Stappaerts1980, Sabella2015]. Benchmarking against better known values of other materials [Basiev1999b], or the Kerr nonlinearity [Sabella2015], are valuable ways to increase confidence in the obtained values. For measurements at frequencies approaching the bandgap, a model is needed that includes a description of other nonlinearities occurring in the material (2PA, multiphoton absorption, FCA, ...). It should be noted that measurements of the stimulated scattering threshold often use a chosen exponential gain value G in the vicinity of G = 20, i.e., it is assumed that for typical spontaneous seeding, an irradiance growth of e^G with G ∼ 20 leads to a readily observable stimulated scattering signal [Grasyuk1998, Ippen1972]. However, there is no universally accepted value for G which may lead to some variability between reported values. In waveguides, two-color pump-probe techniques have been used to better separate the gain signal from background noise and other complications [Yang2020, Renninger2016].
• The gain coefficient may be determined indirectly by measuring the total cross-section for spontaneous scattering and its peak linewidth (T₂ dephasing time) [Basiev1999a], which are indicated in the tables, by the terms 'SpRS/SpBS cross-section' and 'SpRS, SpBS linewidth'. Absolute measurements of the cross section are rare due to the increased practical difficulties associated with precise photometric measurements; a problem often mitigated by benchmarking against better known materials. It has also been shown that IRS of a probe beam at the anti-Stokes frequency also gives access to absolute values without benchmarking [Schneebeli2013]. For Brillouin scattering, the cross-section may be determined from the photoelastic tensor [Ippen1972]. Determination of the linewidth may be readily achieved using a spectrometer provided that the instrument width is sufficiently small or is adequately deconvolved. The linewidth may also be determined from direct measurements of T₂ by using fast pump–probe optical pulses [Waldermann2008].

Third-order polarization rotation refers to techniques where the change of polarization of an intense pump beam, or of a probe beam, is monitored to determine the real (or imaginary) part of the nonlinear susceptibility χ⁽³⁾, which may also yield the nonlinear refractive index n_2,eff or a nonlinear absorption coefficient, e.g., α₂. Examples are ER measurements [Maker1964, Owyoung1972, Miguez2014], specifically for linearly isotropic materials such as liquids (see section 3.2 on solvents), or Optical Kerr Gating [Duguay1969, McMorrow1988]. If the polarization is changed solely by refractive index changes, this refers to the quadratic Kerr effect. These techniques are most often used for bulk materials, such as isotropic liquids or linearly isotropic or cubic symmetry solids, but can also be applied to other material categories such as fiber materials and on-chip waveguiding materials.

Besides the general best practices listed in section 2.1, third-order polarization rotation measurements also require special attention to precisely knowing the polarization of the source, or sources in the case of pump-probe experiments where knowledge of the relative polarization is also needed. The example of ER is illustrative. In that method for isotropic materials, due to symmetry, there are three unknown susceptibility elements. The key in an ER experiment is to create an elliptically polarized beam of well-known orientation. This can be done, e.g., by starting with a linearly polarized beam and introducing a quarter-wave plate, and then carefully measuring the transmission of the beam through a polarizer as a function of orientation, both before and after propagation through the NLO medium.

For pump-probe experiments, the optical Kerr gate is a good example. Starting with linearly polarized pump and probe beams with a relative polarization at 45 degrees, the pump induces a birefringence in the nonlinear sample that rotates the transmitted polarization. Monitoring the transmittance of the probe through a polarizer crossed to give zero transmittance without the pump gives a sensitive method for measuring the induced refractive index or absorption changes [Stegeman2012, Duguay1969, McMorrow1988]. Again, careful attention to the polarization is needed in these experiments.

In beam distortion measurements, one quantifies the spatial variation of the beam due to a NLO process by measuring the power that passes through a fixed reference (e.g., pin-hole) or using a spatially resolved detector (e.g., quad-cell). For beam absorption measurements, one seeks to measure the nonlinearity-dependent power reflected/transmitted (also referred to as 'nonlinear R/T') from the material, with the transmission dependent on both nonlinear absorption and scattering. In either case, it is possible to utilize single-beam or pump-probe style measurements. While more complex, the latter case provides more versatility to explore the temporal, polarization, and angular dependences of the nonlinearity as well as degenerate and non-degenerate spectral dependences.

Beam distortion and absorption refer to techniques such as Z-scan [Sheik-Bahae1989, Sheik-Bahae1990b, Balu2004, de Araújo2016], I-scan [Taheri1996], nonlinear reflection/transmission (loss) [Bloembergen1962, Clays1991, Sutherland2003, Cheng2018, Dong2018, DaSilva-Neto2020], BD [Ferdinandus2013, Ferdinandus2017], and power limiting [Siegman1962, Smirl1975, Soileau1983, Gu2006, Maas2008, Liaros2013, Riggs2000]. Note that the I-scan technique had previously been used before being named as such (see, for example, [Bechtel1976, Van Stryland1985]). As these techniques involve spatial distortion/pointing and/or loss of the beam, they are generally used for the extraction of nonlinear parameters in free space. As a result, bulk materials, 0D–1D–2D materials, and metamaterials are commonly evaluated using these techniques. However, nonlinear loss measurements can also be used in waveguides and fibers, and is common for characterizing, e.g., 2PA and saturable absorption [Sorin1984, Colin1996, Set2004, Jiang2018a].

In addition to the general best practices listed in section 2.1, beam distortion techniques are the simplest to analyze using clean Gaussian beam profiles as this is currently an assumption of many subsequent analysis techniques [Sheik-Bahae1990b, Gu2005] to extract the NLO coefficients; however, by using reference samples of known nonlinearity, such techniques can be calibrated for non-Gaussian beams [Bridges1995]. Furthermore, for thick samples and/or short pulses, it is useful to characterize the pulse chirp to correct for modifications in the temporal response of the nonlinearity due to varying group velocities. We also note that nonlinear beam distortion and loss, when used together, provide a complete methodology for characterizing lossy and/or lossless materials, enabling accurate evaluation of the magnitude, sign, and dispersion of the nonlinear refractive index n₂, n_2,eff, α₂, α_2,eff, thermal index (dn/dT and dk/dT), etc through the fitting of experimental quantities. With subsequent analysis [DelCoso2004, Christodoulides2010], the complex value and sign of, e.g., a pure χ⁽³⁾ can be determined as well. When only one of the techniques is used, the information is often restricted. Beam distortion alone is sometimes sufficient to discern the sign and magnitude of the complex nonlinear index and susceptibility in the case of lossless (or low-loss) materials. However, in the case of very lossy materials (Im() ∼ Re()), the terms of the complex susceptibility are inextricably coupled in measurements of both nonlinear losses and NLR. The use of beam distortion (Z-scan or other methods) alone results in an effective coefficient which combines contributions from both nonlinear absorption and refraction [Del Coso2004].

In general, it can be difficult to separate the different contributions using a single methodology (e.g., Z-scan requires both open and closed aperture experiments), although information on the sign of the respective nonlinearities can often be obtained. Beam absorption alone is typically sufficient to discern the sign and magnitude of nonlinear absorption but does not give information on the NLR, although in some cases the sign of the NLR can be predicted [Smith1999] (here negative absorption would correspond to either absorption saturation or gain depending on circumstances). Since both components cannot be individually interpreted, absorption alone also cannot provide complete information on the χ⁽³⁾ although it may be possible to discern the sign of the terms. Pump-probe experiments such as BD [Ferdinandus2013] again require simultaneous transmittance and deflection signals to determine both absorptive and refractive components [Ferdinandus2017]. These restrictions can be easily understood by comparing the number of unknowns (NLR and absorption coefficients) for a given material with the constraints (distortion and absorption measurements) imposed by the experimental information. In the case of an under-constrained problem such as when attempting to extract two coefficients with a single measurement, the sign can be argued based on Kramers–Kronig causality if one knows the underlying mechanism(s) and/or the spectral response of one of the nonlinear terms. Lastly, we reiterate that although in some cases a singular technique is sufficient to obtain information on the nonlinearity of a material, if one assumes a pure χ ⁽³⁾ process, it is rare for this assumption to be absolutely accurate (e.g., higher-order processes may provide non-negligible contributions). In this scenario, a single measurement will return only an effective nonlinear response, comprising multiple underlying physical mechanisms. Thus, the use of several analysis techniques is highly encouraged. This will provide more data to constrain analysis and elucidate the response of various contributing effects.

The purpose of NLO interferometry is usually to measure the real part of the nonlinear susceptibility χ^(2,3), but this technique can also be used to measure both the real and imaginary parts by suitable experimental modifications [Chang1965, Sutherland2003]. When using a wavelength of operation at which the NLO material is transparent, the most important part is the non-resonant Re(χ⁽ⁿ⁾). In contrast, if we are interested in the enhancement of χ⁽ⁿ⁾ by resonance effects, information on both Re(χ⁽ⁿ⁾) and Im(χ⁽ⁿ⁾) spectra is needed. Although dispersion relations similar to the Kramers–Kronig relations hold for NLO coefficients [Sheik-Bahae1990a, Bassani1991, Sheik-Bahae1991, Hutchings1992], it is practically impossible to strictly apply the Kramers–Kronig relations and there are restrictive conditions under which the Kramers–Kronig relations hold in time-resolved spectroscopy [Tokunaga1995]. Thus, direct measurement of the real and imaginary parts is often required. NLO interferometry is most often used for bulk materials, but can be applied to other material categories as well. Although interferometric methods are also modified as heterodyne detection to be employed in SFG [Ostroverkhov2005, Nihonyanagi2013, Wang2017] (otherwise only |χ⁽²⁾| is obtained), most of the targeted nonlinearities measured by NLO interferometry are third-order nonlinearities in a pump-probe setup. In this case, a NLO interferometer is often based on pump, probe (signal), and reference beams. There are multiple interferometric configurations possible. They can be categorized in two-arm types (Michelson, Mach–Zehnder, Sagnac, etc, or two-beam interference) and resonator types (Fabry-Perot, or multiple-beam interference), and hybrids of them, e.g., Laser Interferometer Gravitational-Wave Observatory combining Michelson and Fabry-Perot [Abbott2016]. There are also other classifications such as common path (Sagnac) and non-common path (Michelson) for the two arms, and also beam division methods are specified (space, time, direction, polarization, etc).

In addition to the general best practices outlined in section 2.1, it is important to evaluate the measurement sensitivity of the NLO interferometric method used and to account for a possible surface reflectance change due to changes in both the Re(χ⁽³⁾) and Im(χ⁽³⁾) parts. The bulk NLR can also change the propagation time of pulses. With pulsed lasers, it is convenient to use time division (i.e., in the same sample path, reference, pump, and probe pulses arrive at the sample in that order) and spectral interference [Tokunaga1995, Chen2007]. In addition to time division, common path interferometry such as division by direction [Misawa1995] or by polarization [van Dijk2007] (where the reference and probe share a common optical path) is advantageous for obtaining stable interference, but otherwise active control to stabilize the optical path length is often required [Cotter1989]. In order to cancel instabilities of the laser irradiance, balanced detection can also be employed by dividing the probe beam in two to be detected with an identical detector for subtractive detection [Waclawek2019]. Resonance cavity effects with Fabry-Perot interferometers are also useful for enhancing the NLO effects by increasing the light-matter interaction [Birnbaum2005, Fryett2018, Wang2021].

Third-order nonlinear imaging techniques, most notably THG imaging [Squier1999, Woodward2016, Karvonen2017], can be meaningfully applied to material categories exhibiting spatially varying third-order nonlinear parameters (0D–1D–2D materials, metamaterials, on-chip waveguiding materials, and hybrid waveguiding systems). Third-order nonlinear imaging allows one to characterize the magnitude of χ⁽³⁾ along with the conversion efficiency associated with the material and the performed experiments.

While the general best practices listed in section 2.1 also apply for third-order nonlinear imaging, we further stress the importance of carefully characterizing the NLO excitation parameters (the peak power/irradiance/energy, beam size, pulse width and polarization) inside the material. This is particularly important, because optical components such as microscope objective lenses may considerably modify the properties of incident pulses.

The bulk materials category is a highly diverse segment that presents NLO data on macroscopic 3D-material samples (ranging from thin films of ∼100 nm thickness to crystalline material samples on the scale of cm's), where material structuring (see metamaterials category in section 3.4) and quantum size effects (see 0D–1D–2D materials category in section 3.3) are not important. The properties of bulk materials are typically measured using free-space diffracting beams (usually one or two-beam experiments). Most experiments are performed with sample lengths less than the diffraction length (Rayleigh range) to allow for simple analysis.

Many bulk materials themselves are key for applications including frequency synthesis, stimulated scattering, ultrafast pulse characterization, and lasers. In addition, bulk materials generally serve as a starting point for the investigation of new materials before they are integrated into devices for applications across fields such as optoelectronics and fiber optics including uses in sensing, optical communications, and fiber lasers, as well as other types of active fibers and integrated photonics technology.

The listed materials, which include high-purity as well as impurity-doped materials, have been grouped into three sub-categories of insulators, semiconductors, and conductors. Within these groups, we focus our efforts on the development of inorganic materials only. Solvents are included in a separate section of this article (see section 3.2), and we also draw attention to section 3.8 on the characterization of material nonlinearities in the THz regime, many of which are bulk materials. These works have been separated to account for their unique considerations and in view of the rising area of THz NLO. However, works which utilize optical beams to generate THz waves remain in this section.

In keeping with this article's focus on post-2000 developments, the listed NLO materials emphasize those featured in research since 2000 but is not all-inclusive, giving priority to those publications that contain the most experimental information (see best practices in section 2).

3.1.1.2.1. Background for insulators/dielectrics

3.1.1.2.2. Background for semiconductors

3.1.1.2.3. Background for conductors

For solid materials, attention should be paid to thermal effects when using high-repetition-rate lasers or with materials showing linear absorption. Thermal effects may build up collectively over many pulses or arise within the pulse width depending upon the loss of the material. As a result, one must clearly identify the different roles of thermal and non-thermal nonlinearities [Bautista2021, Falconieri1999]. Also, the role of the pulse duration is very important in determining the origin of the nonlinearity being exploited [Christodoulides2010, De Araújo2016] and should be considered accordingly. Additionally, in many cases, bulk materials are explored as thin films grown on substrates. In this scenario, researchers should carefully characterize and remove the response of the substrate. Works should contain a detailed description of how the substrate contribution was removed as well as a report of the substrate-only response for comparison. In addition to these specific considerations for bulk materials, also the best practices described in section 2 should be taken into account when performing NLO measurements.

Tables 2 A and B show a representative list of, respectively, the second- and third-order NLO properties reported since 2000 for our range of bulk materials, across our three subcategories. The works included in tables 2A and B were selected based upon the best practices in section 2 and the considerations outlined above. Table 2B lists, besides third-order nonlinearities, also 3PA coefficients. Each entry in the tables contains information about the material and its properties, linear optical properties, NLO technique used, the excitation parameters, and the resulting NLO parameter(s).

The entries are arranged in alphabetical order, and within each material, entries are ordered by NLO technique (referred to as 'Method' in the tables). The works that report dispersive datasets or dependences of the NLO parameter on multiple parameters (e.g., thickness, doping, composition, etc) are denoted by '^{a b}' respectively, in the Tables. The papers included in the Tables nominally report data obtained at room temperature, unless denoted otherwise by '^c'. The works reporting different compositions or material variations are split into different entries where applicable. For papers with multiple thicknesses of the same material, the thickest material is reported. The data is sub-divided into 'Material properties', 'Measurement details', and 'Nonlinear properties'. Measurements using two beams have the properties of the pump and probe beam listed separately, where provided, while single beam measurements are contained within the pump column. Units are contained within the table unless noted in the header. Within each column the information is given in the order of the header description. If dispersive values for the NLO parameter were provided, the cited value represents the peak value for the material within the stated measurement range. For works that report tensor components, the component is shown as a leading number followed by a colon and the value (e.g., χ⁽²⁾ ₁₅ = 1 × 10⁻¹² (m V⁻¹) → 15: 1 × 10⁻¹² (m V⁻¹)). Indices are shown as listed in the original paper. Lastly, some papers relied on non-standard NLO measurement techniques (e.g., method listed as 'Other') or have notes associated with their measurement/analysis (e.g., identifying the definition used for the conversion efficiency η). These values and information are listed within the 'Comments' column with the value and unit identified.

3.1.2.1.1. Advancement and challenges for insulators/dielectrics

3.1.2.1.2. Advancement and challenges for semiconductors

3.1.2.1.3. Advancement and challenges for conductors

As lasers and various methods of measurement continue to advance, our ability to characterize materials evolves in parallel. Systems with several tunable excitation beams, supercontinuum probes, variable repetition rates, etc are expanding characterization capabilities and allowing more flexibility, which before 2000 was difficult to put into practice. We encourage future NLO works to embrace these new capabilities, exploring characterization and datasets which include more than just single wavelengths and pulse widths. Among various parameters, temporal and spectral information are perhaps the most useful information to be added for bulk materials, although this depends upon the application area of the material. In addition, the more thorough characterization will support an improved understanding of the NLO interactions at play, helping to delineate the roles of unwanted effects such as thermal nonlinearities and nonlinear scattering, and improving the reliability of results.

Another area of recommendation for the community is in the reporting of key experimental metrics. In many cases, papers are missing certain parameters of the experiment (peak/average power, beam waist, repetition rate, pulse width, etc, see tables) or do not report the method for calculation (flat top, Gaussian, full-width-half-max, etc). This can lead to difficulty when interpreting the results as well as invoke questions as to the experimental rigor. By meticulously reporting the metrics of experiments, we can improve the trust in the results, aid in reproducibility, and help to avoid errors.

Lastly, the new millennium has seen an explosion in new materials available for study. In many cases, it can be observed that works jump directly to the demonstration of example applications without performing rigorous characterization of the material's NLO response. While such studies are useful, we remind the community that carefully conducted experiments and reporting of inherent NLO coefficients (χ^(2,3), n₂, g_Raman, etc) is of key importance to the field as new materials and applications continue to be explored.

Among the ways to categorize nanomaterials or NSMs, at least two approaches can be followed [Ngo2014]. 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 NSMs, as used in this text, is shown in figure 3 (adapted from [Sajanlal2011]). 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 [Fox2001]. In this case, the optical properties of the material would become size-dependent. 0D, 1D and 2D materials presenting quantum confinement effects are called QDs, quantum wires, and QWs, respectively.

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There is a myriad of NLO and other applications for 0D–1D–2D NSMs, which can be revisited in [Liu2017, Zhang2017a, Xu2017, Autere2018a, Jasim2019, Eggleton2019, Liu2019a, Wang2020, Ahmed2021]. 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 [Zhang2021] when they take place in nanoscale dimensions. Although the practical applicability of 0D–1D–2D NSMs is at times limited by e.g. fabrication quality issues and/or losses, the NLO properties of 0D–1D–2D NSMs 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 [Stockman2011]. 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 λ/2, where λ 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 [Barbillon2017, Panoiu2018, Krasavin2019]. 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 NSMs at optical wavelengths, further details on the THz nonlinearities are provided in section 3.8.

The history of NSMs 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 [Heiligtag2013]. But the general nanoscience and nanotechnology scientific history and first publication date back to 1857 with Michael Faraday's work [Faraday1857]. Then came Feynman's lecture (1959), Taniguchi's 'nanotechnology' (1974) and Drexler (1981), as reviewed in [Heiligtag2013]. 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

3.3.1.2.2. Background for 1D materials

3.3.1.2.3. Background for 2D materials

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, a matrix, or a substrate. It is very important to characterize the NLO properties of the solvent, matrix 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 [Jiang2018b] 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 χ⁽²⁾/χ⁽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 [Autere2018a]. Finally, we point out that the nonlinearities can also be expressed in terms of conductivities σ⁽²⁾/σ⁽³⁾ rather than susceptibilities [Cheng2014]. 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.

Tables 4A and B 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 B 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. 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 '^{1 2 3 4 5 6 7}' 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 [Reyna2017], they are not covered here.