Harmonic Generation at the Nanoscale

Nonlinear photon conversion is a fundamental physical process that lies at the basis of many modern disciplines, from bioimaging and theranostics in nanomedicine to material characterization in material science and nanotechnology. It also holds great promises in laser physics with applications in information technology for optical signal processing and in the development of novel coherent light sources. The capability to efficiently generate harmonics at the nanoscale will have an enormous impact on all these fields, since it would allow one to realize much more compact devices and to interrogate matter in extremely confined volumes. Here, we present a perspective on the most recent advances in the generation of nonlinear optical processes at the nanoscale and their applications, proposing a palette of future perspectives that range from material characterization and the development of novel compact platforms for efficient photon conversion to bioimaging and sensing. T


INTRODUCTION
Although multiphoton processes had already been predicted in the 30s [Goeppert-Maier (1931)], experimentalists had to wait the introduction of the laser [Maiman (1960)] before they could provide the first proof of harmonic generation [Franken et al. (1961)], an observation that marked the outbreak of nonlinear optics. In a perturbative approach, nonlinear optical phenomena are classified according to the number of photons simultaneously involved [Boyd (1984)]. Lower order nonlinear phenomenainvolving only three or four photonsare the most widely investigated and exploited for non-invasive optical studies of matter, owing to their detectable yields granted by the large values of the corresponding second-and third-order nonlinear susceptibilities, ⃡ (2) and ⃡ (3) . Second harmonic generation (SHG) is a ⃡ (2) -related process, whereby two impinging photons of the same energy, interacting with matter, generate a third photon at twice the energy in a coherent fashion (Figure 1a). If the energy degeneracy is lifted, the energy of the generated photon may be either the algebraical sum or the algebraical difference of the impinging photons energies. These phenomena are known as sum and difference frequency generation (SFG and DFG), respectively (Figure 1b and c). Similarly, the excitation of matter with a single impinging photon can trigger the emission of two output photons with total energy equal to the one of the exciting photon ( Figure 1d). This process is known as spontaneous parametric down conversion (SPDC). Instead, thirdorder nonlinear processes involve the interaction of 4 photons. Therefore, one can attain a variety of mixing processes, commonly known as Four Wave Mixing (FWM). Examples of such processes are (i) two impinging photons producing two photons at different energies and (ii) three impinging photons generating a single outgoing photon (Figure 1f and g). The FWM process where three incoming photons having the same energy generate a fourth 3-fold more energetic photon is also known as Third Harmonic Generation (THG) (Figure 1e), the third-order equivalent of SHG. The above described processes are also defined as parametric, since the initial and final quantum-mechanical states of the system interacting with light are the same.
single crystals with reasonably large size combined with good optical and dielectric quality. This problem was alleviated in 1968, when Kurtz and Perry proposed the powder technique [Kurtz and Perry (1968)].
In this perspective, we will address the most recent advances in harmonic generation at the nanoscale and the most promising applications deriving from the engineering and enhancement of nonlinear optical effects in extremely confined volumes. Section 2 presents a survey of the physical mechanisms at play in harmonic generation at the nanoscale, while Section 3 shows how the generation of harmonics can be employed as a local probe to investigate the nanoscale properties of matter. Section 4 focuses on the most recent efforts in This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 engineering light-matter interaction at the nanoscale to artificially enhance the harmonic generation and on the perspective of engineering metasurfaces to obtain new ultra-flat nonlinear materials. Section 5 gives an overview on the application of harmonic-generating nanostructures as effective probes for bio-oriented applications (i.e. optical sensing and bio-imaging). Finally, Section 6 presents a perspective concerning the employment of parametric amplification to enhance the harmonic generation in extremely confined volumes by means of an external seed beam to foster its application in bio-oriented experiments.

HARMONIC GENERATION AT THE NANOSCALE
In nanoscale systems, the phase matching condition for the efficient generation of harmonics cannot be met since the size of the nano-object, L, is typically much smaller than the excitation field wavelength . One can instead exploit the local field enhancement of the incident fundamental and emerging harmonic intensities. In particular, resonances in nanoscale structures, for instance localized surface plasmon or Mie resonances, lead to electromagnetic field localization and amplitude enhancements. Therefore, at the nanoscale, symmetry also becomes a critical criterion to identify non-zero macroscopic susceptibility tensors to enhance even-order nonlinear processes such as SHG. For instance, because of their central symmetry, liquids do not often sustain sizeable SHG as in common second-order nonlinear media. Let alone symmetry, the magnitude of the elements of the third-rank ⃡ (2) tensor depends on the nature of the material, be it organic, inorganic or metallic. Efficient materials for frequency doubling are usually of dielectric nature with large band gaps to avoid re-absorption of the SHG wave. The most common materials employed for SHG are potassium dihydrogen phosphate (KDP) or trihydrogen phosphate (KTP) [Eckardt et al. (1990)] because of their relatively large bandgap (350 nm -4500 nm) and phase-matching range (984 nm -3400 nm) covering the whole visible range up to the near infrared [Yariv (1995)]. KTP is also among the first materials that has been synthetized into nanoparticles for bioimaging , Mayer et al. (2013)]. In fact, ensemble measurements of SHG efficiency performed by HRS on colloidal suspensions of non-centrosymmetric nanocrystals generally provide values consistent with the bulk ones (at least for larger particles) [Joulaud et al. (2013)]. Therefore, materials known for their good nonlinear properties and with This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.  [Kasel et al. (2019)].
Although in bulk centrosymmetric media, such as metals, ⃡ (2) is expected to vanish, the presence of interfaces and nanoscale defects is often associated with sizeable second-order nonlinear effects. Indeed, surfaces, interfaces as well as local defects are spatial regions where the central symmetry is removed [Bloembergen and Shen (1966), Chen et al. (1981), Sipe et al. (1980)]. Low-dimensional systems, moreover, may sustain strong electromagnetic field gradients, which introduce strong contributions from higher order multipoles, breaking inversion symmetry and thus allowing for the recovery of non-zero susceptibilities and, hence, of sizeable SHG responses. While this concept was first proposed for extended systems, it holds important implications and effects in nanoscale systems, where the ratio between surface area and bulk volume dramatically increases. Consequently, despite the bulk material central symmetry and strong absorption in the UV-visible-IR region of the electromagnetic spectrum, metallic (e.g., gold and silver) nanostructures have been proposed for enhancing second-order nonlinear processes at the nanoscale.
Unexpectedly, high SHG yields and large second-order susceptibilities have been recently reported [Zhang et al. (2011), Celebrano et al. (2015], thanks to the large resonant enhancement of the electromagnetic fields provided by localized surface plasmon resonances (LSPR), which compensate for the abovementioned disadvantages. In nanostructures, the resonant enhancement of SHG can be achieved both This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 at the fundamental or the harmonic frequency. This possibility arises from the dramatic dependence of the LSPR wavelength on the material, size, shape, and environment of the nanostructures. This line of research, which will be described in detail in Section 4.1, is still receiving considerable attention and will likely be pursued further as the possibility of realizing finely designed nanostructures is enabled by the recent improvements of standard nanofabrication methods, such as focused ion beam (FIB) and e-beam lithography (EBL). In particular, the resolution of FIB is recently benefiting from the employment of helium ions for the milling beam, which allows fabricating features below the nanometer [Boden et al. (2011)].
Such a resolution is, at the moment, out of reach of EBL, which is however less expensive and time consuming for the realization of extended structures.
Yet, matching the fundamental and harmonic frequencies with LSPR frequencies is not the only criterion to design efficient nanostructures. It is also important to achieve mode matching, namely the best overlap of the charge distributions of the fundamental and harmonic modes associated with the corresponding LSPRs. Mode matching in plasmonic nanostructures replaces the older concept of the Miller's rule operating at the macroscopic scale for many different materials [O'Brien et al. (2015), Butet et al. (2016), Celebrano et al. (2015)]. Miller's rule states that the nonlinear susceptibility tensor ⃡ (2) scales with the ratio of the linear susceptibility tensors ⃡ (1) at the fundamental and harmonic frequencies. Mode matching in plasmonic nanostructures can only be obtained with a careful design of the nanostructures and further improvement can be obtained along this line with more advanced structures, possibly with the aid of computer-assisted methods [Malkiel et al. (2018)]. In this frame, the exploitation of Fano profiles has for instance been proposed, where the combination of a broadband and a narrow resonance may lead to favorable sharp peak-valley spectral features [Bachelier et al. (2008)]. Besides mode matching, polarization matching must also be fulfilled before any resonance enhancement can occur. The role played by this rule has been clearly underlined in a study of the SHG resonance from supported gold nanorods [Hubert et al. (2007)].
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HARMONIC GENERATION AS A PROBE FOR NANOSCALE PROPERTIES
The nanoparticle size is intimately hooked to the mode description of the SHG response, which therefore may become a fundamental tool for characterizing nanoscale colloids. In particular, as size grows, the electric dipole approximation is no longer valid and higher order electromagnetic modes appear with specific charge distributions and therefore specific angular and polarization patterns in the far field for the SHG response. Therefore, switching from one mode to another may lead to dramatic changes in the SHG intensity and its angular or polarization distribution. A specific example may be illustrated with SHS from spherical gold nanoparticles dispersed as a liquid suspension. For the same couple of fundamental and harmonic frequencies, for instance for 800 nm and 400 nm, respectively, the dipolar mode will dominate for small sizes of about 20 nm with the strongest SHS intensity collected for both the fundamental and harmonic light polarized perpendicular to the plane of scattering, whereas, for sizes larger than about 100 nm, the strongest SHS intensity will be obtained for second harmonic light polarized perpendicular to the Since SHG in metallic nanoparticles is due to the surface, the shape of the nanostructures plays a crucial role. For unsupported nanostructures, like liquid suspensions of nanoparticles, this feature is essential in predicting the properties of SHG light. Nanostructures with a centrosymmetric shape like spheres, rods or cubes will therefore exhibit a strongly retarded response due to the cancellation of the local electric dipole contribution to the SHG response. Retardation is the property accounting for the time delay between all contributions to the total SHG response. It is often described through a multipolar expansion where the electric dipole constitutes the first order dominating the next orders and in particular the electric quadrupole one. For nanoparticles with a centrosymmetric shape, the electric dipole contribution vanishes, and the next orders of the multipolar expansion become clearly visible (see Figure 2). It is however expected that, for non-centrosymmetric nanostructures like prisms, retardation would play a much less important role.

Therefore, competition between symmetry and retardation translates into a competition between size and
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 shape, leading to characteristic SHG light properties and corresponding angular and polarization patterns, as seen in Figure 2 [Duboisset et al. (2019)]. In perspective, this simple method may become a routine analytical method for the characterization of nanoparticles samples.
Recently, more advanced nanostructures have been proposed, thanks to the versatility of lithographic techniques. Arrays of such nanostructures, also termed metasurfaces [Krasnok et al. (2018)] (see Section 4.2), have hence been designed with a great variety in their SHG response. Interestingly, the substrate, which breaks the initial symmetry of the structures, plays a fundamental role in determining the nonlinear optical properties of the structure and provides it with another degree of complexity that may be used for refined engineering. Likewise, with structures dispersed in a homogeneous matrix, complex geometries have been synthesized from multi-material core-shell to metallic urchins-like nanostructures. researchers recently demonstrated that it is possible to dramatically increase its THG efficiency by electric doping [Soavi et al. (2018)]. On the other hand, the lack of SHG in SLG can be exploited as a powerful tool, since appearance of SHG directly reveals its assembly in multilayers [Dean et al. (2010)]. Despite the negligible SHG expected in SLG, it has been theoretically predicted that light confinement in graphene nanoflakes could significantly enhance SHG in this nanoscale systems [Yu et al. (2016)], making graphene This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093

Enhanced harmonic generation at the nanoscale
In the past decades, researchers dedicated intense efforts in devising nanostructures to further boost the weak nonlinear optical processes at the nanoscale. One of the most successful approaches to compensate for the lack of phase matching conditions in sub-wavelength regions [Kauranen et al. (2012) ]. This strategy, along with a broken-symmetry geometry, allows obtaining a (2) that is as high as 150 pm/V [Celebrano et al. (2015)], which is comparable with the most efficient nonlinear nanocrystals [Rogov et al. (2015b)]. It is here worth stressing that the reported This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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(2) in plasmonic nanostructures is a lower-bond value, since in these systems only the surface plays a role in SHG.
Given the remarkably high efficiency attained at the nanoscale, individual plasmonic nanoantennas were recently exploited also for ultrafast pulse characterization [Extermann et al. (2008), Accanto et al. (2014), Gennaro et al. (2018)]. In perspective, the possibility of finely tuning the geometry of plasmonic nanoantennas using modern lithographic techniques would allow one to further enhance nonlinear processes at the nanoscale through cascaded nonlinear processes [Celebrano et al. (2019)] and envision their employment also as nanoscale parametric amplifiers [Zhang et al. (2016)].
However, plasmonic materials (i.e. metals) display large losses at optical frequencies, which are associated with relatively low damage thresholds that can mitigate the benefits of the field enhancements in nanoantenna realizations. In this respect, intense efforts have been recently dedicated to devise nanoantennas based on dielectric materials, because of the extremely low losses at optical frequencies to THG in 2019 [Carletti et al. (2019a)], this approach was only very recently confirmed experimentally [Koshelev et al. (2019)].
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Another effective route to enhance the harmonic generation at the nanoscale consists in combining the field enhancements in plasmonic antennas with the large bulk nonlinearities of dielectric materials. This strategy, which allows to circumvent both the limitations imposed by the losses in metals and the low field enhancements in dielectrics, is based on hybrid plasmonic-dielectric nanoscale systems. Two research groups simultaneously applied this concept, where a plasmonic gap nanoantenna is employed to confine the impinging pump fields to the spatial location of individual nonlinear nanocrystals, to enhance their THG This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 As mentioned in Section 3, an intriguing solution for harmonic generation at the nanoscale resides in 2D materials, thanks to their unique band structure [Yoshikawa et al. (2017), Cox et al. (2017)]. In particular, while THG conversion efficiency has been already characterized in single layer graphene [Kumar et al. (2013)], it has been recently shown that THG can be sizably enhanced by doping the system through external gating [Soavi et al. (2018)], opening up further perspectives in the possibility of electrically tailoring the nonlinear response at the nanoscale.
Thus far, we limited our discussion to individual nanoantennas and materials whose dimensions are confined below the diffraction limit of light, where the nonlinear emission often emerges from subwavelength hot spots. Subwavelength nonlinear emitters are indeed of utmost importance to attain the highest level of integration for application as active optical logic elements in all-optical communication networks. The record conversion efficiencies attained with dielectric and hybrid nanostructures, which exceed 10 -5 for both SHG [Gili et al. (2016)] and THG [Shibanuma et al. (2017)], allow picturing these platforms as ideal candidates for perspective applications in parametric conversion and photon-by-photon logic operation at the nanoscale. Here, the low photon yields compared to typical nanoscale single photon sources, such as single molecules and quantum dots, could be easily mitigated by the low damage thresholds, the higher photostability and the room-temperature operation. In this framework, it was recently demonstrated for the first time a source of entangled photon pairs based on a nanoscale AlGaAs platform, which works at room temperature and can be integrated in silicon-compatible devices [Marino et al. (2019a)].

Enhanced harmonics generation and manipulation with nonlinear metasurfaces
Despite the large nonlinearities in terms of susceptibilities reported for nanoscale antennas, extreme integration comes at the expenses of the absolute conversion efficiency, which is hindered by the small volume of the active material and the impossibility of applying phase matching conditions. In this framework, the ability of realizing nonlinear metasurfaces, where the nonlinear emission of individual efficient nanoantennas (i.e. nonlinear meta-atoms) is combined with collective interactions, allows This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 achieving an improved efficiency along with new optical functionalities [Yu et al. (2015), Krasnok et al. (2018)]. Optical metasurfaces are engineered 2D ensembles of ordered [Lin et al. (2014), Zheng et al. (2015)] or disordered [Jang et al. (2018)] sub-diffraction objects (i.e. meta-atoms) with tailored optical properties and constitute the basis for future ultraflat photonics. By carefully tailoring the interaction between meta-atoms, it has been shown that metasurfaces can provide superior and unprecedented optical functionalities [Khorasaninejad et al. (2016)], such as negative refraction [Nanfang et al. (2011)], compared to bulk optical elements, and in a much more compact realization. In this framework, metasurfaces of nonlinear nanoantennas have been shown to be an effective means to enhance and steer harmonic generation. A comprehensive overview on the perspective applications of nonlinear semiconductor metasurfaces was recently published by M. Shcherbakov and co-workers [Shcherbakov et al. (2019)]. Here we will focus on the enhancement of harmonic generation provided by dielectric and hybrid metasurfaces.
To date, a plethora of nanoscale platforms that act as extremely efficient nonlinear meta-atoms are at hand.
Although their size is obviously larger than the individual nonlinear oscillating dipoles (e.g., atoms, molecules, unitary cells) in nonlinear materials, these nanostructures have the potential to be engineered and arranged at will in planar geometries to further enhance the nonlinear optical effects (see Figure 4a).
An efficient approach to engineer densely packed arrays of plasmonic nanoantennas consists in periodically arranging the nonlinear emitters to exploit the coherent nature of the nonlinear signal [Gentile et al. (2011), Boardman and Zayatz (2014)]. The construct can be thought as the 2D equivalent of periodical poling, where instead a periodic inversion of the nanoantennas allows obtaining enhanced nonlinear emission in specific angular directions as a result of constructive interference of the nonlinear light (see Figure 4b) [Segal et al. (2015)]. The possibility of engineering the lattice modes of a periodic array of nanostructures to direct and beam the nonlinearly emitted light in preferential directions has been also recently applied by G. Marino et al. [Marino et al. (2019b)] to a purely dielectric metasurface. This result, combined with that published by the same group [Marino et al. (2019a)], allows envisioning the possibility of beaming entangled photons into very small angles. This is extremely attracting from a purely technological point of This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093 view, since it would allow one to efficiently couple entangled photons into low numerical aperture optical fibers.
A more intriguing approach suggested by Czaplicki and colleagues [Czaplicki et al. (2016) ]. Hybrid metasurfaces obtained by coupling plasmonic nanohole arrays to 2D materials are also extremely promising to attain efficient focusing of the nonlinear light [Chen et al. (2018)]. An interesting perspective of nonlinear metalenses is their potential exploitation in medical endoscopy [Lee et al. (2020)], to convert in situ a high-power infrared light into a focused beam in the visible range to perform local diagnosis or to be employed as optical tweezers in a non-invasive way.
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To circumvent the limitations imposed by losses and low damage threshold in plasmonic antennas, another intriguing possibility resides in nanostructuring the surface of the crystal itself. Although the refractive index of the nanostructures would not show any discontinuity with respect to that of the substrate, which would be desirable to better confine the fields in the nanostructures, it has been theoretically demonstrated that this approach could lead to significant enhancement in the SHG from thin layers of LiNbO 3 (see , as we will describe in the next section.

HARMONIC GENERATION AS A TOOL FOR BIO-ORIENTED APPLICATIONS
Since the first nonlinear optical phenomena were reported in plasmonic nanostructures [Lippitz et al. (2005)], plasmon-enhanced nonlinear sensing has always been sought as a natural application for such phenomena. In fact, the nonlinear dependence on the illumination intensity and the inherent rejection of the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. Thanks to the coherence of the processes, nonlinear signaland hence sensitivitycan be further enhanced via an additional external seed beam (see Section 6). Therefore, although these first studies are still far from clinical applications, the availability of novel cheap ultrafast lasers sources working in the transparency window of tissues, the sensitivities attained thus far and the recent advances in nonlinear signal enhancement set the ground for a swift change in paradigm for nonlinear plasmonic sensing platforms in the next years.

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In bio-imaging and bio-medical applications, the adoption of nanoparticles as labels for is often motivated by one or more of their properties: i) better photostability in comparison to dye molecules and fluorescent proteins, ii) access to multimodal read-out (e.g., optical and magnetic), and iii) possibility to take advantage of additional physical properties (e.g., photothermal effect, selective accumulation in tumors), in some cases iv) the possibility to apply multiplexed detection schemes. To date, most of the proposed imaging approaches are based on luminescent nanoparticles, such as semiconductor quantum dots, gold colloids, and up-conversion nanoparticles. The use of harmonic generating nanostructures as markers is still at its early stages of development, the main practical obstacle being the requirement of a pulsed femtosecond laser as light source. However, the increasing availability of rugged and cost-effective ultrafast sources working outside the classical Ti:Sapphire spectral region is progressively stimulating efforts in this direction. The main benefits of the harmonic approach for imaging can be summarized as: i) The inherent absence of bleaching and blinking, as only virtual states are involved in the harmonic process provided that the particles are excited out of their absorption bands (see Fig. 1).
ii) Wide excitation/emission flexibility throughout the transparency range of the nanoparticle material.
iii) Simultaneous emission at multiple harmonic orders with spectrally narrow bands.
iv) Coherent and polarization-sensitive response.
The combination of properties i) to ii) provides a unique asset for applications requiring long-term monitoring of nanoparticle-labelled structures in optically congested samples, such as biological tissues. In particular, the possibility to tune the excitation towards longer wavelengths assumes a great significance in the light of the extended transparency windows in the infrared recently proposed: NIR-II (1100-1350 nm) and NIR-III (1600-1870 nm). These spectral regions correspond to local minima of water absorption at wavelengths beyond the classical transparency window (NIR-I, 650-950 nm). Although water absorption in these regions is comparatively higher than in NIR-I, the scattering efficiency is severely reduced and the overall light penetration in tissue is favored. For a comparison, Sordillo et al. have reported that the total attenuation length (a parameter closely related to the penetration of ballistic photons) is three-fold longer This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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for NIR-III than in NIR-I for a healthy prostate tissue (600 µm vs 200 µm) although for other samples the difference is less significant (e.g., breast cancer) [Sordillo et al. (2014)]. Notably, it has been demonstrated that working in these extended NIR windows allows performing multiphoton imaging in brain through the intact skull ]. It is worth reminding that lower scattering along the excitation path which preserves the ballistic photon component is essential for the success of a nonlinear interaction where intensity, which depends both on the spatial and temporal structure of the pulse at the focal spot, plays a crucial role. On the other hand, in the detection path, scattering is less problematic as for scanning imaging techniques, the collection angle is large and even multiple scattered signal photons can be collected.
In most of the nonlinear imaging applications proposed to date, the detection of noble metal nanoparticles relies on their photoluminescence properties (especially for Au particles). Operation with harmonic emission [Lippitz et al. (2005), Tai et al. (2007)] is quite uncommon for plasmonic particles and substantially less exploited than in the case of bio-imaging by dielectric particles, where the harmonic approach has been pursued for silicon [Jung et al. (2009)] and for various metal oxides. A peculiar characteristic of non-centrosymmetric metal oxides, also known as Harmonic Nanoparticles (HNPs), is that the harmonic emission (and a fortiori the lowest even order, SHG) originates from the bulk rather than the surface, at least for diameters beyond 20 nm [Kim et al. (2013)]. It follows that the nonlinear emission intensity scales as the volume squared, rendering rather large HNPs in the 50-150 nm range quite efficient as nonlinear biomarkers. HNPs have been applied in vitro, [Nakayama et al. (2007)] ex vivo, and in vivo [Pantazis et al. (2010)] over the years. An example of multiphoton imaging of a fluorescently stained zebrafish loaded with BaTiO 3 nanoparticles injected during its development is presented in Figure 5.
A very recent development for imaging is the simultaneous collection of second and third harmonic upon NIR-II excitation at 1300 nm. This multi-order detection scheme is effective in increasing imaging selectivity by suppressing the hindrance from auto-fluorescence and from the harmonic background emitted by endogenous structures such as collagen (SHG) and lipids (THG) (see Fig. 6a) [Dubreil et al. (2017), ]. The extended imaging penetration enabled by NIR-II and NIR-III combined with the This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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sub-cellular spatial resolution proper to optical approaches (and not accessible to MRI, CT and other whole body imaging techniques) constitutes an undeniable asset for tracking individual cells, with straightforward applications for the assessment of regenerative therapies based on the engraftment of stem cells [Dubreil et al. (2017)] and for studies on immune-response reactions [Ramos-Gomes et al. (2019)].
Harmonic imaging presents other distinctive features when compared to fluorescence imaging, for instance because of the phase-coherent nature of the collected signals. In fact, the interferences taking place among the emissions stemming from multiple harmonic objects in a sub-diffraction limited spot should be properly considered to compute the actual spatial resolution and the optical transfer function of the set-up (see Fig.   6b), which will differ from the classical (incoherent) case. The Psaltis and Depeursinge groups independently demonstrated that, by measuring the interference with respect to a reference secondharmonic field, the axial localization of a BaTiO 3 HNP embedded in a biological medium could be determined in a scan-less fashion with applications for object tracking in three dimensions [Hsieh et al. (2009), Shaffer et al. (2010]. The Ameloot group has addressed the field of fluorescence correlation spectroscopy/microscopy and developed a theoretical model accounting for the coherent intensity fluctuations of second harmonic objects diffusing in the focal volume of a laser [Slenders et al. (2015)].
They successfully applied this model to study the mobility of LiNbO 3 50 nm nanoparticles in human cells.
Notably, they were able to quantify the diffusion coefficient from a single spectroscopy measurement without the need of external reference calibration [Slenders et al. (2018)].
In the case of dielectric nanoparticles, as the harmonic intensity depends on the elements of the nonlinear susceptibility tensor stimulated by the excitation field, the polarization analysis of the emission provides access to the orientation of the crystallographic axes of individual nanoparticle. This approach has been exploited over the years by several groups to distinguish small aggregates from mono-crystalline objects [Brasselet et al. (2004) The works reviewed in this section testify the liveliness of the field of harmonic-nanostructure imaging.
Despite the exciting potential of the approach, these remain to-date proof-of-concept demonstrations. One must be aware that harmonic imaging markers cannot compete with luminescent nanoparticles in terms of brightness and widespread accessibility of the necessary experimental requirements. A key advantage of harmonophores compared to fluorophores relies on the ability of researchers in physics to identify the specific niches where this approach can be competitive and highlight to the life-science community the advantages of harmonic emission in terms of better imaging penetration and higher selectivity against This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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background, for instance exploiting their flexible spectral properties. Besides, letting aside the vast literature on metal NPs, several works have highlighted the low in vitro cytotoxicity of various metal-oxide particles used as harmonic probes in selected biological systems [Hsieh et al. (2009, Li et al. (2016), Dubreil et al. (2017), Sugiyama et al. (2017)] and also identified some exceptions [Staedler et al. (2012)]. In general, because of the lack of heavy metals in the composition of many metal-oxide NPs, they might be preferred to quantum dots containing Hg or Cd. However, we emphasize that it is not possible to provide a general assessment in this respect as cytotoxicity anda fortioribiocompatibility of nanoparticles critically depends on their size, shape, surface modification, concentration, uptake mechanisms involved, and cell lines tested [Albanese et al. (2012)]. The volume squared dependence of signal intensity (at least for bulk harmonic generation) implies that the particles used for imaging are rather large (30-100 nm) compared to other approaches (quantum dots 10 nm, up-conversion nanoparticles 20 nm), therefore these markers should be more efficiently employed for cell tracking applications over extended time periods, taking advantage of their inherent photostability, rather than for intracellular studies where smaller size is often an essential requirement. Most of the metal-oxide synthesis protocols, although very cost-effective, result in rather large size-and morphology-dispersed samples. In terms of control of these parameters, nonlinear metal nanoparticles present clear advantages.
In the near future, we will likely witness a flourishing of imaging protocols based on phase-coherent detection schemes. In particular, the coherent nature of the process can be exploited to amplify the signal by means of either an external local oscillator or a seed beam in a homodyne-like or stimulated configuration. Apart from coherent-amplification procedures, which will be detailed in Section 6, holography-type approaches by means of an external phase-reference may allow obtaining spatial information along the propagation axis in a scan-less fashion, thus increasing the volume imaging speed.
Another and yet unexplored possibility could rely on the full recovery of the spectral phase of large bandwidth pulses [Extermann et al. (2008)] interacting with biological samples, exploiting the particle harmonic emission as a proxy. This method could in principle provide access to spatially resolved This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

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information about the optical properties of the sample (complex refractive index and dispersion properties).
More importantly, the analysis of the coherent image formation can be potentially exploited to infer information on sub-diffraction features in nanostructures as reported in Figure 2 [Duboisset et al. (2019)].
This approach might be extended to investigate structural properties of endogenous harmonophores, as demonstrated by a recent study on the organization of microtubules in the cell cytoskeleton, which can be used to study the effects of microtubule-targeting drugs and to detect conformational changes in tubulin during neuronal maturation connected to the onset of degenerative diseases [Van Steenbergen et al. (2019)].
In situ nonlinear photo-triggering by UV and visible emission of harmonic particles upon NIR-I, II, III excitation could also be an enabling evolution of the field, relying on the high nonlinear conversion efficiency, in particular for particle sizes of the order of 100 nm. Very recently, two works have provided proof-of-principle demonstrations of the therapeutic integration of this scheme. In a first embodiment, the SHG emission from BaTiO 3 HNPs upon irradiation at 1040 nm resulted in the excitation of nearby photosensitizers (rose Bengal) used in photodynamic therapy [Sun et al. (2019)]. In a second setting, the SHG emitted by Bismuth Ferrite nanoparticles was used as a stimulus to trigger the photo-uncaging of a molecular cargo (tryptophan) [Vuilleumier et al. (2019)] (see Fig. 6c). Within this nonlinear phototriggering approach, the tunability of harmonic generation in structures smaller than the coherence length can provide an efficient decoupling between the therapeutic photo-interaction (photosensitizer activation, drug release) and the diagnostic (imaging) procedure, which can be safely performed at longer wavelengths.
Following a similar line of thinking, visible photons generated by THG upon NIR-II excitation can be absorbed in a one-photon process by a close-by molecule with much smaller n-photon (n≥3) cross section.
Molecules of interest would include many fluorescent proteins (GFP, RFP, etc…) and also channelrhodopsin or other light-activated proteins used in optogenetics [Ao et al. (2019)].
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093

APPLICATIONS
As discussed in Section 5, one of the most promising perspectives in label-assisted bioimaging using HNPs resides in the possibility to extend their operational wavelength to the extended NIR wavelength range. In particular, the fact that nonlinear optical processes do not involve resonant optical transitions in matter makes harmonic generation largely independent on the wavelength range. This constitutes, along with the absence of photobleaching, a major advantage with respect to common fluorophores, since these tags can be tracked within the transparency window of tissues. Still, to attain sizeable SHG yields, their sizes need to be larger than those of fluorescent labels, therefore they are expected to have limited interaction modalities and to remain confined to specific cell compartments once uptaken. The possibility of enhancing the SHG signal becomes therefore crucial, since it would allow one to considerably decrease the size of the HNPs and eventually enable even label-free NIR imaging of intracellular dynamics and in general of biological samples far from resonant optical transitions. An intriguing possibility is to exploit the coherent nature of nonlinear parametric processes to amplify them via heterodyne techniques [DeLange et al. (1968)] or by stimulated effects.
Homodyne-like interference has been effectively exploited in recent years to amplify the elastic scattering of nano-objects. Within this approach, where the scattering of the object is enhanced by the constructive interference with the light that is partially reflected by the substrate, which is exploited as a local oscillator (LO), it was possible to detect single quantum dots in their dark state [Kukura et al. (2010)] and even individual non-fluorescent molecules at room temperature [Celebrano et al. (2011)]. In the nonlinear regime, this paradigm has been also applied to amplify SHG, exploiting a frequency-doubled external nonlinear LO (NLO) in an optical homodyne configuration [Yazdanfar et al. (2004), Le et al. (2006)] (see Figure 7) and, very recently, extended to THG [Stock et al. (2020)]. Interferometric amplification is an extremely versatile approach that does not rely on the optically active resonances in the sample or exogenous labels and could potentially benefit from further enhancements by exploiting resonances in plasmonic nanoparticles employed as labels [Masia et al. (2009), Zhang et al. (2016]. Although This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.

PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093
interferometric amplification can be affected by phase decoherence and wavefront deformation, due to its sensitivity to the relative phase between the NLO and the signal and to the spatial overlap between their wavefronts, label-free imaging in biological samples via interferometric SHG has been recently reported in the visible range [Bancelin et al. (2017)]. Stimulated processes were also reported as an attractive alternative to interferometry to amplify SHG [Goodman et al. (2015), Gao et al. (2018)]. Stimulated SHG allows performing amplification directly at the sample location. Therefore, compared to interferometric SHG, it is less sensitive to wavefront deformation, since the phase and the spatial modes of the stimulating beam (seed) and that of the SHG signal need to overlap only at the sample location, and can be potentially detected background-free by exploiting a non-collinear geometry for the SHG signal and the seed beams. In Indeed, external amplification would enable a dramatic decrease of the integration times and, hence, the observation of faster dynamics in biological environments with extremely long observation periods. In nonlinear sensing, amplification would also allow one to enhance by orders of magnitude the sensitivity and resolution, which are ultimately related to the overall signal yield in background-free conditions.

CONCLUSIONS
We provided an overview of the most recent advances in harmonic generation at the nanoscale and framed it in the wider context embracing the efforts of the scientific community to enhance nonlinear optical processes at the nano-and meta-scale. In recent years, the field of nonlinear photonics in extremely confined volumes rapidly reached a mature stage and first applications have been already demonstrated or are within reach, offering a palette of future perspectives in many fields, ranging from nonlinear bioimaging and This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.  This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093  [ Wolf et al. (2015)] under the terms of the Creative Commons CC BY license.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0006093   process. SHG amplification can be attained both in homodyne-like configuration using an optical nonlinear local oscillator (NLO) and by stimulated effects using a seed beam. In both configurations the amplifying beam is degenerate in energy with the emitted photons. c) A sketch of the pump, seed/NLO and SHG signal beams at the sample.
This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset. This is the author's peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.