1. Introduction

The search for exceptional nonlinear materials is as long as the field of nonlinear optics itself. The list of once promising nonlinear media is lengthy and in the end, a majority of them did not live up to their promise for one reason or another as most practical nonlinear devices continue to rely upon the well-established ionic crystals (LiNbO₃, KTP) and on silicon- or gallium-compounds (Si, SiO₂ as in fiber, SiN, GaAs). Typically, the nonlinearity in these materials is weak and for this reason they require either a long length, or the help of photonic structures such as microresonators. To this date, nonlinear optical fiber remains most widely used platform for nonlinear optics for this reason.

ENZ materials [1,2] – a material or spectral region where |Re{ɛ}| < 1 – are the newest entry into the catalogue of nonlinear materials demonstrating exceptional properties from large refractive index tuning [3,4], enhanced harmonic generation [5], and near unity generation of phase conjugate waves [6]. As a result, they have been projected to become the building blocks for current and future all-optical and electro-optical systems. Yet, it would be prudent to pose a question: is there as anything exceptional about ENZ materials that would let them succeed in the brutal nonlinear optics arena where so many others failed ignominiously? It is our opinion that the answer is affirmative, and, moreover, homogeneous ENZ materials such as transparent oxides possess not one but two salient features that make them stand out in the crowded field – a high degree of electric field confinement (what we refer below as extrinsic enhancement) , as well as an ideal balance of strength and speed suited for many applications (to which we refer as intrinsic enhancement) .

Let us explain this point in a laymen fashion: If one adapts the language of Feynman diagrams [7], optical nonlinearity can be explained as light generating material excitations (typically electron transitions between or within the energy bands) which in turn affect light for as long as they exist. Considering that within the UV to mid-infrared range, the strength of light matter interaction as represented by an optical dipole is more or less constant according to the oscillator sum rule [8], our goal is to maximize the interaction time – either intrinsically by operating close to absorption resonance, or extrinsically as in a number or diverse schemes united by a common moniker “slow light”. When it comes to ENZ the enhancement of nonlinearity follows from the simple relation between the dielectric constant $n = {\varepsilon ^{1/2}}$ which when differentiated leads to $\delta n\sim \delta \varepsilon /2n$ – hence when the permittivity is reduced, a small change results in a large change in refractive index and phase velocity. The change of dielectric constant can be written as $\delta \varepsilon \sim {\chi ^{(3)}}{E^2}$ where ${\chi ^{(3)}}$ is third order nonlinear susceptibility, which is an internal material characteristic and is not linked to the fact that refractive index approaches zero. At the same time, the electric field is related to the intensity of light I as $I = n{E^2}/2{\eta _0}$ where ${\eta _0} = 377\Omega $ is the vacuum impedance. Therefore, the overall optically induced index change becomes $\delta n\sim ({{\chi^{(3)}}{\eta_0}/{n^2}} )I\sim {n_2}I$ and the nonlinear index of refraction ${n_2}$ gets enhanced by a factor ${n^{ - 2}}$. With these points in mind we can consider the enhancement of nonlinearity in ENZ having two factors – intrinsic (or material) ${\chi ^{(3)}}$, and extrinsic $1/{n^2}$.

Let us first deal with the intrinsic factor, susceptibility. It has been noted in [9] that to fully exploit the strength-bandwidth compromise it is desirable to have the response speed of the effect to be just a couple of times faster than the operational speed of the device one is trying to achieve. Going faster would be an overkill – it would not affect the speed of device and only reduces the performance/efficiency. For many applications of interest, a time scale of a few hundreds of femtoseconds is ideal as this enables fast transients while providing a strong overlap with widely available ultrafast laser systems. It just so happens that ENZ materials such as Al:ZnO and Sn:InO have demonstrated nonlinear effects with characteristic times on the order of 300 fs to 700 fs when driven by a near-infrared or ultraviolet excitation [10]. This is achieved, for example, through optical transitions within the (typically conduction) band of the material where the carriers relax through the continuum of energy states by sequential single phonon processes (timescale 1–10 fs) that are readily available. Of course, such fast nonlinearities associated with intraband processes had been previously observed in semiconductors [11,12], but ENZ materials combine these optimized ${\chi ^{(3)}}$ effects with the large “extrinsic” enhancement factor $1/{n^2}$ that can be tuned to any desired wavelength and to which we now turn our attention.

To explicate the “extrinsic” factor $1/{n^2}$ in a proper framework we shall take a look at the ENZ material described by the Drude formula

 

Fig. 1. This figure shows dispersion curves of light in the dispersive material in the vicinity of the pole (1/ɛ∼0) at resonant frequency ω₀ and zero (ɛ∼0) at frequency ω_p straddling the gap where ɛ<0. The lower curve commonly referred to as lower polariton branch in the vicinity of the pole experiences reduction of group velocity vg but the electric field does not get enhanced as the energy is contained in material excitation. This is “conventional” slow light medium. The upper curve or upper polariton branch also experiences reduction of group velocity but this reduction is accompanied by strong enhancement of the electric field. This enhancement is maximum when ω₀ approaches zero, i.e. when the electrons are free as they are in doped semiconductor ENZ materials described by Eq.1.

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Thus, ENZ materials facilitate enhancement of nonlinearity by $n_g^2$ just like in structured slow light schemes [12,13] including microresonators. Unlike the latter, ENZ enhancement does not require any sophisticated fabrication and can easily be maintained on a subwavelength scale, while doped semiconductor materials provide enhancement in the intrinsic ${\chi ^{(3)}}$ of the material due to a more ideally lengthened nonlinear relaxation and an overlap of these properties with spectral ranges of interest such as the telecommunication and thermal sensing regions. Of course, the loss will inherently always be higher in ENZ than in all-dielectric photonic structures (but less than in metal plasmonic ones [15]), yet the simplicity of fabrication, easy availability and ability to operate on very small scale can more than outweigh the disadvantages of ENZ and make them materials of choice for various nonlinear devices in the future.