In the continuously evolving realm of nonlinear optics, epsilon near zero (ENZ) materials have captured significant scientific interest, becoming a compelling focal point over the past decade. During this time, researchers have shown extraordinary demonstrations of nonlinear processes such as unity order index change via intensity dependent refractive index, enhanced second harmonic generation, saturable absorption in ultra-thin films and more recently, frequency shifting via time modulation of permittivity. More recently, remarkable strides have also been made in uncovering the intricacies of ENZ materials' nonlinear optical behavior. This review provides a comprehensive overview of the various types of nonlinearities commonly observed in these systems, with a focus on Drude based homogenous materials. By categorizing the enhancement into intrinsic and extrinsic factors, it provides a framework to compare the nonlinearity of ENZ media with other nonlinear media. The review emphasizes that while ENZ materials may not significantly surpass the nonlinear capabilities of traditional materials, either in terms of fast or slow nonlinearity, they do offer distinct advantages. These advantages encompass an optimal response time, inherent enhancement of slow light effects, and a broadband characteristic, all encapsulated in a thin film that can be purchased off-the shelf. The review further builds upon this framework and not only identifies key properties of transparent conducting oxides that have so far made them ideal test beds for ENZ nonlinearities, but also brings to light alternate material systems, such as perovskite oxides, that could potentially outperform them. We conclude by reviewing the upcoming concepts of time varying physics with ENZ media and outline key points the research community is working toward.

The study of the interaction between light and matter is a fundamental process that underpins many technologies and scientific disciplines, including communication, sensing, imaging, and more recently, quantum computing. From the ancient use of mirrors and prisms to the modern development of lasers and optical fibers, humans have always sought to manipulate light to their advantage. However, the ability to control light–matter interaction is often limited by the intrinsic properties of materials, such as their refractive index, absorption, and dispersion. More recently, various techniques have been developed to enhance, manipulate, and control the interaction between light and matter, such as plasmonics,1–3 metamaterials,4–8 photonic crystals,9,10 and epsilon-near-zero (ENZ) systems.11 Plasmonics, for example, involves the coupling of electromagnetic waves (EM) to the motion of free electrons on the surface of metals, to confine and enhance EM fields at the nanoscale. Metamaterials, on the other hand, are artificial structures that exhibit exotic optical properties not found in natural materials, such as negative refractive index,12,13 hyperbolic dispersion,14,15 and cloaking.5,16,17

In recent years, ENZ systems have emerged as another promising platform for enhancing the interaction between light and matter. ENZ systems are characterized by a vanishingly small value of their permittivity (|Re{ε}| ≤1), which leads to peculiar optical properties, such as enhanced electric field (E-field), slow light, wavelength expansion, alongside diverging group and phase velocities.18 While the ENZ condition is only focused on the |Re{ε}|, large losses, given by Im{ε}, can often limit the effects. Due to this, several works focus on minimizing not only |Re{ε}|, but also the Im{ε}, leading to a vanishingly small refractive index, or near-zero-index (NZI) condition. Initial attempts to build a material with less than unity index date back to the 1950s, when their use was proposed for antenna applications for enhancing microwave radiation directivity.19 However, the interest in systems with tailored permittivity and permeability, including but not limited to ENZ, took off in the early 2000s after pioneering works by Engheta and co-workers.20–25 These works explored ENZ systems for their whole host of peculiar properties such as resonance pinning,26 photon tunneling,23,27 supercoupling,24 enhanced control over emission,20 and extreme nonlinear interactions.28,29 Out of these, nonlinear optics plays a critical role in a broad set of applications, such as telecommunications, spectroscopy, biomedical imaging, and more recently, quantum computing and communication,30,31 making it particularly interesting.

Given the inherent ability of ENZ systems to achieve benefits such as slow light propagation,32 field confinement,33 ultrafast response,28 and relaxed phase matching conditions,34,35 they have been heavily explored for their nonlinear properties29,36–38 over the last decade. While earlier works focused on the large magnitude of the intensity-dependent refractive nonlinearities,28,39 more recently the focus has shifted to understanding the mechanism and intricacies of the nonlinearity,40–42 while expanding the capabilities using nanostructured materials and specialized excitation conditions.43–45 This review provides a summary of the various types of nonlinearities observed in ENZ media, elucidates the role of dispersion in enhancing nonlinearities, and contextualizes ENZ systems within the larger landscape of nonlinear optical materials. More specifically, Sec. II provides an overview of nonlinearities and various enhancement mechanisms, namely, intrinsic and extrinsic. Section III introduces ENZ systems and details the role of dispersion in the enhancement of nonlinearities using a Drude-based homogenous film, as an example. Section IV provides a detailed review of the nonlinear mechanisms in homogenous ENZ systems, followed by Sec. V, which gives an overview of nonlinearities in structured ENZ media. The review ends with Sec. VI, which highlights emerging materials and applications for ENZ systems, followed by concluding remarks.

The goal of this review is to bridge the gap between the optics and materials research communities. While transparent conducting oxides are a well-studied class of materials and have been tailored for several optoelectronic applications, they have only recently been explored for their nonlinearities under the ENZ condition. Due to this, the most readily available films, ITO and doped ZnO, have been the most widely used.28,32,39,46,47 However, now that the community has a better understanding of the mechanism behind the enhanced nonlinearities within ENZ media, the possibility of developing new materials that are tailored for enhanced nonlinearities is wide open. In this sense, for readers new to this area, Secs. II–V provide details of the nonlinear mechanism, and for readers well-versed with ENZ nonlinearities, Sec. VI provides trends in emerging materials and applications.

Nonlinear optics has been a captivating field of study since the 1960s and was sparked by the invention of the laser. Since then, a whole host of material systems have been explored for nonlinear processes such as harmonic generation, frequency mixing, intensity-dependent refractive index, multi-photon absorption, self- and cross-phase modulation, and others.48,49 Typically, these materials exhibit a nearly instantaneous, but weak optical nonlinearity arising from the strong polarization of the electron cloud by an incident field. This can be modeled using an anharmonic potential which describes the binding potential U(r) of an electron in a chemical bond, of size a and binding energy Uo48,50 [Fig. 1(a)]. Upon the application of weak electric fields, the electron oscillates near the bottom of the potential, which is parabolic. This gives rise to a linear electron displacement for a given applied field. Upon the application of stronger electric fields, the electron moves further away from the bottom of the potential and reaches the non-parabolic region, where the electron displacement is no longer linearly proportional to the applied field. This, however, requires electrons to be displaced on the scale of a, which necessitates the application of electric fields on the order of atomic electric field, given by E 0 U 0 / q a, where q is the electron charge Considering that the binding energy for most materials is a few eV and the inter-atomic distance is between 0.16 and 0.21 nm, atomic fields are on the order of  1011 V/m.51 Hence, electron polarization-driven nonlinearities are typically weak (e.g., they require strong electric fields to generate). Despite this fact, these processes have been key enablers of wide-ranging applications, such as optical frequency combs, optical rectification, nonlinear imaging techniques, parametric oscillators, supercontinuum sources, and others for decades.52 However, new materials with stronger nonlinear responses, whose optical properties can be significantly altered by a low-power optical field, are highly desired to achieve devices with advanced functionalities, especially for burgeoning applications such as terahertz switching and routing, single/entangled photon generation, high-efficiency frequency conversion, and quantum repeaters for communication.53–55 To do so, researchers have explored several methods, which can be broadly categorized into two categories: intrinsic and extrinsic factors.56 The role of the intrinsic and extrinsic factors can be distinguished by describing the nonlinear coefficient χ 3 = E F × χ o 3, where χ o 3 is the intrinsic nonlinearity of the material and process at play (e.g., the nonlinearity of the electron potential well) and E F (extrinsic factor) is any applicable extrinsic factor enhancement (e.g., multiple round trips enabled due to the nonlinear material residing inside a cavity).

FIG. 1.

(a) Anharmonic potential modeling the instantaneous nonlinearity exhibited by a polarized electron cloud. (b) Strength of a third order nonlinearities, characterized by n2 (or similar nonlinear response) vs the characteristic time of various materials. These are an order of magnitude estimates and depict an increasingly strong nonlinearity as the characteristic time reduces. [Ga:ZnO (GZO), Al:ZnO (AZO), and InxSn1−xO3 (ITO)].

FIG. 1.

(a) Anharmonic potential modeling the instantaneous nonlinearity exhibited by a polarized electron cloud. (b) Strength of a third order nonlinearities, characterized by n2 (or similar nonlinear response) vs the characteristic time of various materials. These are an order of magnitude estimates and depict an increasingly strong nonlinearity as the characteristic time reduces. [Ga:ZnO (GZO), Al:ZnO (AZO), and InxSn1−xO3 (ITO)].

Close modal

The intrinsic nonlinearity represents the inherent internal nonlinearity of the process at hand. In our example, the nonlinearity of the electron potential is represented by a third-order nonlinear coefficient χ o ( 3 ), which is a material- and process-specific property. As such, it can be altered by shifting the material or nonlinear process being used. For electronic polarization nonlinearities, χ o ( 3 ) can be increased by moving to materials with smaller band gaps, since it is inversely proportional to the bandgap of a material.57 Larger bandgap materials are typically strongly bound (high Uo) and have a short inter-atomic distance (small a). This results in a stronger inter-atomic field ( Eo), necessitating larger incoming electric fields to drive the electron into the nonlinear regime, and thus leading to a smaller nonlinear coefficient compared to narrower bandgap materials. For example, the transparent region of fused silica in the near infrared (NIR) exhibits a χ ( 3 ) on the order of 10−22 m2/V2, vs that of Si which is on the order of 10−20 m2/V2.48 

The magnitude of the intrinsic nonlinearity is also dictated by the nonlinear process that is exploited. Shifting, for example, from electronic polarization (virtual transitions) to e-h pair generation (real transitions) for nonlinear control of the refractive index can result in substantial modifications to the nonlinear coefficient. Based on the anharmonic oscillator model, the change in the index is proportional to the energy density stored or absorbed within a material.48,50 In the case of absorption-driven effects (often the case near a resonance), this leads to the approximation Δ n ( ω ) α ( ω ) n ( ω ) I τ / 2 N U o, where Δ n ( ω ) is the optically induced index change, n ( ω ) is the linear refractive index, and α ( ω ) is the absorption coefficient all at ω, I is excitation irradiance, τ is the characteristic time of the nonlinearity (e.g., recombination time), N is the density of states, and U o is the binding energy.50,58 Other than the irradiance of light I, τ is the only parameter in this expression that can be varied by orders of magnitude. This implies that moving to nonlinearities with a longer characteristic time results in a larger nonlinear effect—a well-known effect.50 

Real transitions achieve this as carriers must move between physical states, and thus require a finite amount of time to return to equilibrium. The characteristic time of real transitions in various materials can vary between femtoseconds, for intra-band processes, and milliseconds for thermal diffusion processes, thus resulting in a comparably large range for their respective nonlinear index coefficients n 2. This coefficient, which describes the linear dependence of the refractive index n on optical intensity I (according to n = n o + n 2 I) follows a typical bandwidth-strength trade-off [Fig. 1(b)]. As an example, above-bandgap nonlinearities (real transitions), given by χ ( 3 ), in semiconductors such as Si and GaAs are   4 orders of magnitude larger than non-resonant nonlinearities (virtual transitions) of fused silica, as well as   4 orders of magnitude slower.

If the characteristic time τ is fixed, obtaining stronger index tuning via intrinsic effects is relegated to increasing peak irradiance, I, or optimizing application-specific parameters such as the interaction length to maximize an induced phase shift, Δ Φ N L = Δ n k o L. Indeed, over the past several decades, millimeter-scale and longer nonlinear crystals have been utilized along with high-power lasers to enable several applications such as parametric oscillation generation and amplification, Kerr-based mode locking, ultra-fast pulse shaping, and more.

Beyond the ability to modify an intrinsic nonlinearity through material or process, extrinsic factors such as employing structure or tailored excitation schemes are complementary tools that can increase the overall efficiency of a nonlinear process. The primary objectives in leveraging these factors include enhancing the field intensity, increasing the interaction length, and introducing strong dispersion.

One example of extrinsically enhanced NL is using high-Q ring resonators (or Fabry Perot cavities), fabricated using materials such as fused silica, Si3N4, or MgF2 to generate frequency combs.59 Low power onset (  3 mW) of frequency combs in these structures is primarily achievable due to the large Q-factor, even though these materials do not possess large nonlinear coefficients.60–62 Several round trips of the photon across the ring resonator increases the interaction length, thus leading to a higher effective nonlinearity. Another method of enhancing nonlinearity is by using slow light in highly dispersive media, where the transit time is given by τ t r = L n g / c. Here L is the length, and the ng is the group index, which can be quite large in highly dispersive media such as near the band edge of a photonic crystal,63,64 in the vicinity of bound plasmonic surface modes,65,66 or in the case of structures displaying Fano resonances.67 Yet another method entails enhancing the local electric field and optical intensity through the use of structures such as plasmonic waveguides or nanoantennas.68–70 Practically, however, systems of choice usually rely on a combination of increasing the interaction time, enhancing the electric field, and engineering slow light via strong dispersion. Over the last decade, several sophisticated resonant schemes have been explored such as electromagnetically induced transparency, several nanoantennae and metasurfaces, bound states in the continuum, anapoles, and topological modes.71–75 

Although many efforts rightfully blur the distinction between intrinsic and extrinsic effects using a measured χ ( 3 ) of the system for simplicity (e.g., treating it as a black box), it is useful to distinguish extrinsic effects from the intrinsic nonlinearity to compare materials and gain insight into optimization techniques. This is because extrinsic effects can always be employed to improve the overall efficiency of a nonlinear process whether it be naturally occurring such as group velocity enhancement due to a high material index, or by design such as employing a cavity, but they do not fundamentally modify the intrinsic nonlinearity of a material. For example, operating a nonlinear switch made of silicon near the critical angle is a useful tool to improve the absolute change in reflection/transmission, but this specific excitation scenario does not modify the intrinsic nonlinear coefficient of silicon. It simply improves the sensitivity of the approach to the induced nonlinear modifications.

While extrinsic factors have been incredibly successful in improving the overall efficiency of nonlinear processes, the success generally comes at the cost of increased complexity, higher cost, and quite often limited spectral bandwidth. As a result, one generally benefits from using a material or process with a higher intrinsic nonlinearity restricted by limits on loss and the desired speed, while extrinsic factors can be employed to further tailor the process to specific application conditions. It is in exactly this sense, ENZ materials are unique because they offer several of these extrinsic benefits naturally and do so without a significant impact on the spectral bandwidth of operation.18,56

Vanishingly small permittivity can be realized in several ways, which can be broadly categorized into two categories: natural materials which exhibit ENZ locally (sub-wavelength), and structured materials which exhibit an effective ENZ condition (over several wavelengths). As an example, we consider materials that exhibit ENZ homogenously via the collective resonance of free carriers near the plasma frequency. The optical properties of these materials (such as metals and doped semiconductors), that have a large free carrier density (1 × 1019–5 × 1022 cm−3) are often described by the Drude model,76 yielding a relative permittivity of
ε ω = ε ω p 2 ω 2 + i Γ ω ,
(1)
where ε is the high-frequency permittivity contribution, ω p is the plasma frequency, and Γ is the damping rate. The plasma frequency is related to the density of free carriers, Nc, the electron charge, q, and the electron effective mass, m*, according to
ω p 2 = N C q 2 ε 0 m * .
(2)
Most metals or metallic compounds, such as Au, Ag, Al, TiN, and ZrN, exhibit a crossover in the UV to visible spectral range.77–88 While some, such as transition metal nitrides, offer large tunability of the crossover wavelength, all of these are accompanied by high losses ( Im { ε } > 0.5) and high carrier scattering rates.85,89–91 This limits their applicability and has led to only limited demonstrations of enhanced nonlinearities due to ENZ effects.92–94 In contrast, transparent conducting oxides (TCO) such as doped In2O3, ZnO, and CdO, have been heavily explored for their nonlinearities.28,39,95–97 These materials exhibit lower carrier densities, on the order of ≈ 1020 cm−3–≈ 1021 cm−3. Therefore, the crossover wavelength occurs in the near to mid-infrared (MIR) spectral range and exhibits lower losses due to reduced scattering rates,98–102 and the realization of NZI properties.

To illustrate the benefit to nonlinear properties of introducing the ENZ point, Fig. 2(a) plots the permittivity of doped ZnO in the NIR spectrum and compares it to fused silica and nominally undoped ZnO. The third order susceptibilities of ZnO103 and fused silica104 are χ 3 ≈ 5 × 10−20 and ≈ 5 × 10−22 m2 V−2, respectively. While two orders of magnitude apart, they follow the scaling law of χ ( 3 ) E g 1 (observed in dispersion free spectral regions),105 due to which the difference in third order nonlinearity is expected. However, doping of ZnO with Al ions increases the free carrier density from  1018 to 5 × 1020 cm−3. This results in the NIR spectrum being dominated by the Drude response, which introduces strong dispersion into a region that otherwise would have very little. Consequently, electromagnetic energy at wavelengths corresponding to ENZ spectral region experiences a confluence of several peculiar effects, such as a slow group velocity due to large dispersion, enhanced electric field within the ENZ media, and relaxed phase matching conditions in sub-wavelength films due to a small n (more details to follow). These lead to enhanced nonlinearity, as seen by an enhanced effective intensity dependent refractive index (IDRI) coefficient n 2 , eff, when an ENZ material is pumped and probed in the ENZ spectral region [Fig. 2(b)]. The change in refractive index as a function pump intensity I is given by Δ n = n 2 , eff I.

FIG. 2.

(a) Free electrons induce strong dispersion in a region otherwise minimal change in the index while exhibiting low carrier scattering induced optical losses. (b) Predicted values of the n2 coefficient of Al-doped ZnO (AZO), when pumped at four different wavelengths (800, 1300, 1550, and 2000 nm) and probed from 700 to 2000 nm. The largest enhancement is observed when the pump and the probe are within the ENZ region, defined here as the spectral range spanning a range of ±100 nm about the crossover wavelength.

FIG. 2.

(a) Free electrons induce strong dispersion in a region otherwise minimal change in the index while exhibiting low carrier scattering induced optical losses. (b) Predicted values of the n2 coefficient of Al-doped ZnO (AZO), when pumped at four different wavelengths (800, 1300, 1550, and 2000 nm) and probed from 700 to 2000 nm. The largest enhancement is observed when the pump and the probe are within the ENZ region, defined here as the spectral range spanning a range of ±100 nm about the crossover wavelength.

Close modal

The most intuitive ENZ effect is the local field enhancement which follows from the boundary condition of the electric field ( E) at the interface of different media. At a source-free boundary, the tangential electric field remains the same ( E | | 1 = E | | 2) and the electric field component normal to the interface must fulfill the relation ε 1 E 1 = ε 2 E 2 to maintain the continuity of electric flux density. Following this, as the permittivity of one material approaches zero, the normal electric field within this material must increase to maintain equality while the tangential field component remains unaffected. As a result, there is an overall increase in the magnitude of E within the ENZ layer. Since the condition occurs at the boundary it extends to films of thickness much smaller than the skin depth, facilitating subwavelength confinement [Fig. 3(a)].18 However, for free space applications, the reflection coefficient begins to dominate, especially at large angles, minimizing the amount of light entering the material and reducing the enhancement effect. This field enhancement within an ENZ medium can provide a significant improvement to the overall nonlinear response since it is intrinsically related to powers of the electric field ( P N L = χ ( 2 ) E 2 + χ ( 3 ) E 3 + ).29,39,106 This effect is also robust and applies even in the case when loss in the material is quite high (e.g., Im ε > 2, achieving ENZ but not NZI). However, since only the field normal to the interface is affected, leveraging this effect in free space requires illuminating the sample at an off-normal incidence angle, which typically comes at the cost of reduced transmission due to the large index contrast experienced between air and the near-zero permittivity condition.

FIG. 3.

Depictions of various effects associated with an ENZ condition. (a) Enhancement of normal field component observed in a hybrid (zero index material layer on top of Silicon) waveguide compared to a pure dielectric waveguide. The ENZ layer, placed on top of the waveguide, induces a strong field confinement at the interface of the layers.18 Reproduced with permission from Kinsey et al., Nat. Rev. Mater. 4(12), 742–760 (2019). Copyright 2019 Springer Nature Limited. (b) An experimental demonstration of wavelength expansion effect observed at an ENZ condition. The excitation of counterpropagating modes over a range of wavelengths through a metamaterial-based waveguide leads to interference nodes, the distance between which is proportional to the effective wavelength of the medium. As the input approaches near zero effective index condition, the node length expands, and effective wavelength tends to be infinity (  1627 nm). The figure shows this interference of nodes across the length of the waveguide for a range of wavelengths.107 Reproduced with permission from Reshef et al., ACS Photonics 4(10), 2385–2389 (2017). Copyright 2017 American Chemical Society. (c) Theoretically, in a nonlinear four wave mixing process within an ENZ medium, sources acquire no phase as they propagate, guaranteeing a constructive interference and an increase in the signal in both directions. Panel (c) shows a general four wave mixing process (DFWM).34 Reproduced with permission from Suchowski et al., Science 342(6163), 1223–1226 (2013). Copyright 2013 The American Association for the Advancement of Science. (d) General dispersion relation of an ENZ material under lossless and lossy conditions. As the frequency of the mode approaches the ENZ resonance, the slope of d ω / d k approaches zero marking the slow light region. However, material loss decreases the extent of flat dispersion, thus limiting the effective group index to a finite value.

FIG. 3.

Depictions of various effects associated with an ENZ condition. (a) Enhancement of normal field component observed in a hybrid (zero index material layer on top of Silicon) waveguide compared to a pure dielectric waveguide. The ENZ layer, placed on top of the waveguide, induces a strong field confinement at the interface of the layers.18 Reproduced with permission from Kinsey et al., Nat. Rev. Mater. 4(12), 742–760 (2019). Copyright 2019 Springer Nature Limited. (b) An experimental demonstration of wavelength expansion effect observed at an ENZ condition. The excitation of counterpropagating modes over a range of wavelengths through a metamaterial-based waveguide leads to interference nodes, the distance between which is proportional to the effective wavelength of the medium. As the input approaches near zero effective index condition, the node length expands, and effective wavelength tends to be infinity (  1627 nm). The figure shows this interference of nodes across the length of the waveguide for a range of wavelengths.107 Reproduced with permission from Reshef et al., ACS Photonics 4(10), 2385–2389 (2017). Copyright 2017 American Chemical Society. (c) Theoretically, in a nonlinear four wave mixing process within an ENZ medium, sources acquire no phase as they propagate, guaranteeing a constructive interference and an increase in the signal in both directions. Panel (c) shows a general four wave mixing process (DFWM).34 Reproduced with permission from Suchowski et al., Science 342(6163), 1223–1226 (2013). Copyright 2013 The American Association for the Advancement of Science. (d) General dispersion relation of an ENZ material under lossless and lossy conditions. As the frequency of the mode approaches the ENZ resonance, the slope of d ω / d k approaches zero marking the slow light region. However, material loss decreases the extent of flat dispersion, thus limiting the effective group index to a finite value.

Close modal

The ENZ condition also enables a large increase in the phase velocity, v p = ω / k = c / n , where smaller values of refractive index n result in an “expanded” wavelength λ ENZ = λ o / n [Fig. 3(b)]. In the case of very small optical loss in the ENZ material, such that 0, the wavevector k also tends to zero. This results in minimal phase accumulation throughout the ENZ medium which effectively transforms light waves into spatially static or “DC” fields.11 Minimization of phase accumulation aids in relaxing the phase-matching condition for nonlinear processes, such as second harmonic frequency generation or four-wave mixing,34 where maintaining phase matching is a necessary condition to prevent destructive interference of the nonlinear fields as they propagate through a medium. Within ENZ media, the lack of phase accumulation throughout the propagation length greatly relaxes this constraint irrespective of incoming photon direction. This stands in contrast to traditional methods which typically achieve phase matching only in a specific direction [Fig. 3(c)].11 However, all waves participating in the nonlinear process must belong to the ENZ region to retain the advantage of phase matching (e.g., for a second-order process k 1 , ENZ ± k 2 , ENZ = k 3 , ENZ 0). This is practically challenging to accomplish due to the bandwidth limitation of the ENZ condition in homogenous materials [see, for example, Fig. 2(a)]. In fact, if this is not achieved phase mismatch may be exacerbated.

Achieving ENZ naturally generates dispersion in the spectral vicinity. This is consequential when considering the group velocity ( v g = ω / k), which governs the energy and information transfer velocity. When dispersion is strong, v g becomes very small, leading to the phenomenon known as “slow light.”108–111 As shown in the ω- k dispersion plot of a homogeneous ENZ medium [Fig. 3(d)], the strong dispersion near the permittivity crossover produces a minimal slope ( ω / k 0) and thus a reduced group velocity that is limited only by the loss within the material. The reduced group velocity then enhances the nonlinear interaction, as photons spend increased time inside the material, τ prop = L / v g.110,111 Slow light also contributes to enhancing the electric field and energy density u = I O / v g because of energy compression in the slow propagation region.

The three mechanisms mentioned above act as extrinsic factors that enhance the nonlinearity of the ENZ medium. What is intriguing is that they occur naturally in homogeneous ENZ materials, such as doped ZnO, without the need for complex design and fabrication. At the same time, common homogeneous ENZ materials are fundamentally wide bandgap semiconductors, and the introduction of the ENZ condition via strong doping [Fig. 2(a)] unlocks real transitions in the ENZ region via free carrier interactions that would otherwise be prohibited (e.g., ω E g). Intense light excitation is then able to modify the free carrier density N c via interband excitation, or the effective mass m * of free carriers within the conduction band via intraband excitation, leading to a change in ε ω according to Eqs. (1) and (2) (see Sec. IV for more details). These processes have been shown to have a characteristic time on the order of 0.1–1 ps and are naturally stronger than virtual transitions due to their finite relaxation time. Thus, the nonlinearity of homogeneous ENZ films is not only extrinsically enhanced but also intrinsically strong.

This unique combination of natural intrinsic and extrinsic enhancement has led to some spectacular demonstrations of index change, reaching the order of unity over the last decade.39 Furthermore, the fact that the ENZ region is passively tunable across a wide spectrum by controlling material deposition,98,99,112 actively tunable via electrically gating with a capacitive structure,68,113,114 and dynamically tunable by pumping the materials optically (in both UV and IR regions),115 has resulted in the widespread popularity of Drude-based homogeneous materials in device demonstrations.114,116–118 However, the introduction of the ENZ condition via doping introduces optical losses and only provides enhancement over a limited bandwidth spectral region near the permittivity crossover.

In this sense, ENZ is not a “free” enhancement, and the cost of utilizing it should be weighed within the scope of a desired application (see the paragraph “Slow Free Carrier, Intraband Nonlinearities” in Sec. IV for details). In many cases, the broad nature of the ENZ region (  400 nm bandwidth in the 1–2 μm spectral window) does not severely limit applications. Similarly, while the losses typically limit the interaction length to less than 1 μm, the absolute nonlinearity (third harmonic efficiency or Δ R , Δ T) can be comparable to or higher than bulk material platforms.29,39,93

As a concluding note for this section, it is important to point out that ENZ in naturally occurring materials can also be achieved through the interaction of light with bound electron motion, lattice vibrations, or molecular vibrations. All these responses can be modeled using a Lorentz oscillator, which represents physical processes such as interband, band edge absorptions, and vibrational resonances. For example, one important such process is surface phonon polariton (SPhP) excitation in polar dielectric crystals. Such polarons result from atomic vibrations generated from the interaction between polar optical phonons and long wavelength irradiation in the mid-IR (e.g., for SiC) to sub-10 THz regions (e.g., for GaAs, and CaF2), within the well-known Reststrahlen band, i.e., between longitudinal (LO) and transverse optic (TO) phonon frequencies.119,120 If the strength of the resonance is sufficient, it can result in a spectral region where the permittivity is negative (Fig. 4). The Reststrahlen band is characterized by exceptionally low losses due to the slow scattering rates of optical phonons, which typically occur on the timescale of picoseconds, in comparison to the faster scattering of free carrier plasmons.120,121 Moreover, upon TO phonon resonance the permittivity crosses zero once again (from negative to positive, Fig. 4). While intriguing due to the extreme dispersion, this epsilon-near-pole (ENP) region is also characterized by an extremely high loss, which limits applications.120 

FIG. 4.

Lorentz-induced ENZ response observed in SiC due to phonons. Adapted from Ref. 120.

FIG. 4.

Lorentz-induced ENZ response observed in SiC due to phonons. Adapted from Ref. 120.

Close modal

Although phononic ENZ materials, or some of the more recently explored organic ENZ materials,122–124 have commonalities with free-carrier ENZ materials in the linear regime,125 the nonlinear processes at play are fundamentally distinct owing to the stark difference between phonons, polarons, and free carriers (e.g., the presence of a restoring force). The remainder of this review is focused on free-electron effects, but additional information on phononic ENZ materials can be found in a recent review by Foteinopoulou et al.126 

Section IV A provides details into the nonlinear processes exhibited by Drude-based ENZ materials, broadly categorized into two categories, those having fast and slow nonlinearities. Fast nonlinearities encompass instantaneous (typically sub ≈1 fs)127 processes like electronic polarization (virtual transitions) and the collective oscillation of the free electron gas within the conduction band (real transition).58 They are responsible for driving processes such as harmonic generation. Electronic polarization nonlinearities in semiconductors are well covered in various nonlinear textbooks48,128 and will not be repeated here. On the other hand, slow nonlinearities involve real transitions that have a decay time, typically on the order of hundreds of femtosecond to 1 ns, and lead to processes such as the intensity-dependent refractive index28,39,46,129–131 and adiabatic frequency shifts.33,70,93,132 Of the slow response, we describe two components, the interband and intraband driven response. Figure 5(a) presents a flow chart categorizing the above-mentioned nonlinearities.

FIG. 5.

(a) A flow chart categorizing the different types of nonlinearities commonly observed in a Drude based ENZ system. The first row classifies the nonlinearity on the nature of transition: real or virtual. The next row classifies it based on speed and finally, the third row classifies it based on the energy at which they occur, inter-band [(3/4)Eg] and intra-band [(1/2)Eg], where Eg corresponds to the bandgap of the material. (b) Epsilon-near-zero nonlinearities spectrally distributed, based on type of nonlinearity. Around the bandgap of the material (λ ≈ 300 nm) real, slow inter-band processes occur. In the spectrally transparent region (λ ≈ 500–1100 nm) virtual, fast processes occurs. Finally, in the NIR region (λ > 1100 nm) real, fast and slow intra-band processes occur.

FIG. 5.

(a) A flow chart categorizing the different types of nonlinearities commonly observed in a Drude based ENZ system. The first row classifies the nonlinearity on the nature of transition: real or virtual. The next row classifies it based on speed and finally, the third row classifies it based on the energy at which they occur, inter-band [(3/4)Eg] and intra-band [(1/2)Eg], where Eg corresponds to the bandgap of the material. (b) Epsilon-near-zero nonlinearities spectrally distributed, based on type of nonlinearity. Around the bandgap of the material (λ ≈ 300 nm) real, slow inter-band processes occur. In the spectrally transparent region (λ ≈ 500–1100 nm) virtual, fast processes occurs. Finally, in the NIR region (λ > 1100 nm) real, fast and slow intra-band processes occur.

Close modal

With reference to the wavelength of the excitation beam (assuming the probe is fixed to the permittivity crossover), Fig. 5(b) highlights the spectral regions where each of these responses occur in a typical ENZ material such as ITO. First, for excitation energies E > (3–4) Eg [Fig. 5(b), dark gray region on left side of permittivity vs wavelength graph], slow interband nonlinearities reliant on band-to-band absorption tend to dominate. The change in carriers in turn affects the free-carrier density of the Drude term which leads to index changes in the NIR part of the spectrum. Second, for energies between 2ℏωp < E < (3/4)Eg [Fig. 5(b), medium gray region in middle] lies a transparent low loss region. This region is dominated by fast virtual transitions and leads to electronic polarization nonlinearities such as Kerr index modulation. Third, for excitation energies E < 2ℏωp [Fig. 5(b), light gray region on right], free-carrier effects term begins to dominate the optical response. In this spectral region, depending on the experimental condition, either a fast or a slow intraband process can dominate. Slow intraband processes that lead to excitation of hot carriers in a non-parabolic conduction band (CB) have been the most widely explored. The fast intraband nonlinear processes are driven by the movement of free electrons around the Fermi level in the non-parabolic CB and tend to benefit from slow light and electric field enhancement that occur around the ENZ spectral region.

While slow free carrier effects are most often discussed in ENZ literature, a fast free-carrier process in Drude ENZ materials exists at all wavelengths and can dominate the response for ℏω < (3/4)Eg. Khurgin et al.58 describe this process as the incident energy driving the oscillation of free electrons within a non-parabolic conduction. Under small driving fields, the electrons oscillate predominantly in the parabolic region of the band where the electron velocity grows linearly with momentum—giving rise to a current density J = −qk v(k,t) that is linear. However, under large driving fields, electrons reach the nonlinear regions of the band, where the velocity is no longer proportional to the change in momentum—giving rise to a nonlinear current density [Figs. 6(a) and 6(b)]. In this case, the total current density can be expanded into powers of the electric field E, similarly to the Taylor series expansion for the polarization P(t) = χ1 E(t) + χ2 E2(t) + χ3 E3(t) + ⋯.

FIG. 6.

(a) Parabolic and non-parabolic approximation of the E–k band structure dictating the fast and slow nonlinearity in materials. (b) Carrier velocity in the parabolic and non-parabolic conditions corresponding to the bands in (a).

FIG. 6.

(a) Parabolic and non-parabolic approximation of the E–k band structure dictating the fast and slow nonlinearity in materials. (b) Carrier velocity in the parabolic and non-parabolic conditions corresponding to the bands in (a).

Close modal
The velocity of the electron is described through the first (α) and second derivative (β) of effective mass in a non-parabolic band as
v z ( k , t ) = [ m t ( 1 ) δ k ( t ) + 1 2 α δ k 2 ( t ) + 1 6 β δ k 3 ( t ) ] ,
(3)
where k is the momentum, and mt is the transport effective mass. Taking the time average velocity under harmonic excitation, the first- and third-order current densities can be written as
J ( 1 ) ( ω 1 , 2 ) = ( i N q 2 m t ( 1 ) ) ( ω 1 , 2 ) + i y E 1 , 2 e ( i ω 1 , 2 , t ) + c . c . , ( J ( 3 ) ( ω 1 , 2 ) = i N q 4 β 6 2 ( E 1 e i ω 1 t ω 1 + i y + E 2 e i ω 2 t ω 2 + i y + c . c . ) ) 3 ,
(4)
where the second-order term cancels in the case of a symmetric conduction band about the band minimum. This can be connected to a nonlinear susceptibility for a four-wave-mixing process:
χ fast 3 ( ω 3 = ω 1 ω 2 + ω 1 ) = χ f c ( 1 ) ( ω ) e 2 f β 2 2 k 0 2 ( ω 2 γ 2 ) ,
(5)
where fβ (kF) =  k 0 2 ⟨β⟩⟨mt ⟩. For a typical ENZ material such as ITO, Nc = 1.1 × 1021 cm−3, kF = 3.2 nm−1, and k0 ≈ 2 nm−1 (via fitting). This results in χ fast ( 3 ) = 2.5 × 10−19 m2 V−2 for an excitation at 1 eV, typical of an ENZ region.

This value of χ fast ( 3 ) is similar in strength to that of the bound electron nonlinearity found in most materials (e.g., Al2O3: χ3 ≈ 10−20 m2 V−2).133 Thus, while the process is entirely different, free carrier motion does not give rise to a nonlinearity that is significantly stronger than that of bound electrons. Yet, ENZ provides field enhancement and slow-light effects to increase the magnitude of electric field E and provide longer light–matter interaction time. This combination can lead to a higher effective nonlinearity compared to that observed in conventional crystals without nanostructuring (see Sec. IV B for details). However, the ability to increase the overall conversion efficiency of processes such as harmonic generation and four wave mixing by increasing the interaction length (such as applied to fiber- and LiNbO3-based systems) is ultimately limited by loss in the ENZ regime and the limited transparency bandwidth of ENZ materials. As such, efficiency-sensitive applications would continue to favor low-loss materials, despite the enhancements available via ENZ, while space-constrained applications or scenarios dependent on complex phase matching could benefit from fast nonlinearities in thin ENZ films.

Fast nonlinearities in ENZ materials have not been studied as extensively as slow nonlinear effects, neither theoretically nor experimentally. Fast nonlinearities are of increasing interest for frequency-control applications. Fast processes stand to particularly benefit from the near-zero effects such as slow light, electric field enhancement, and relaxed phase matching conditions. Several reports have demonstrated second- or third-harmonic generation using ENZ films.33,70,93,121,136–139 This has been achieved leveraging surface-symmetry breaking, phonon response, or structuring. In 2015, Capretti et al.93 described the first use of ITO for second harmonic generation, while comparing it to metallic TiN thin films. This sparked interest in fast nonlinearities in ENZ materials, leading to demonstrations of small, yet technically interesting conversion efficiencies of η SHG 1 0 13. Consequently, Yang et al.33 demonstrated high harmonic generation, of up to the ninth harmonic, using doped-CdO with the pump positioned at λENZ ≈ 2100 nm. Subsequently, Tian et al.134 began with a pump at a higher frequency pump (λENZ ≈ 1050 nm), than Yang et al., and demonstrated high harmonics reaching UV wavelengths with high efficiencies of, 10−4 and 10−6 for fourth and fifth harmonics, respectively [Fig. 7(a)]. Finally, with the use of second and third harmonic generation in ENZ films, Jaffray et al.135 presented a method of ultrafast pulse characterization based on frequency resolved optical gratings (FROG) [Figs. 7(b) and 7(c)]. Such demonstrations move ENZ films closer to the realm of practical niche applications.

FIG. 7.

(a) ITO film (≈ 103 nm thick) illustrating second through-fifth harmonic generation (HG).134 Reproduced with permission from Tian et al., Photonics Res. 9(3), 317 (2021). Copyright 2021 Authors, licensed under a Creative Commons Attribution Unported License. (b) Frequency resolved optical grating (FROG) trace from an ITO film showing second harmonic generation (SHG) for pulse characterization.135 (c) Recovered temporal amplitude and phase profiles. Reproduced with permission from Jaffray et al., Nat. Commun. 13(1), 3536 (2022). Copyright 2022 Authors, licensed under a Creative Commons Attribution Unported License.

FIG. 7.

(a) ITO film (≈ 103 nm thick) illustrating second through-fifth harmonic generation (HG).134 Reproduced with permission from Tian et al., Photonics Res. 9(3), 317 (2021). Copyright 2021 Authors, licensed under a Creative Commons Attribution Unported License. (b) Frequency resolved optical grating (FROG) trace from an ITO film showing second harmonic generation (SHG) for pulse characterization.135 (c) Recovered temporal amplitude and phase profiles. Reproduced with permission from Jaffray et al., Nat. Commun. 13(1), 3536 (2022). Copyright 2022 Authors, licensed under a Creative Commons Attribution Unported License.

Close modal

For refractive index modulation in degenerately doped semiconductors, real transitions involving free carriers dominate the permittivity tuning effect in the Drude region. The modified Drude equation with bound electron high-frequency contribution is employed to model the free-carrier effects [Eq. (1)]. The tunable parameters within the Drude model are Nc and m*, which can be altered by optical excitation. Photon absorption can occur both above and below the band edge leading to two separate processes that can modify permittivity. It is important to note that the damping rate Γ can also change in the Drude model with excitation, however because it is often the case for TCOs that Γ ≪ ωp, any change of Γ is negligible and the change in permittivity is dominated by Nc and m*. However, for high loss materials this may not be true, and modifications to Γ may be equally important. These tuning mechanisms are not unique to ENZ materials. As an example, pumping a low doped Si film (Nc ≈ 1 × 1014 cm−3) at energies above the bandgap will lead to index modulation in a similar manner, although the ENZ condition may not be reached. Various modeling efforts by Liberal,11 de Leon,42 Sivan,41 Wang,140 and Secondo,40 have successfully matched experimental measurements for both intra- and inter-band measurements of ENZ materials and have been extensively discussed and built upon over the past decade.

1. Intraband absorption

First, for case of doped semiconductors, when the excitation energy is less than the bandgap (typically, E < 2ℏωp) intraband, or free carrier, absorption tends to dominate the optical response.40 The effect can be modeled directly from the Fermi-Dirac statistics by following carrier and energy conservation. For intraband absorption the number of carriers in the system must remain constant before and after optical excitation
f ( E , μ ( I ) , T e ( I ) ) ρ ( E ) d E = N ,
(6)
where f ( E , μ ( I ) , T e ( I ) ) is the Fermi–Dirac distribution, E is the energy of the carrier, μ is the chemical potential, and Te is the electron temperature, ρ(E) is the density of states, and N is the number of carriers. The second condition is that the change in energy in the system must be equal to the absorbed energy
( f ( E , μ ( I ) , T e ( I ) ) E ρ ( E ) d E f ( E , μ 0 , T e 0 ) E ρ ( E ) d E = δ Uabs A L τ d ,
(7)
where A is the absorptivity, I is the intensity, τ is the temporal FWHM of the laser, and d is the thickness of the film. In conjunction, the non-parabolic conduction band results in the effective mass being a function of energy m* (E) (Fig. 8 right side of band diagram).
FIG. 8.

Figure describes the inter-band (band diagram, left) and intra-band (right) excitation process and the corresponding effects on ε. Inter-band pumping leads to the excitation of electrons from Ev to Ec, resulting in an increase in density of electrons. This results in a blue shift of the plasma frequency, or in other terms, a decrease in the ε at a given wavelength. Figure on the bottom left depicts this effect, where the dashed line corresponds to the unmodulated ε of the thin film, and blue curve corresponds to the modulated ε, after inter-band pumping. On the other hand, intra-band pumping results in excitation of electrons to higher levels of the conduction band, which tend to be characterized by heavier effective masses (color bar, varying from white to peach, corresponds to m* varying from low to high, respectively). This results in a red shift of the plasma frequency, or an increase in ε at a given wavelength. Purple circles: electrons, White circles: holes, EF: fermi level, dUabs: pump energy, Ec: conduction band, Ev: valence band, m*: effective mass, ε: permittivity.

FIG. 8.

Figure describes the inter-band (band diagram, left) and intra-band (right) excitation process and the corresponding effects on ε. Inter-band pumping leads to the excitation of electrons from Ev to Ec, resulting in an increase in density of electrons. This results in a blue shift of the plasma frequency, or in other terms, a decrease in the ε at a given wavelength. Figure on the bottom left depicts this effect, where the dashed line corresponds to the unmodulated ε of the thin film, and blue curve corresponds to the modulated ε, after inter-band pumping. On the other hand, intra-band pumping results in excitation of electrons to higher levels of the conduction band, which tend to be characterized by heavier effective masses (color bar, varying from white to peach, corresponds to m* varying from low to high, respectively). This results in a red shift of the plasma frequency, or an increase in ε at a given wavelength. Purple circles: electrons, White circles: holes, EF: fermi level, dUabs: pump energy, Ec: conduction band, Ev: valence band, m*: effective mass, ε: permittivity.

Close modal

When an electron absorbs energy, it is promoted to a higher energy state, E+dE, where it is no longer in equilibrium with the rest of the electron population. The electron quickly relaxes back to the Fermi level through electron–electron scattering. As a result, the absorbed energy dE is then distributed throughout the sea of electrons raising their overall temperature Te. The heated electron sea then scatters with lattice phonons (at a rate dictated by the electron–phonon coupling coefficient G), transferring the energy to the lattice and raising its temperature Tl until Te = Tl. The system then relaxes to its initial state through thermal conduction (phonon–phonon scattering), radiation, etc.

The temporal nature of this process is captured by the two-temperature model (TTM).141 This phenomenological model tracks the electron temperature in the band as it sharply rises with a fs-pulse then decays according to heat transfer between electrons and the lattice
C e ( T e ) T e t = z ( κ T e z ) G ( T e T l ) + H ,
(8a)
C l ( T l ) T l t = G ( T e T l ) .
(8b)
Equations (8a) and (8b) describe the electron and lattice temperatures (Te and Tl), the volumetric heat capacities of electrons and lattice (Ce and Cl), the thermal conductivity κ, the electron–phonon coupling constant G, and the source term H.142–144 
Under high intensity, the resulting free-carrier absorption can lead to a significant increase in the electron temperature. This tends to broaden the Fermi–Dirac distribution in energy, increasing the occupation probability of higher energy states. Combining this with the Fermi statistics equations, the average effective mass in the conduction band becomes
m avg * = ( 1 N f ( E , μ , T e ) m * ( E ) ρ ( E ) d E ) 1 .
(9)
The final piece to the puzzle is the Drude model. Equation (1) shows the plasma frequency ωp depends on electron effective mass m*. If m* is modified by the absorption process, so are the ωp and relative permittivity ε(ω). Given that the effective mass typically increases at higher energies, the occupation of higher energy tends to increase the overall effective mass of the electron sea, resulting in an increase in the permittivity of the material in the ENZ region.
Under the assumption that the effective mass of the carriers is approximately the effective mass at the Fermi level, the effective intensity dependent refractive index (IDRI) coefficient evaluated within the ENZ region at the screened plasma frequency (ωp) can be described as40 
n 2. e f f ( ω p ) = ε α 0 n g τ 2 n ( ω p ) N E F ,
(10)
where α0 is the linear absorption coefficient, ng is the group index, τ is the excitation time, n′(ωp) is the real part of the refractive index at the plasma frequency, and EF is the Fermi energy. For a film with ENZ wavelength λENZ = 1550 nm, α0 = 1 × 105 m−1, ng = 6, τ = 100 fs, N = 1021 cm−3, and EF = 4.5 eV, the calculated value of the index coefficient is n2.eff ≈ 10−16 m2/W, which is an order of magnitude higher than that of semiconductors such as Si or GaAs (n2 ≈ 10−17 m2/W), and four orders of magnitude higher than that of dielectrics such as SiO2 or Al2O3 (n2 ≈ 10−20 m2/W), as shown in Fig. 1(b).

Finally, it is important to note that in most cases the lattice temperature rise of ENZ materials is small (a few degrees K). However, if the material is not given enough time to relax between excitation pulses (commonly nanoseconds to microseconds in metals), the lattice temperature can continue to rise and generate a strong thermal index gradient in the film, or even damage it.142,145 Thus, ENZ materials cannot be driven fast without corrective action. As such, dissipating thermal energy quickly and lowering the irradiance threshold are important frontiers to being able to push the boundaries of ENZ materials in terms of speed.

2. Interband absorption

If the excitation energy is larger than or equal to the bandgap ( ω E g) interband absorption occurs. This results in the formation of e–h pairs, modifying the carrier density in the material until the carriers recombine. The density of carriers can be tracked in time according to the two-level rate equation
d N d t = G t Δ N e t / τ n ,
(11)
where G ( t ) is the carrier generation rate, τ n is the characteristic relaxation time of the carriers (typically on the order of picosecond to nanosecond scale, although it can be smaller in the presence of significant defects), and Δ N is the difference in electron density between the excited state and the steady state.28,146,147
For interband transitions, the number of free carriers can change as well as the overall energy. In this case, the conservation equations governing energy and number of carriers in the system are given by Secondo et al.148 
f E , μ I , T e I E ρ E d E f E , μ 0 , T e 0 E ρ E d E = δ U abs δ N μ + δ N ρ μ ,
(12a)
N = N 0 + δ N = 0 f E , μ , T e ρ E d E ,
(12b)
where δ N is the produced excess electron density and N 0 is the base electron density. Also, in this case, the density of additional carriers must be related to absorbed energy density of the pump.
Based on Eqs. (1) and (2), an increase in the density of free carriers will lead to an increase in plasma frequency and corresponding decrease in the permittivity of the film in the ENZ region. The effective IDRI coefficient for interband can then be equated to Refs. 40 and 48,
n 2. eff ω p = q 2 τ α 0 2 ε n ω p m avg * ω 3 .
(13)
Finally, it is important to note a few additional effects that can arise under interband excitation. First, when pumping above the Fermi level ( ω E F > 0.1 eV), photon generated carriers will initially occupy states above the Fermi energy but will quickly scatter to equilibrate with the rest of the electron sea near the Fermi-level. In this scenario, both the free carrier density and effective mass can be modified since any energy that is larger than the bandgap would translate to an increase in the electron temperature. These processes tend to have opposed effects on the permittivity and therefore should be simultaneously considered by combining the descriptions above. Second, strong interband excitation can also modify the bandgap through renormalization, (due to Coulombic interaction between the charged particles) and the Moss-Burstein shift (due to band filling as carriers populate the lowest available states). These processes tend to spectrally broaden the absorption edge and modify its amplitude. When modeling the permittivity of the doped material via Drude–Lorentz oscillators which include the band edge, these effects alter the properties of the Lorentzian term [equivalent to variation in ε in Eq. (1)], which in turn affect the permittivity in the ENZ spectral region. While similar effects have been explored for crystalline semiconductors such as Si and GaAs,150–152 physical models of these processes for common TCO materials such as doped ZnO and doped Sn2O3 are currently lacking in the literature and provide an opportunity to significantly improve interband modeling.

Numerous studies have demonstrated, TCOs such as doped ZnO, doped In2O3 and CdO, exhibit large and rapid intensity dependent index changes.28,39,95,97,129,153 Building from works of Chang et al.154 on nonlinear effects in oxide nanowires as well as ENZ effects in thin films and plasmonic systems, Kinsey et al.28 studied pumping a highly doped AZO film in the UV to generate free carriers in the conduction band [Figs. 9(a) and 9(b)]. The authors observed the index at crossover to change from 0.5 + 0.25i to 0.33 + 0.45i. Moreover, the recombination time was found to be ≈ 100 fs, which was considerably faster than that typically exhibited by compounds such as Si and LT-GaAs (>10 ps).155,156 This was attributed to defect states within the bandgap that facilitated recombination,98,157 like in the case of low-temperature-grown (LT) GaAs.158,159 Subsequently, a similarly large nonlinearity was observed in ITO by Alam et al.158,159 This involved degenerate pumping of the film in the NIR to generate hot carriers. Since the pump was also positioned in the ENZ region, enhanced absorption was observed which led to a record index change of 0.7 (at off-normal angle of incidence). This was followed by a study by Clerici et al.,129 who utilized both inter-band and intra-band pumps to generate opposing effects on the plasma frequency [Fig. 9(c)]. By appropriately spacing the two pumps in time and space, unique and ultra-fast effects were realized. Some of the more recent works have expanded such studies to Ga-doped ZnO and CdO [Fig. 9(d)].148,149

FIG. 9.

(a) and (b) Al:ZnO film pumped in the UV showing transient reflection and transmission.28 Reproduced with permission from Kinsey et al., Optica, 2(7), 616 (2015). Copyright 2015 Optical Society of America. (c) Two color excitation of an Al:ZnO film showing measured wavelength shifts with varied UV and IR excitation pulses.129 Reproduced with permission from Clerici et al., Nat. Commun. 8, 15829 (2017). Copyright 2017 Authors, licensed under a Creative Commons Attribution Unported License. (d) Doped CdO film change in optical density vs probe wavelength and delay time.149 Reproduced with permission from Yang et al., Nat. Photonics 11(6), 390–395 (2017). Copyright 2017 Springer Nature Limited.

FIG. 9.

(a) and (b) Al:ZnO film pumped in the UV showing transient reflection and transmission.28 Reproduced with permission from Kinsey et al., Optica, 2(7), 616 (2015). Copyright 2015 Optical Society of America. (c) Two color excitation of an Al:ZnO film showing measured wavelength shifts with varied UV and IR excitation pulses.129 Reproduced with permission from Clerici et al., Nat. Commun. 8, 15829 (2017). Copyright 2017 Authors, licensed under a Creative Commons Attribution Unported License. (d) Doped CdO film change in optical density vs probe wavelength and delay time.149 Reproduced with permission from Yang et al., Nat. Photonics 11(6), 390–395 (2017). Copyright 2017 Springer Nature Limited.

Close modal

With an understanding of the nonlinear processes in Drude-based ENZ materials, we can revisit the description of intrinsic and extrinsic effects to place the benefits of ENZ materials in context. Here, we will focus the discussion on the slow intraband nonlinearities that modify the index as they are the most often investigated in the literature.

Figure 10(a) shows examples of the effective intensity dependent refractive index coefficient, n 2 , eff, measured for popular ENZ materials, indexed by the respective conditions of absence, separate presence, or joint presence of the excitation and probe optical beams. Values included here are selected to maintain parameters such as excitation pulse width, repetition rate, and normal incidence, while allowing variation in other conditions that are key to the comparison, namely, the probe and excitation wavelengths.

FIG. 10.

(a) Effective intensity dependent refractive index coefficient, n2.eff, for popular materials showing intrinsic enhancement from group index when pumping and probing at the ENZ condition, compared to that of typical semiconductors Si and GaAs. Values for this plot are denoted in table S1 of the supplementary material section. The labels on the x-axis denote whether the pump and/or probe beams are within the group index enhancement region, expressed as 1±0.1λ/λENZ. (b) Group index of a typical ENZ material normalized to the ENZ wavelength. A sharp enhancement region is denoted by the shaded region where pumping or probing in this region would yield a 3–5 times enhancement of the nonlinearity as shown in (a).

FIG. 10.

(a) Effective intensity dependent refractive index coefficient, n2.eff, for popular materials showing intrinsic enhancement from group index when pumping and probing at the ENZ condition, compared to that of typical semiconductors Si and GaAs. Values for this plot are denoted in table S1 of the supplementary material section. The labels on the x-axis denote whether the pump and/or probe beams are within the group index enhancement region, expressed as 1±0.1λ/λENZ. (b) Group index of a typical ENZ material normalized to the ENZ wavelength. A sharp enhancement region is denoted by the shaded region where pumping or probing in this region would yield a 3–5 times enhancement of the nonlinearity as shown in (a).

Close modal

Focusing first on the condition where both pump and probe are ENZ-detuned, the measurements illustrate intrinsic free-carrier nonlinearities of TCOs that are comparable in strength to the free-carrier nonlinearities of conventional semiconductors (e.g., Si and GaAs) within the same spectral range. These values are  3 orders of magnitude larger than those of bound electronic nonlinearities of common reference materials such as SiO2 or Al2O3, but this is expected (see Sec. II) due to the shift from below-gap bound electronic processes (virtual transitions) to above-gap free-carrier processes (real transitions). When one beam is placed near the permittivity crossover of the film, a rise in the average value of the nonlinearity is observed. This can be understood by examining Eq. (10). Here, we note that all terms remain approximately constant with respect to the excitation/probe wavelength except n g and n ω p. As a result, we can expect that placing one of the optical beams within the ENZ region would provide an improvement of n g ω p (note that at ENZ, 1 / n ω p n g ω p).110,148,160 As illustrated in Fig. 10(b), the value of n g is in the range of 3–5 for the transparent conducting oxides.18,58 and corresponds quite well to the increase in nonlinearity observed in experiments [Fig. 10(a)]. From the same argument, when the second beam is located near the permittivity crossover of the film, another rise in the nonlinear coefficient is observed that is again proportional to n g.

As a result, ENZ in TCO materials at normal incidence can be viewed as facilitating two simultaneously occurring features: (1) an intrinsic excitation of stronger free-carrier nonlinearities in the near infrared spectral region as opposed to excitation of weaker below-gap virtual transitions, had the ENZ wavelength not been present in NIR spectrum; (2) an extrinsic slow-light enhancement wherein an enhancement factor of up to n g 2 is introduced. The result, when compared to Si and GaAs that utilize free-carrier effects in the same spectral window, is an overall improvement in the nonlinearity of one–two orders of magnitude ( n 2 , eff of 10 5 to 10 3 cm2 GW−1 depending on the exact material and experimental conditions161), predominantly due to the added dispersion and slow-light effects achieved by introducing the ENZ condition. In this case, the slow-light effects provide two distinct enhancements: (1) to the absorptivity of the film—which facilitates more efficient use of the excitation intensity, (2) to the spectral sensitivity of index—which maximizes the observed change in index for a given shift of the crossover frequency [see Eq. (10) and Fig. 10(b)]. These effects can be utilized separately or in combination by appropriately defining the operational wavelengths.

It is important to reiterate that while Drude-based ENZ does not radically improve the intrinsic free-carrier nonlinearity of the material, it achieves enhancement through the introduction of slow light effects. This extrinsic effect is bound by the typical trade-off between strength and bandwidth, wherein a steep dispersion (e.g., high n g) can only be achieved in a limited spectral range. As the ENZ condition in most TCOs is relatively broadband (  200 nm), it therefore provides only a moderate enhancement ( n g n / ω) to the nonlinearity of the film, as shown in Fig. 10(b). As a result, large irradiance values (>10 GW cm−2) are still required to achieve the strong index modulations on the scale of Δ n  0.1–1 reported in the literature.47,97,154,162–165 Although the broad enhancement is beneficial, unfortunately it necessitates the introduction of loss within the film via the introduction of a high density of free carriers. While this impacts the efficiency for transmissive applications, the loss in most popular TCOs remains sufficiently low ( Im n  0.1–0.5 at crossover)18 such that sub-micron films can achieve transmissivity values of 80% and higher. More critically, under extreme irradiances or high-repetition rates, the energy absorbed through free-carrier processes leads to a collective increase in the lattice temperature of the material. This can produce a slow (1  μs to 1 ms scale) and strong thermal index effect that limits the potential speed of operation and can lead to material degradation.166–169 Thus, while the nonlinearity in Drude ENZ relaxes on the scale of picosecond, operating films at a high-repetition rate requires the mitigation of the thermal energy buildup either through the reduction in the required irradiance or through thermal engineering; this remains an open area of study.

Despite these limitations, Drude-based ENZ materials offer several key benefits. Primarily, ENZ materials offer the ability to simultaneously employ intrinsic and extrinsic enhancements over a broad spectral range naturally, without requiring nanofabrication. Moreover, the materials employed are well-developed, can be readily tuned during deposition to meet specific requirements, and can be integrated with various device platforms.68,98,116,170–176 Additionally, the TCOs have been shown to exhibit quite impressive damage thresholds in excess of 1 TW cm−2 for thin films.32,134 Thus, while the nonlinearity may not be significantly enhanced, the damage threshold is so high that ENZ materials unlock the potential of extreme refractive index changes on the scale of unity that are not possible with typical semiconductors or metals.

To reduce the irradiance required to achieve a desired nonlinear response, such as strong index shift or frequency conversion, boosting the extrinsic enhancement of ENZ films has been pursued. As noted in Sec. II, these techniques seek to employ structure or specific excitation schemes to improve confinement, generate strong dispersion in reflection, transmission, and absorption, interaction length, or interaction time, providing a multiplier for the intrinsic nonlinearity of the constituent materials. For homogeneous ENZ films, both the fast and slow nonlinearity can benefit from these effects. Researchers have explored the prospect of coupling metasurfaces to ENZ films,43,44,70,75,177,178 by utilizing highly confined modes such as the ENZ or Berreman mode,65,66,69,149,179,180 or by placing ENZ structures inside cavities.106,181–183 Ultimately, by adding structure to a bulk film one can modify the material dispersion and light confinement with additional resonances or anisotropy.

One of the early approaches to achieving this was by tuning highly confined modes in thin homogeneous films such as the ENZ and Berreman modes.65,66,187 These modes, often excited by tuning the incident angle, or by providing the necessary momentum using gratings, lead to a sharp and strong absorption resonance that can effectively modulate properties like transmission and reflection. As shown in Fig. 11, both modes are characterized by flat dispersion and low group velocity, resulting in slow light and enhanced absorption within the thin films. While both these modes exhibit similar effects, they often require different coupling conditions. Since the ENZ mode is a high-momentum, non-radiative mode, prism/grating couplers are needed to provide the momentum necessary for mode excitation. On the other hand, the Berreman mode lies to the left of the light line, is radiative in nature and can be excited from free space. Additionally, Berreman modes have been heavily focused in ENZ materials for showing enhanced nonlinearities at higher angles of incidence.65,66,121 Perfect absorption can indeed be achieved by these films,179 which requires specific properties and geometric constraints, leading to improved nonlinear index tunability.

FIG. 11.

ω–k dispersion curve for a 20 nm thick AZO film (representative ENZ material). The curve is highly dispersive in the vicinity of the ENZ spectral region. As a result, group velocity dω/dk tends to 0. Depending on the position of the mode in the momentum space, either a Berreman mode or and ENZ mode can be excited. While both these modes result in slow light, only the Berreman mode can be coupled to from free space. The ENZ mode on the other hand requires gratings or prisms to impart the additional momentum required for excitation.

FIG. 11.

ω–k dispersion curve for a 20 nm thick AZO film (representative ENZ material). The curve is highly dispersive in the vicinity of the ENZ spectral region. As a result, group velocity dω/dk tends to 0. Depending on the position of the mode in the momentum space, either a Berreman mode or and ENZ mode can be excited. While both these modes result in slow light, only the Berreman mode can be coupled to from free space. The ENZ mode on the other hand requires gratings or prisms to impart the additional momentum required for excitation.

Close modal

To broaden the design space to improve nonlinearities, researchers also considered utilizing nanoantenna structures coupled to homogeneous ENZ layers. For example, frequency conversion was achieved using a structured medium consisting of titanium nitride (TiN) split ring resonator arrays [Fig. 12(a)]. The TiN nanoantennas exhibited a double resonance effect in which the SHG was enhanced due to the simultaneous presence of a plasmonic resonance at the excitation frequency, and an ENZ resonance at the second harmonic frequency.184 Counteracting the enhancement from the dual resonance, was the relatively large optical losses of TiN. Since TCOs are known to exhibit lower losses than metals at their crossover wavelength, J. Deng et al.70 utilized gold nanoantenna coupled to a 15 nm thick ITO film. They demonstrated a 104 fold enhancement of SHG by coupling into the ENZ mode. More information on harmonic enhancement observed in ENZ media can be found in the report by Solis et al.188 

FIG. 12.

(a) SEM image of TiN split-ring resonators (SRR) on a sapphire substrate. To enhance SHG, this structure combines a plasmonic resonance at the pump frequency and ENZ resonance of TiN at the SHG frequency.184 Reproduced with permission from Wen et al., ACS Photonics 5(6), 2087–2093 (2018). Copyright 2018 American Chemical Society. (b) SEM image of 27 nm thick gold dipole nanoantenna, on a 23 nm thick ITO coated glass substrate. The ENZ resonance coincides with the plasmonic resonance of the gold nanoantenna, leading to strong coupling. As a result of modified dispersion and improved coupling of free space light, values for n2,eff as high as  3.5 cm2/GW are reported.44 Reproduced with permission from Alam et al., Nat. Photonics 12(2), 79–83 (2018). Copyright 2018 Authors, licensed under a Creative Commons Attribution Unported License. (c) Graph showing the rotated angle of the polarization azimuth as a function of irradiated fluence. The structure consists of rectangular gold nanoantenna on a 310 nm thick ITO film. Due to the anisotropic response of the system, p-polarized light is selectively transmitted with a phase delay based on the pump fluence, whereas s-polarized light is transmitted irrespective of pump fluence.185 Reproduced with permission from Wang et al., ACS Photonics 8(9), 2791–2799 (2021). Copyright 2021 American Chemical Society. (d) Schematic of an ultra-fast THz wave generator. The device consists of gold SRRs coupled to a  20 nm thick ITO film on a glass substrate. Upon resonant pumping with NIR femtosecond laser pulses. this metasurface yields a peak THz intensity that is enhanced by a factor of  1 × 104 times compared to that of a similar metasurface for which the ITO film is absent.186 Reproduced with permission from Lu et al., Nano Lett. 21(18), 7699–7707 (2021). Copyright 2021 American Chemical Society. (e) Schematic of a high-Q perfect absorber based on a simulated doped CdO film coupled to a Bragg microcavity. The dimensions are: tSi = 291 nm, tSiO2 = 714, tCdO = 10 nm, and tf = tb = 586 nm. While the high Q resonance theoretically yields a large ΔR (94%), the bandwidth is limited to  10 nm.181 Reproduced with permission from Hu et al., Opt. Exp. 27(19), 26405 (2019). Copyright 2019 Optical Society of America.

FIG. 12.

(a) SEM image of TiN split-ring resonators (SRR) on a sapphire substrate. To enhance SHG, this structure combines a plasmonic resonance at the pump frequency and ENZ resonance of TiN at the SHG frequency.184 Reproduced with permission from Wen et al., ACS Photonics 5(6), 2087–2093 (2018). Copyright 2018 American Chemical Society. (b) SEM image of 27 nm thick gold dipole nanoantenna, on a 23 nm thick ITO coated glass substrate. The ENZ resonance coincides with the plasmonic resonance of the gold nanoantenna, leading to strong coupling. As a result of modified dispersion and improved coupling of free space light, values for n2,eff as high as  3.5 cm2/GW are reported.44 Reproduced with permission from Alam et al., Nat. Photonics 12(2), 79–83 (2018). Copyright 2018 Authors, licensed under a Creative Commons Attribution Unported License. (c) Graph showing the rotated angle of the polarization azimuth as a function of irradiated fluence. The structure consists of rectangular gold nanoantenna on a 310 nm thick ITO film. Due to the anisotropic response of the system, p-polarized light is selectively transmitted with a phase delay based on the pump fluence, whereas s-polarized light is transmitted irrespective of pump fluence.185 Reproduced with permission from Wang et al., ACS Photonics 8(9), 2791–2799 (2021). Copyright 2021 American Chemical Society. (d) Schematic of an ultra-fast THz wave generator. The device consists of gold SRRs coupled to a  20 nm thick ITO film on a glass substrate. Upon resonant pumping with NIR femtosecond laser pulses. this metasurface yields a peak THz intensity that is enhanced by a factor of  1 × 104 times compared to that of a similar metasurface for which the ITO film is absent.186 Reproduced with permission from Lu et al., Nano Lett. 21(18), 7699–7707 (2021). Copyright 2021 American Chemical Society. (e) Schematic of a high-Q perfect absorber based on a simulated doped CdO film coupled to a Bragg microcavity. The dimensions are: tSi = 291 nm, tSiO2 = 714, tCdO = 10 nm, and tf = tb = 586 nm. While the high Q resonance theoretically yields a large ΔR (94%), the bandwidth is limited to  10 nm.181 Reproduced with permission from Hu et al., Opt. Exp. 27(19), 26405 (2019). Copyright 2019 Optical Society of America.

Close modal

For refractive index modulation, efforts have primarily involved the use of ITO and gold.44,125,185,189 Alam et al.44 reported a gold dipole antenna array on top of a 23 nm thick ITO layer [Fig. 12(b)]. The nanoantennas were designed such that their resonance overlapped with the ENZ point of the ITO film at 1420 nm, providing a 50 fold enhancement of peak intensity. Additionally, Wang et al.185 showed that anisotropy can be modified in these materials with Au nanoantennas on an ENZ layer [Fig. 12(c)]. The temporal response displayed a change in polarization of a probe beam due to pump absorption aided by the nanoantennas. Yu et al.186 showed enhanced terahertz (THz) conversion efficiency when using ITO split ring resonance [Fig. 12(d)]. Finally, researchers also considered embedding these films within cavities, such as in the work by Hu et al.181 To improve the Q-factor of the broad ENZ resonance, a Dy:CdO film was embedded in a Brag microcavity [Fig. 12(e)]. While the study achieved a large ΔR of 94% at 7 μJ/cm2 of pump fluence, it came at the cost of a greatly reduced bandwidth (<10 nm). Other, more exotic approaches to enhancing ENZ nonlinearity have consisted of exciting whispering gallery modes in dielectric nanospheres coated with ITO,190 or achieving momentum matching between light and thin-film ENZ modes using a prism.97 

When fixing key output quantities, such as change in reflection/transmission or conversion efficiency, adding extrinsic enhancement via structure to a homogeneous ENZ film enabled researchers to reduce the required irradiance. This lessens the impact of the drawbacks of homogeneous ENZ materials, namely, optical loss and subsequent thermal dissipation, while providing a larger design space through which to optimize the response. The drawback of these approaches comes in the form of a limited bandwidth (e.g., introduction of resonance) and fabrication complexity, and thus typically must be tailored to a specific application or operation condition.

In addition to homogeneous materials, vanishing permittivity can also be achieved by engineering an effective medium based on structural dispersion where the composite medium exhibits ENZ response as a whole, rather than exhibiting an ENZ condition for each individual constituent material.160 These intrinsically structured systems provide access to new spectral regions and more freedom in design to meet specific application conditions when compared to homogeneous materials. An important difference between structured and natural homogeneous systems, in the context of ENZ interactions, is that structured materials exhibit their optical effects as an average across several unit cells, while naturally occurring materials do so at the nanoscale. Thus, optical interactions in structured ENZ may differ greatly between the wavelength and sub-wavelength scales.

Perhaps one of the most straightforward methods to achieve an effective ENZ condition is through the formation of alternating layers of metal and dielectrics. The resulting permittivity is derived from effective medium theory (EMT) according to Ref. 4,
ε = ε m d m + ε d d d d m + d d and 1 ε = d m ε m + d d ε d d m + d d ,
(14)
where ε and ε are the effective permittivity components perpendicular and parallel to the plane, and dm (dd) and εm (εd) are the thickness and dielectric constant of the metallic (dielectric) layer, respectively. A judicious mixture of materials with opposite signs of permittivity (i.e., positive dielectric and negative metallic materials) and subwavelength thicknesses will act effectively as an ENZ medium while offering flexibility in the design process to tune the ENZ crossover across the spectrum. Implementations in various forms such as lamellar sheets, rods and inclusions,14,192–194 have been shown to generate exceptionally strong anisotropy in the permittivity by achieving ENZ only along certain axes, even achieving the hyperbolic dispersion [Fig. 13(a)]—in other words a hyperbolic metamaterial (HMM).85 Note that, due to the large mismatch between the dielectric and metallic permittivity, very thin uniform layers of metal are often needed—which can be challenging to realize, especially in the case of noble metals.195 Yet, the epitaxial growth of ultrathin (  5 nm) ultra smooth layers has been reported as a viable approach to reduce loss and achieve better optical performance.196 
FIG. 13.

(a) Anisotropic optical response of Ag/TiO2 stack metamaterial calculated using effective medium theory, with a 35% fill ratio of Ag. (b) Degenerate electric monopole, longitudinal magnetic dipole and transverse magnetic dipole induce a Dirac-like material response resulting in effective permittivity and permeability crosses zero at the degeneracy point between the 3 resonances. Adopted from Ref. 191. (c) Waveguide near cutoff, achieving ENZ condition.

FIG. 13.

(a) Anisotropic optical response of Ag/TiO2 stack metamaterial calculated using effective medium theory, with a 35% fill ratio of Ag. (b) Degenerate electric monopole, longitudinal magnetic dipole and transverse magnetic dipole induce a Dirac-like material response resulting in effective permittivity and permeability crosses zero at the degeneracy point between the 3 resonances. Adopted from Ref. 191. (c) Waveguide near cutoff, achieving ENZ condition.

Close modal

Another approach to achieve an effective zero permittivity it to utilize Dirac-like cone dispersion-based metamaterials. These Dirac cone zero index metamaterials (DCZIM) are often all dielectric in nature.191,197 They are formed by the accidental threefold degeneracy of two linear bands (conical dispersion) and a quadratically dispersive band, occurring at the center of the Brillouin zone. Because of this degeneracy, the group velocity of the two linear bands changes signs, implying that the effective index crosses 0. Furthermore, the effective permittivity and permeability also cross zero [Fig. 13(b)], which can be shown via rigorous effective medium homogenization approaches. Accordingly. these metamaterials are also referred to as epsilon-mu-near-zero (EMNZ). As an example of a DCZIM, Li et al. demonstrated a square lattice of Si nanopillars embedded in a SU-8 (see disclaimer section) matrix.191 The presence of spatially and spectrally degenerate electric monopole, transverse magnetic dipole, and longitudinal magnetic dipole resulted in Dirac-like cone photonic band structure [Fig. 13(b), inset]. While the zero effective ε and μ aids in impedance matching, these structures do not exhibit slow light enhancement, critical to optical nonlinearities. Due to the linear dispersion, DCZIMs are characterized with a finite group velocity at the frequency where effective ε and μ properties vanish.56 As a result, DCZIMs have largely been used in the demonstration and manipulation of linear optical effects as their lack of slow light largely precludes enhancement to nonlinear processes.191,197

Yet another approach is to take advantage of dispersion engineering with the help of a composite metal-dielectric waveguide. For example, consider a rectangular metallic waveguide.198 For the waveguide supporting the fundamental transverse electric TE10 mode, the propagation constant is expressed as
β = ( 2 π n f / c ) 2 ( π / w ) 2 ,
(15)
where w is the width for the corresponding magnetic field, n is relative refractive index of the plasma, c is the speed of light in vacuum, and f is the operating frequency. The relative effective index for the TE10 guided mode can be expressed as199,
ε eff / ε 0 = n 2 c 2 / ( 4 f 2 w 2 ) .
(16)
As implied by Eqs. (15) and (16), when the waveguide is operated at the cutoff frequency of its TE10 mode (i.e., β = 0, indicating an infinite phase velocity), the waveguide appears as an effective medium having effective zero permittivity [Fig. 13(c)].198,199 In the optical regime, a nanoscale plasmonic waveguide with hollow dielectric core surrounded by metallic cladding could mimic an ENZ medium when operated at the cut off frequency of its fundamental TE10 mode. The propagation constant for the resulting mode becomes zero. Such structures are quite challenging to realize and thus far have not been extensively explored for their nonlinear properties, although energy squeezing and tunneling of electromagnetic waves have been experimentally realized.199,200

Among the techniques mentioned above, achieving an effective ENZ condition using metal-dielectric composites has been the most widely used technique to modify nonlinear optical interactions. Such metamaterials have been successful in extending the ENZ and NZI condition into the visible spectral range (500–1000 nm)200,201 where existing homogeneous ENZ materials are too lossy to exhibit NZI and the strong index dispersion needed for slow-light enhancement (see the review by Kinsey et al.18).

Rashed et al.194 reported a multilayer metamaterial sample composed of 4 alternating layers of 16 nm of Au and 32 nm of TiO2, resulting in a 605 nm ENZ wavelength, along with 30% and 10% modulation of the transmitted (ΔT/Tlin) and reflected (ΔR/Rlin) probe signals, respectively. The total increase in Δεeff,real ≈ 0.16 (420% change) and the temporal nonlinear dynamics related to the relaxation of hot electrons was investigated by a two-temperature model (2TM). To further confirm the enhancement of the nonlinear optical response in ENZ metamaterials, Suresh et al.201 fabricated a sample consisting of five bilayers of Ag and SiO2 (with respective thicknesses of 16 and 65 nm) that resulted in an ENZ wavelength of 509 nm [Fig. 14(a)]. The experimental results showed the advantages of an ENZ metamaterial over a bulk metallic thin film, achieving an effective nonlinearity of n2,eff = (1.2 ± 0.1) × 10−12 m2 W−1 and lower loss βeff = (−1.5 ± 0.2) × 10−5 m W−1. Subsequently, Genchi et al.202 demonstrated that the nonlinear optical parameters of Au/Al2O3 ENZ multilayer metamaterials could be continuously modulated in a broad range of the visible spectrum at different incidence angles with transverse electric (TE) or transverse magnetic (TM) polarized light.

FIG. 14.

(a) Effective parallel permittivity and cross-sectional image of the fabricated Ag-SiO2 multilayer stack.201 Reproduced with permission from Suresh et al., ACS Photonics 8(1), 125–129 (2021). Copyright 2021 American Chemical Society. (b) Sketch of ITO/SiO2 multilayer, the permittivity and the optical phase diagram.208 Reproduced with permission from Xie et al., Opt. Express 30(12) 21634–21648 (2022). Copyright 2022 Optical Society of America. (c) Cross view images of the HMM sample, the angular-wavelength transmission spectrum and the spectra of the calculated effective permittivity tensor components.204 Reproduced with permission from Kolmychek et al., Opt. Lett. 47(22), 6009 (2022). Copyright 2022, Optical Society of America. (d) The proposed bistable all-optical device based on an ENZ ITO waveguide and corresponding simulated bistable switching as a function of optical power.209 Reproduced with permission from Gosciniak et al., Laser Photonics Rev. 17(4), 2200723 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.

FIG. 14.

(a) Effective parallel permittivity and cross-sectional image of the fabricated Ag-SiO2 multilayer stack.201 Reproduced with permission from Suresh et al., ACS Photonics 8(1), 125–129 (2021). Copyright 2021 American Chemical Society. (b) Sketch of ITO/SiO2 multilayer, the permittivity and the optical phase diagram.208 Reproduced with permission from Xie et al., Opt. Express 30(12) 21634–21648 (2022). Copyright 2022 Optical Society of America. (c) Cross view images of the HMM sample, the angular-wavelength transmission spectrum and the spectra of the calculated effective permittivity tensor components.204 Reproduced with permission from Kolmychek et al., Opt. Lett. 47(22), 6009 (2022). Copyright 2022, Optical Society of America. (d) The proposed bistable all-optical device based on an ENZ ITO waveguide and corresponding simulated bistable switching as a function of optical power.209 Reproduced with permission from Gosciniak et al., Laser Photonics Rev. 17(4), 2200723 (2023). Copyright 2023 Authors, licensed under a Creative Commons Attribution Unported License.

Close modal

In addition to stacked metal-dielectric layers, embedding metallic pillars into a positive permittivity dielectric can also result in an ENZ condition. Neira et al.203 electrodeposited Au into nanoporous AAO (anodized aluminum oxide) templates on a glass substrate to achieve a plasmonic nanorod metamaterial sample with the ENZ wavelength around 600 nm. The nanorod metamaterial geometry provided more than two orders of magnitude larger intensity dependent index nonlinearity than the interband nonlinearity of the plasmonic metal itself at any designed wavelength. Next, Kolmychek et al.136,204 demonstrated an order of magnitude enhancement of second-harmonic generation (SHG) for p-polarized fundamental wavelength of 820 nm at the angle of incidence of 15°136 and self-focusing/defocusing204 [Fig. 14(c)] effects in ordered arrays of Au nanorods in porous aluminum oxide. The observed phenomena were tightly associated within the spectral vicinity of the ENZ wavelength where the transition occurred from the elliptic to hyperbolic dispersion and the maximum |n2,eff| and |βeff| were, respectively, 4 × 10−13 m2 W−1 and 10−6 m W−1, for an incidence angle of 30°.

Although the prevailing ENZ metamaterials are polarization dependent, they can also be designed to be polarization insensitive. A free-standing metal-dielectric fishnet nanostructure that had polarization- insensitive NZI properties was demonstrated by Yun et al.205 By optimizing the metal-dielectric fishnet structure to simultaneously balance the electric and magnetic responses, both the effect permittivity and permeability were close to zero at 1.55 μm, leading to produce a zero index and matched impedance to free space. It should be noted that these structures do not always possess slow light effects which are crucial for enhanced nonlinearities, due to a lack of rapid dispersion.

Furthermore, a metal-insulator-metal (MIM) cavity with engineered effective permittivity exhibited multiple ENZ points and could simultaneously increase absorption and the Purcell factor at the excitation and emission wavelengths, respectively.183 These structures can be readily designed to exhibit ENZ wavelengths which are responsive to both TE- and TM-polarized excitation. Additionally, their spectral positions can be adjusted by controlling the refractive index and thickness of the embedded dielectric. On this basis, Kuttruff et al.206 demonstrated an all-optical,  3 ps switching of the reflectance of a MIM nanocavity approaching a relative modulation depth of 120% in the VIS-NIR spectral range.

In addition to the metal-dielectric stack structure, all-dielectric metamaterials can also enhance the nonlinearity while opening a route to tailor the loss. Although not ENZ in the passive state, negative permittivity and free-carrier losses can be optically induced, allowing a route to control the impact of loss while using structure to further enhance the nonlinearity. However, the corresponding ENZ wavelength is generally located in the IR range. Kelly and Kuznetsova207 numerically studied the pump-probe ultrashort pulse modulation in the AZO/ZnO metamaterial whose ENZ wavelength was 1778 nm. While the structure was dielectric in the passive state, UV pumping the metamaterial introduced free carriers and resulted in an ENZ condition at shorter wavelengths. At the permittivity crossover, the amplitude and FWHM duration of the pump pulse were the main factors affecting ultrashort probe pulse propagation. The short pump pulse (100 fs) with the highest investigated pump pulse amplitude (restricted by the damage threshold) caused a rapid and significant change in optical permittivity, leading to a highly asymmetric shape of the output probe pulse. Additionally, Xie et al.208 theoretically investigated the ultrafast dynamic switching of optical response using tunable hyperbolic metamaterial (HMM) that consisted of 5 pairs of alternating layers of ITO and SiO2 [Fig. 14(b)]. The proposed HMM platform showed the largest intensity dependent index nonlinearity ( n 2 , eff ≈ 5.87 × 10−16 m2 W−1 and β eff ≈ −1.27 × 10−9 mW−1) at an ENZ wavelength of 1454 nm, higher than that of the bulk ITO.

Beyond ENZ metamaterials, the ENZ and NZI condition can also be realized using the structural dispersions of waveguides. With this approach, a nonlinear plasmonic waveguide with effective ENZ condition was designed by Li and Argyropoulos207 to achieve efficient tunable nonlinear coherent perfect absorption response at the nanoscale. In this case, the local electric field at the ENZ coherent perfect absorption (CPA) point was intense (≈ 35 times stronger) and uniform. This led to the prediction that nonlinearities such as the intensity dependent index could be improved and ultimately triggered by reduced input intensities.210 Also, as shown in Fig. 14(d), Gosciniak et al.209 proposed a bistable resonator-free all-optical device based on ENZ ITO material in a slot waveguide. A bistable nonlinear activation function, developed based on this configuration, showed a prediction accuracy of 98% for MNIST (Modified National Institute of Standards and Technology), 89% for Fashion MNIST, and ≈52% for CIFAR-10 (Canadian Institute for Advanced Research) datasets.

To conclude this section, the formation of an effective ENZ condition via structured systems is a collective property and is of course dependent upon the constituent materials and their structure. This can be designed to provide extrinsic enhancements via slow light effects and confinement, like in the case of homogeneous ENZ materials. Yet, the intrinsic nonlinearities, and their subsequent process (e.g., virtual/real), strength, and relaxation characteristics, are founded in the properties of the constituent materials used to form the structure. The effective nonlinearity can thus be viewed as a combination of nonlinear processes occurring within the constituent materials ( χ o ( 1 , 2 ) 3), weighted by the ratio of the electric field overlap with, and multiplied by an extrinsic enhancement due to the structural response [ Δ ε eff E F ( χ o , 1 3 E 1 2 + χ o , 2 3 E 2 2 ) ]. As an example, a lamellar composite of Ag and SiO2 can be engineered to form an effective ENZ condition in the visible spectral range, despite neither constituent material being ENZ. The nonlinear processes ( χ o 3 ) will be equivalent to the nonlinear processes in the bulk versions of the constituent materials at the same excitation wavelength, a free-carrier Fermi-smearing nonlinearity for Ag and a virtual nonlinearity for SiO2. Upon excitation, an incident field will then be distributed within the material, leading to absorption within Ag (which drives the free-carrier Fermi-smearing nonlinearity), and propagation within SiO2 (which drives a virtual Kerr process). The distribution of this field can be altered by the structure or excitation condition, enabling coupling to confined plasmonic modes on thin silver films, while also facilitating an overall slow light effect due to the effective material dispersion. Relaxation of the system would then be driven by thermal relaxation within the Ag films while the Kerr process is instantaneous. Although this discussion is somewhat obvious, we find it a useful picture. Only observing the material in the homogenized view blurs the physical insight into the nonlinearities at play and preludes key points through which the response may be tailored. For example, utilizing materials with higher intrinsic nonlinearities would of course be beneficial, while engineering field confinement can be used to focus light onto lower loss materials or higher nonlinearity materials, while still utilizing an overarching slow light effect provided through an effective ENZ condition. Unlike homogeneous materials, structured approaches provide significant flexibility to engineer these conditions through an expanded material pallet as well as geometric design that can be particularly powerful at improving the strength of the nonlinearity within the constraints of the intrinsic and extrinsic effects being employed. However, this comes at the cost of increased fabrication complexity and excitation conditions. This trade-off should be weight based on the intended application.183 More information on nonlinearities in structured materials, can be found in Refs. 201 and 211–216.

With an understanding of the origins of nonlinearities in ENZ materials as well as how they can be tailored by various techniques, we can now turn our attention to a consideration of the opportunities that ENZ materials provide. This section will focus on two aspects: (1) the potential to explore new materials and balance performance tradeoffs and (2) the potential use of the large and fast index contrast to study space-time optical effects.

As seen from Table S1, and recent demonstrations, most of the studies on homogeneous ENZ nonlinearities have been limited to three materials, doped In2O3, ZnO, and CdO.80,98,102,148,149,217,218 This is largely because degenerately doped ITO is readily commercially available219 and high solubility of Al in ZnO220 leads to straightforward sputtering221,222 and pulsed laser deposition processes.223,224

While these films have led to demonstrations of exceptionally large and ultrafast intensity-dependent index nonlinearities at ENZ frequencies, the high optical pump power requirements have posed a significant challenge to practicality. This naturally leads to the question—are there materials better suited to exhibit larger or more efficient nonlinearities around the technologically relevant wavelengths?

Considering that TCOs have been researched for several decades225–227 and tailored for specific applications such as flat panel displays,228 solar cells229,230 and LEDs,231 it would be short sighted to dismiss the possibility of materials better suited for ENZ nonlinearities.18 This section of the review is aimed leveraging the understanding of nonlinearities in ENZ materials to highlight other material systems that could potentially serve as an alternative to (and perhaps even outperform) the conventional TCOs. It begins by highlighting the ideal characteristics of a material with a crossover in the NIR, followed by introducing material systems.

1. Ideal characteristics

In the case of Drude materials, a zero crossing of the permittivity occurs when, physically, the contribution of free electron resonance becomes strong enough to counter high frequency resonance(s) (bound electrons, interband absorption, excitonic absorptions). Mathematically, this occurs when the ε term equals in magnitude the Drude term. From Eq. (1), it is evident that the ratio of N c / m * and ε are key to achieving the zero crossing at technologically relevant wavelengths. This implies that materials with a low ε , high N c and low m * are well suited to exhibit ENZ and NZI properties in the NIR. Out of these, ε and m * are material specific and can be tuned by changing the composition of ternary or quaternary compounds making the process intricate. Free carriers ( N c) on the other hand can be readily tuned, up to the dopant's solubility limit in the host matrix. Since tuning free carriers is generally easy, we can consider how many carriers are needed for a given ε and m * to generate a permittivity crossover at a desired wavelength. Figure 15 highlights the range of ε and m * (indicated by points within the colored band) that gives rise to a permittivity crossover between 1400 and 1600 nm for various free carrier densities (indicated by the colored bands) at a mobility of 40 cm2/(V s). While ε and m * both affect the crossover wavelength, it is evident from Fig. 15 that m * has a larger effect (especially in the range of low m * materials which are important for NIR applications) and should be critically considered when exploring alternative materials.

FIG. 15.

Based on the Drude model, this graph displays a range of crossover wavelengths between 1400 and 1600 nm (shown by the red band), as a function of high frequency permittivity ε , electron effective mass m * and free carrier density N c (indicated by the shade of red, where lighter shades indicate higher free carrier densities). Mobility is fixed at a commonly found value for materials of 40 cm2/(V s). In the case of low m * (  0.2–0.3), free carrier densities of 3 × 1020 cm−3 suffice to achieve a crossover of  1500 nm. However, as m * approaches   0.6 or higher, depending on ε , instances exist where even 1 × 1021 cm−3 will not result in a  1500 nm crossover. Thus, amongst the two material specific properties, m * plays the critical role in determining the crossover wavelength.

FIG. 15.

Based on the Drude model, this graph displays a range of crossover wavelengths between 1400 and 1600 nm (shown by the red band), as a function of high frequency permittivity ε , electron effective mass m * and free carrier density N c (indicated by the shade of red, where lighter shades indicate higher free carrier densities). Mobility is fixed at a commonly found value for materials of 40 cm2/(V s). In the case of low m * (  0.2–0.3), free carrier densities of 3 × 1020 cm−3 suffice to achieve a crossover of  1500 nm. However, as m * approaches   0.6 or higher, depending on ε , instances exist where even 1 × 1021 cm−3 will not result in a  1500 nm crossover. Thus, amongst the two material specific properties, m * plays the critical role in determining the crossover wavelength.

Close modal
In addition to these parameters, other parameters such as the bandgap, conduction band non-parabolicity and mobility also have an influence, especially on the nonlinear performance. For example, in the case of intraband nonlinearities, a high CB non-parabolicity parameter d m / d E is desirable, as an increase in average effective mass is the primary driver of index change. Moreover, since this is an absorption driven affect, a high mobility (μ) is not desirable as this decreases the optical loss ( scattering time , τ = μ m avg * / e) and thus the absorption of the pump (Apump). However, a low mobility also leads to a shallow index dispersion ( d n probe / d m *) which hampers the magnitude of index change. Work by Secondo et al.40 captures this trade-off between steep index dispersion and absorption of the pump, in a figure of merit (FoM) given by
F O M I R = A pump 1 m avg * d m d E 1 N c d n probe d m * ,
(17)
where m avg * is the average effective mass of electron in CB, d m / d E is the change in effective mass as a function of energy levels (in other words, CB non-parabolicity) and d n probe / d m avg * denotes the index dispersion with respect to the average effective mass. On the other hand, in the case of interband nonlinearities, a small CB non-parabolicity parameter is desirable since the effective mass tends to increase for high momentum states in non-parabolic bands, limiting the contribution of the added electron to Drude response. Thus, a quasi-parabolic bound would ensure that each excess electron in the CB experiences an approximately constant m * regardless of its location in band. Moreover, since absorption is dictated by loss at the band edge rather than free carrier absorption in the ENZ region, the absorption-dispersion trade-off of the intraband case is removed such that the loss at ENZ (mobility) is desired to be low (high) to obtain a steep index dispersion. As a result, materials stand to benefit from a lower loss at ENZ and higher absorption at the pumping frequency with the limit arising due to impedance matching when the refractive index at ENZ approaches zero. A recent work by Secondo et al.,148 captures the intricacies of interband pumping with a FoM given by
F O M U V = A pump E g 1 N c d N c d E 1 m * d n probe d N c ,
(18)
where E g is the bandgap and N c, is the free carrier density. Here, the absorption is normalized by the bandgap energy, as wide-bandgap materials require more energy to generate the same number of excess carriers, due to the relation: Δ N c A / ω with ω > Eg. Finally, a large initial free carrier density or a high effective mass tend to decrease the efficiency per unit energy. Table I summarizes the desired materials properties with respect to inter- and intra-band nonlinearities at telecommunication wavelengths. These properties can be readily found in TCOs and perovskite oxides and will be discussed next.
TABLE I.

Inter-band vs intra-band nonlinearities and associated desirable material properties.

Parameter Inter-band excitation Intra-band excitation
Effect on plasma freq. (ωp Blue shifts  Red shifts 
Driving factor  Increase in Nc in the CB  Increase in m avg * of the CB 
Free carrier density  Moderate (ENZ at or > 1550 nm)  High (ENZ at 1550 nm) 
Mobility  High [>60 cm2/(V s)]  Low [  20 cm2/(V s)] 
Effective mass  Low (<0.3  m avg * Low (<0.3  m avg *
High frequency contribution ε  Low  Low 
CB non-parabolicity  Low  High 
Bandgap  Small, but >  1 eV  Not critical, but >  1.6 eVa 
Parameter Inter-band excitation Intra-band excitation
Effect on plasma freq. (ωp Blue shifts  Red shifts 
Driving factor  Increase in Nc in the CB  Increase in m avg * of the CB 
Free carrier density  Moderate (ENZ at or > 1550 nm)  High (ENZ at 1550 nm) 
Mobility  High [>60 cm2/(V s)]  Low [  20 cm2/(V s)] 
Effective mass  Low (<0.3  m avg * Low (<0.3  m avg *
High frequency contribution ε  Low  Low 
CB non-parabolicity  Low  High 
Bandgap  Small, but >  1 eV  Not critical, but >  1.6 eVa 
a

Not critical, but over 2 times that of operating frequency to avoid 2 photon absorption.

2. Transparent conducting oxides

Transparent conducting oxides (TCOs) are a group of semiconductors that are highly conductive, yet transparent, especially across the visible spectrum. They have been used in several applications for over 70 years, with initial studies dating back to early 1900s.234,235 While ITO is by far the most widely used TCO, across all applications, some of the other well explored materials have been impurity-doped ZnO, In2O3, and SnO2, as well as multicomponent oxides composed of combinations of these binary compounds [Figs. 16(a) and 16(b)].232 These materials have been tailored for various opto-electronic applications which favor properties such as high transmission in the visible, high conductivity, high refractive index, reduced use of In (scarce compound in nature) or a combination of the above. On the other hand, ENZ nonlinearities benefit from materials that exhibit a low m *, low ε , and accept high levels of doping, as summarized in Table I.

FIG. 16.

(a) Summary of useful TCO materials such as ZnO, In2O3, SnO2, and their multicomponent oxides.232 Reproduced with permission from Minami, Thin Solid Films 516(7), 1314–1321, (2008). Copyright 2007 Elsevier B.V. (b) Lowest effective mass of each cation plotted in rows and columns of the periodic table for binary and ternary oxides.233 Reproduced with permission from Hautier et al., Chem. Mater. 26(19), 5447–5458, (2014). Copyright 2014 American Chemical Society.

FIG. 16.

(a) Summary of useful TCO materials such as ZnO, In2O3, SnO2, and their multicomponent oxides.232 Reproduced with permission from Minami, Thin Solid Films 516(7), 1314–1321, (2008). Copyright 2007 Elsevier B.V. (b) Lowest effective mass of each cation plotted in rows and columns of the periodic table for binary and ternary oxides.233 Reproduced with permission from Hautier et al., Chem. Mater. 26(19), 5447–5458, (2014). Copyright 2014 American Chemical Society.

Close modal

While there have been no reports that screen all these parameters, a study by Hautier et al.233 does so for electron effective mass and bandgap. They filter the periodic table for binary and ternary oxides with low effective masses and a bandgap higher than 3 eV calculated using the ab initio method. They conclude that while it is unlikely to find oxides with substantially lower effective masses than that of ZnO and In2O3, there exists a whole range of compounds that exhibit similar effective masses, including ZnSb2O6, Cd2GeO4, and Cd2Ge7O16. Unfortunately, all the compounds, except ZnSb2O6, either contain toxic or scarce elements. Some other compounds such as CdGa2O4, MgIn2O4, ZnGa2O4, Ga2O3, and InGaO3 also exhibit low effective masses, but have a bandgap between 1 and 3 eV.233 Although not suitable for use as TCOs, these materials can be explored for their potential ENZ properties in the NIR and MIR. While estimates of low effective mass from ab initio calculations is certainly a step in the right direction, future studies are needed to estimate dopant solubility limits, CB non-parabolicity and other related metrics.

One such work by Secondo et al.40 analyses an eclectic but limited mix of conducting oxides and nitrides for their intraband nonlinearities. They study the band structures of ZnO, ITO, Ga2O3, GaN, and CdO and find that the CB non-parabolicity of all considered materials lies within a factor of 2, with Ga2O3 exhibiting the highest non-parabolicity. As a result, the work concludes that the nonlinearities will be in the same order of magnitude as well. Figure 17(a) elucidates the role of loss in these films. The peak FOM is achieved for ZnO films with mobilities in the range of  15 cm2/(V s) ( ε ω pump 0.6), as high mobility films suffer from decreased absorption while low mobility films suffer from a shallow index dispersion. This is in stark contrast to UV pumping, which benefits from a high mobility as the absorption of the pump is dependent on interband transition and not Ohmic losses.

FIG. 17.

(a) Figure of merit for the selected materials as a function of mobility, based on Eq. (17). Gray dashed lines represent the range of mobilities achieved by films published in literature.40 Reproduced with permission from Secondo et al., Opt. Mater. Express 10(7), 1545 (2020). Copyright 2020 Optical Society of America (b) Figure plots the hall mobility vs free carrier concentration for Sn and Mo doped indium oxide films. In this carrier concentration regime, the electron mobility is limited by ionized impurity scattering.236 Reproduced with permission from Swallow et al., Mater. Horizons 7(1), 236–243, (2020). Copyright 2020 Royal Society of Chemicals.

FIG. 17.

(a) Figure of merit for the selected materials as a function of mobility, based on Eq. (17). Gray dashed lines represent the range of mobilities achieved by films published in literature.40 Reproduced with permission from Secondo et al., Opt. Mater. Express 10(7), 1545 (2020). Copyright 2020 Optical Society of America (b) Figure plots the hall mobility vs free carrier concentration for Sn and Mo doped indium oxide films. In this carrier concentration regime, the electron mobility is limited by ionized impurity scattering.236 Reproduced with permission from Swallow et al., Mater. Horizons 7(1), 236–243, (2020). Copyright 2020 Royal Society of Chemicals.

Close modal

Since mobility is critical to the performance of TCOs generally, especially when pumping at energies above the bandgap, it is imperative to maximize this for a given material. The mobility of degenerately doped TCOs is mainly limited by ionized impurity scattering, i.e., Coulomb interactions between electrons and the dopants.237,238

Furthermore, the dopant also influences the material's band structure and effective mass. Taking the example of Sn-doped In2O3 (ITO), the strong orbital hybridization between the Sn and In 5s states is known to renormalize the CB minima, thus leading to a higher effective mass and in turn, a lower mobility. In addition, since each Sn dopant can only donate a maximum of one excess electron into In2O3, a high doping density of Sn is required to achieve free carrier densities leading to NIR ENZ wavelengths. This illustrates that while ITO has been widely used, Sn is not an ideal dopant for In2O3.239–243 In search of better suited dopants, Xu et al.244 report on Zr, Hf, and Ta as dopants for In2O3. This is due to the size and orientation of the 5d orbital and localized 5d defect states. Recent experimental data also support these findings.245–247 For example, Swallow et al.236 show that Mo doping exhibits higher mobility than Sn doping of In2O3 across carrier concentrations up to 6 × 1020 cm−3 [Fig. 17(b)]. Due to the minimum perturbation of the CB with Mo doping, it also has a narrower CB dispersion with a lower effective mass than ITO. This results in Mo:In2O3 being well suited for both interband and intraband pumping studies. Similarly, InZrO was recently explored for its nonlinearity in a recent work by Ghobadi et al.42 While a comparison of InZrO NL with other TCOs was not in the scope of the work, the films exhibited mobilities on the order of  70 cm2/(V s), higher than what is commonly seen with commercially available ITO, indicating a strong potential for an improved nonlinear response at ENZ.

3. Perovskite oxides

Perovskite oxides are a group of inorganic compounds with an ABO3 type chemical formula, where the A site cation can be an alkaline earth (Mg, Ca), a rare-earth (Nd, Y) or an alkali element (Na, K), the B site cation can be a transition or post-transition metal element from periods 4, 5 or 6 (Cu, Ti) and where O is an oxygen anion.248 A basic perovskite oxide structure is shown in Fig. 18(a), where the B cations reside at the center, A at the corners and O at the middle of the faces. This forms a unit cell of a more complex 3D framework where the B cations are octahedrally coordinated to oxygen atoms, and the BO6 octahedra shares corners across the three directions of the crystal [Fig. 18(b)]. This framework leads to these materials exhibiting a range of interesting properties such as superconductivity,249 metal-insulator transition,250 photovoltaic effect,251 ferroelectricity,251,252 and transparent conductivity.253–257 These wide-ranging properties, which are due to strong interactions between the charge, orbital, structural and spin properties, enable the realization of devices with functionalities beyond those possible with standard semiconductors. Arguably, perovskites represent one of the most important crystal structures among transition metal oxides.

FIG. 18.

(a) Schematic of a basic perovskite unit cell.248 (b) The unit cell is part of a more complex 3D framework where B cations are octahedrally coordinated to oxygen atoms, and the BO6 octahedra shares corners across the three directions of the crystal.248 Reproduced with permission from Sun et al., Adv. Energy Mater. 11(2), 2000459 (2021). Copyright 2021 Wiley-VCH.

FIG. 18.

(a) Schematic of a basic perovskite unit cell.248 (b) The unit cell is part of a more complex 3D framework where B cations are octahedrally coordinated to oxygen atoms, and the BO6 octahedra shares corners across the three directions of the crystal.248 Reproduced with permission from Sun et al., Adv. Energy Mater. 11(2), 2000459 (2021). Copyright 2021 Wiley-VCH.

Close modal

In addition to the above-mentioned properties, some perovskites have been shown to exhibit high room temperature mobilities and can accept large levels of doping.254,256,258 These enhanced electronic properties are mainly the function of relatively large static dielectric permittivity, which promotes screening of the electron-charge interaction.259 The combination of rich electronic properties and the potential to exhibit zero permittivity in the NIR, makes these materials an attractive alternative to conventional TCOs, especially as this could result in devices with greater functionality.248,260–262

Some of the more heavily explored perovskite oxides have been stannates,258 vanadates,263 titanates,264,265 and germanates.266 While most of these can be readily grown with high doping levels, titanates and vanadates are known to exhibit large effective masses (BaTiO3:5.4m0, SrVO3:3.0m0), making them unsuitable for ENZ nonlinearities. The large effective mass primarily arises from strong electron correlations in these materials. Titanates and vanadates have partially filled d-orbitals in their crystal structure, which interact strongly with each other and with the oxygen ions in the structure, via Coloumbic and exchange interactions. Because of this strong interaction, the electrons behave more like heavy particles with high effective mass.256,267,268 On the other hand, stannates and germanates, have been shown to exhibit low effective masses (SrGeO3:0.22m0 and BaSnO3:0.19m0).259,266,269 Unlike titanates, which have the CB minima dominated by the localized 3d states, first-principal calculations show that CB minima is dominated by dispersive Sn and Ge 5s states, leading to smaller effective masses.269 

Among several potential stannates and germanates, doped BaSnO3 is particularly promising for ENZ nonlinearities, due to the following properties: mobilities ranging in between 50 and 350 cm2/(V s), effective masses below 0.25m0, ε of 4.5 and free carrier densities up to the mid 1020 cm−3 range.254,270,273,274 However, there is a dearth of literature exploring the ENZ properties of BaSnO3. More recently however, it was explored for its NIR plasmonic properties as an alternative plasmonic material. Yang et al.272 deposited La doped BaSnO3 using sol-gel method and showed that infrared plasmons sustained by it are superior in spatial confinement ratio compared to those in noble metals, primarily due to their lower carrier densities which places the plasmon energies in the region of interest, IR in this case [Figs. 19(f) and 19(g)]. Similarly, Kim et al.270 demonstrated epitaxial BaSnO3 films on SrTiO3 with a crossover wavelength as low as 2050 nm and a mobility of 65 cm2/(V s) [Figs. 19(a) and 19(b)].270 With crossover wavelengths approaching the telecommunication window, IR pumping also becomes potentially feasible. In this sense, BaSnO3 is very attractive, as the CB non-parabolicity parameter is experimentally determined to be β ≈ 0.37 eV−1, which is over twice as large as that of ITO (β ≈ 0.18 eV) and GZO (β ≈ 0.14 eV), as shown in Fig. 19(c).271 

FIG. 19.

(a) Real (ε1) and imaginary (ε2) parts of permittivity of 100 nm thick LBSO films deposited on SrTiO3 substrates at various temperatures (700–830 °C).270 (b) ENZ wavelength and carrier density of the LBSO films as a function of oxygen deposition pressure.270 Reproduced with permission from Kim et al., J. Phys. D 53(36), 365103 (2020). Copyright 2020 IOP Publishing. (c) Average effective mass of the electron in CB as a function of free carrier density for La:BaSnO3 films.271 Reproduced with permission from Niedermeier et al., Phys. Rev. B 95(16), 161202 (2017). Copyright 2017 American Physical Society. (d) IR reflectance spectrum of La:SrGeO3 polycrystalline pellet.259 Reproduced with permission from Niedermeier et al., Phys. Rev. B 101(12), 125206 (2020). Copyright 2020 American Physical Society. (e) Measured IR plasmon wavelengths in La:BaSnO3 nanoparticles (symbols) compared with calculated values obtained for La:BaSnO3 (red curve) and Ag (black curve) waveguides with a cross section of 50 × 50 nm2. Also shown is the free-space photon dispersion (gray curve) for reference. (f) Confinement ratio obtained from (e) as the ratio of the plasmon excitation wavelength to the free-space photon wavelength.272 Reproduced with permission from Yang et al., Small 18(16), 2106897 (2022). Copyright 2022 Wiley-VCH GmbH.

FIG. 19.

(a) Real (ε1) and imaginary (ε2) parts of permittivity of 100 nm thick LBSO films deposited on SrTiO3 substrates at various temperatures (700–830 °C).270 (b) ENZ wavelength and carrier density of the LBSO films as a function of oxygen deposition pressure.270 Reproduced with permission from Kim et al., J. Phys. D 53(36), 365103 (2020). Copyright 2020 IOP Publishing. (c) Average effective mass of the electron in CB as a function of free carrier density for La:BaSnO3 films.271 Reproduced with permission from Niedermeier et al., Phys. Rev. B 95(16), 161202 (2017). Copyright 2017 American Physical Society. (d) IR reflectance spectrum of La:SrGeO3 polycrystalline pellet.259 Reproduced with permission from Niedermeier et al., Phys. Rev. B 101(12), 125206 (2020). Copyright 2020 American Physical Society. (e) Measured IR plasmon wavelengths in La:BaSnO3 nanoparticles (symbols) compared with calculated values obtained for La:BaSnO3 (red curve) and Ag (black curve) waveguides with a cross section of 50 × 50 nm2. Also shown is the free-space photon dispersion (gray curve) for reference. (f) Confinement ratio obtained from (e) as the ratio of the plasmon excitation wavelength to the free-space photon wavelength.272 Reproduced with permission from Yang et al., Small 18(16), 2106897 (2022). Copyright 2022 Wiley-VCH GmbH.

Close modal

One other perovskite oxide of interest is doped SrGeO3. It has properties like that of BaSnO3, with undoped SrGeO3 predicted to exhibit mobility as high as 390 cm2/(V s).259,275 In a recent work by Nierdermeier et al.,259 single crystal La doped SrGeO3 was grown, which resulted in a mobility of 42 cm2/(V s) at free carrier densities of 1.5 × 1020 cm−3. Furthermore, from the sharp rise of reflection in the reflectivity spectra, we can conclude that the permittivity crossover occurred between wavelengths of 2500 and 3000 nm in these samples [Fig. 19(c)]. More interestingly however, the authors establish a guideline for the exploration of high mobility semiconductors. A combination of small effective mass and large polarization of the hardest LO mode should support high room temperature mobilities despite electron–phonon interactions. While BaSnO3 and SrGeO3 have been actively explored, alkaline earth stannates such as ASnO3 (A = Sr, Ca, Zn, and Cd) have also been predicted to have effective masses and band gaps in the range of 0.15–0.25 and 1.5–4.5 eV, respectively.256,262,269,276–279 However, as evident from the sparse literature available for epitaxial films or single crystals, these materials have yet to be widely investigated.

Several of the perovskite oxides appear to be promising candidates for ENZ nonlinearities, however very few reports have explored these materials. This is primarily because the properties of these materials are not substrate agnostic, often requiring the use of precisely lattice matched substrates to achieve the frequently touted record electrical properties. Work by Kim et al.273 reported that single crystals of BaSnO3 exhibited mobility in the range of 200–320 cm2/(V s) (with carrier densities 1 × 1019 to 4 × 1020 cm−3), while epitaxial films exhibited 70 cm2/(V s). This was mainly attributed to increased carrier scattering due to dislocations and grain boundaries, which were presumably created by the 4% lattice mismatch between the substrate SrTiO3 and La:BaSnO3. Such stringent conditions of lattice matching and careful tuning of the deposition parameters result in scarce availability of the oxides. Thus, from a practical standpoint, a method to achieve high quality films on various substrates is highly desirable. To address this issue, researchers have explored methods to transfer these films using various methods such as sacrificial etching, mechanical exfoliation and spalling.280 For example, the family of water soluble (Ca,Sr,Ba)3Al2O6 materials have been used as a sacrificial layer. Not only do these materials have a good structural and chemical compatibility with perovskite oxides, but their lattice parameters can be tuned from 0.3819 nm (100% Ca) to 0.4124 nm (100% Ba). This presumably could lead to not only easier access to perovskite oxides, but also improved compatibility with device fabrication processes. Finally, to provide a quantitative estimate of improvement, Fig. 20 compares some of the upcoming perovskites to other widely explored TCOs. Figure 20(a) plots the figure of merit of these materials [Eq. (18)] when pumped above the bandgap with enough power to excite a free carrier concentration of 1 × 1020 cm−3. The FoM is plotted as a function of different starting base plasma frequencies (or varying free carrier densities) normalized to the frequency corresponding to 1550 nm. The power needed to excite a free carrier concentration of 1 × 1020 cm−3 ranges from 10 to 50 GW/cm2, if 100% of the incident energy is absorbed. Other assumed parameters are shown in Table S2 and S3. Due to a combination of low base effective mass ( m avg *, lower scattering rate, thus steeper Drude dispersion), moderate band non-parabolicity (dm/dE, lower counteraction to increasing Nc) and high mobility (μ, steeper Drude dispersion), BaGeO3 is predicted to outperform all other materials under the assumed conditions. Furthermore, perovskites are predicted to outperform the commonly used TCOs by a factor of at least 2, within their ENZ spectral region around 1550 nm. Figure 20(b), on the other hand, plots a figure of merit of these materials when they are pumped in the IR, at their respective ENZ wavelengths, as a function of mobility. The carrier concentration of all the materials has been adjusted such that they exhibit an ENZ wavelength of 1550 nm. While conduction band non-parabolicity has been thought of as the metric to determine the strength of IR nonlinearity, it turns out that effective mass is just as important, if not more. This is observed by the fact that, while CdO has the largest dm/dE, it is outperformed by all perovskites with lower non-parabolicities. This is because of low m avg *, especially observed in BaGeO3. Furthermore, although the dm/dE term, of BaGeO3, is much lower than most materials in the graph, it is the low m avg * that drives the outperformance. This leads further credence to the fact that m avg * is an important parameter to consider, as was evident in Fig. 15 as well. Ultimately, both figures highlight the complex interplay that exists between these parameters, underscoring the need for in-depth DFT based modeling and experimental exploration to improve the performance of nonlinear materials utilizing ENZ effects.

FIG. 20.

To compare some of the upcoming perovskites to the commonly used TCOs, this figure plots the UV and IR figures of merits against normalized plasma frequency and mobility, respectively. (a) UV FoM shown against samples with varying plasma frequencies normalized to the frequency ω corresponding to free-space wavelength of 1550 nm, which can also be thought of as samples with varying carrier concentrations. (b) IR FoM shown against samples with varying mobilities. The carrier concentration is adjusted such that each of these materials exhibit a crossover wavelength of 1550 nm. A detailed list of assumptions can be found in Tables S2 and S3.

FIG. 20.

To compare some of the upcoming perovskites to the commonly used TCOs, this figure plots the UV and IR figures of merits against normalized plasma frequency and mobility, respectively. (a) UV FoM shown against samples with varying plasma frequencies normalized to the frequency ω corresponding to free-space wavelength of 1550 nm, which can also be thought of as samples with varying carrier concentrations. (b) IR FoM shown against samples with varying mobilities. The carrier concentration is adjusted such that each of these materials exhibit a crossover wavelength of 1550 nm. A detailed list of assumptions can be found in Tables S2 and S3.

Close modal

In summary, semiconductors such as doped tin, zinc, Cd, and perovskite oxides have been explored for their properties as a transparent conductor. Key to this is transparency in the visible, or a large bandgap. Some of these materials have also found their way into the field of ENZ nonlinearities and led to fascinating demonstrations. However, materials tailored for ENZ nonlinearities, especially in the telecommunication spectrum, do not need to have a large bandgap. In fact, a smaller bandgap would enable interband pumping of these materials in the visible spectrum, which is more readily accessible. Furthermore, parameters such as CB non-parabolicity and effective mass are arguably more important than overall conductivity or transmission through the visible spectrum (which are key to good TCOs). Furthermore, the general push so far has been toward improving mobilities, which is not strictly required, especially considering intraband nonlinearities. Therefore, a significant opportunity lies in moving beyond conventional TCOs, exploring materials that have been tailored for ENZ nonlinearities and not just adopted. Furthermore, with the advent of AI/ML based high throughput material discovery pipelines,281–284 the process of discovering materials is only being accelerated. While the research community might conclude that ENZ nonlinearities using conventional TCOs are the best we can achieve, the possibility of not only drastically improving light–matter interaction, but also bringing it into the realm of practicality, makes this question worth exploring.

In addition to the search for new materials to optimize nonlinear interactions in ENZ, there has been a surge of interest in exploring time-varying optical interactions. These interactions, which we generally classify as time varying effects, occur when a photon interacts with a material or refractive index perturbation moving near the speed of light, and includes perturbation that can have either, temporal or spatiotemporal variations. While early theoretical works considered moving mirrors as a test case,285 the ability of homogeneous ENZ materials to provide a unity-order index change on the sub-picosecond scale is quite an interesting combination. It facilitates the exploration of the unique effects of time-varying interactions without needing to accelerate a material to near the speed of light.

The inclusion of time as an additional degree of freedom in engineering light–matter interaction is of particular interest because of various exotic physical phenomena associated with this approach. One such prominent effect that arises from light propagating in a time-varying medium (TVM) is frequency conversion. Historically, the concept had its roots in plasma physics, with early exploration focused on the realm of relativistic photon propagation286–291 where the interaction of intense laser pulses with dense gaseous mediums yielded a moving ionized plasma front. The generation of free carriers during ionization caused temporal variations in the medium's refractive index along the trajectory of the pulse. The interaction of photons with this dynamic front, acting as a moving “mirror” of sorts (where the increasing free carriers in the Drude term correspondingly decreased the refractive index, leading to heightened reflectivity), resulted in a frequency upshift of the propagating photons. As we advanced into the 21st century, the concept of TVM expanded its horizons to encompass a broader array of optical mediums ranging from tens of kHz to hundreds of THz, through acousto-optics, microwave-based system, opto-mechanics, metamaterials and more recently, epsilon-near-zero (ENZ) films.292 The significance of the frequency conversion achieved through these techniques is noteworthy for several reasons.

For many of the techniques used to achieve frequency conversion, including ENZ films, the process is a non-resonant process. It predominantly acts upon the linear refractive index and facilitates the spectral transference of electromagnetic energy without being bound by conventional phase matching conditions or selection rules, thereby introducing the potential for continuous and deterministic frequency adjustments.291 While conventional nonlinear optical descriptions can be used to describe the effect, e.g., four-wave-mixing, frequency conversion via TVMs is distinct in that it results in the frequency conversion of all photons within the material. Continued development in this realm holds the promise of providing new avenues to address spectral gaps293 that arise from varying operational regimes of different optical network components (emitters, detectors, fibers, etc.). Additionally, unlike conventional χ(2) based systems limited by phase matching conditions,294 TVM-based frequency conversion can dynamically convert photon frequency while preserving quantum states, bridging spectral gaps, and advancing quantum network realization.295–297 

Beyond frequency shifting, another noteworthy time-varying effect is the generation of a time-reversed (TR) wave. Upon interaction with the temporal interface formed with the abrupt perturbation in material index, electromagnetic waves can evolve into a new state with its phase evolving backward in time.298–300 Although modern signal processing techniques have already enabled the generation of time-reversed waves in lower frequency systems,299 its implementation in optical regime is quite complicated as requires independent control over spatial, temporal and polarization degrees of freedom.301 TVM-induced TR generation is attractive in this regard because it does not rely on sampling or recording, as needed for in signal processing. However, the requirement of ultrafast and uniform change of medium properties remains a challenge in realizing optical TR in TVM.302 

TVMs also allow for realizing highly sought after nonreciprocity in linear optical system with broken time-reversal symmetry. Nonreciprocity via temporal modulation is achieved by allowing directional mode conversion between orthogonal traveling modes under phase matching conditions which is fulfilled only for either forward or backward propagating wave, and thus, provides isolation for the restricted direction without bulky traditional magneto-optic biasing.303–305 Overall, the time-varying physics discussed above can lead to versatile application scopes like inverse prisms,306 “perfect lens” for subwavelength imaging,307 imaging in random media,308 optical circulators,309 optical power combiners,310 temporal beam splitters,311 Fresnel drag,312 surface wave generation with temporal grating without subwavelength fabrication,313 temporal waveguides,314 antireflection temporal coatings,315 temporal aiming of pulses,316 all-optical beam steering,317 pulse shaping,318 as well as interesting physics like temporal diffraction and interference319–321 to name a few. For further information on time-varying physics, please see the following review.292 

Clearly, adding the time-dimension to our ability to control light is crucial and unlocks quite unique and impactful opportunities. As we move ahead, we will dig more deeply into how some of these effects can be realized utilizing ENZ media. We will start with one of the hallmarks of TVMs, frequency shifting, highlighting the factors that limit achievable frequency shifts in realistic scenarios. Within this context, we will also review the experimental endeavors related to both ENZ and non-ENZ approaches, aiming to assess the advantages and limitations of each scheme. Following that, we will briefly discuss the dynamics underlying the generation of time-reversed waves, along with an overview of reported implementations in the literature. Finally, we will introduce and briefly evaluate a prominent application direction within TVM, referred to as the “photonic time crystal” while considering its potential for realization within the optical realm.

1. Frequency shifting in TVM

While TVMs offer a broad range of distinct yet unique phenomena to be explored, e.g., photon acceleration,322,323 time-reversed wave generation,298,300,324 photonic time crystals,325–327 a common thread among them is the frequency shift experienced by EM waves. Hence, most of the recent experimental endeavors in realizing optical TVM have been directed toward frequency shifting.

The fundamental physics behind frequency shift can be explained from the standpoint of simple geometric optics. As light crosses a boundary between media with different properties, its trajectory maintains translation symmetry to preserve tangential momentum. In static media, an interaction with the boundary conserves the energy or frequency (ω) but modifies the momentum or wavevector (k), as shown in Figs. 21(a) and 21(b). This process leads to the well-known formula of refraction or general Snell's law to determine the propagation direction of light from the relation n1 sinθ1 = n2 sinθ2 where n1,θ1 (n2,θ2) denotes the index and angle of the incident (refracted) light respectively. In contrast, when light encounters a “temporal” boundary formed through an abrupt change in the refractive index of a homogenous medium at some specific instant of time, the wavevector (k) representing the spatial distribution of light energy—must remain constant. However, the index of the media has been altered. Thus, to maintain this momentum, the frequency, ω = k c / n, adjusts to compensate for the sudden change in refractive index. Moreover, waves interacting with such a “time boundary” produce a “time-refracted” and ‘time reflected/reversed’ wave in a similar fashion as is observed at a spatial boundary. However, since the time axis is unidirectional, the “reflected” wave cannot go back in time, and both the refracted and reflected waves spatially propagate with different frequencies and phases according to ω2 = ω1.( n 1 sin θ 1 n 2 sin θ 2) and ω3 = ω1.( sin θ 1 sin θ 3), where ω1 (θ1), ω2 (θ2), and ω3 (θ3), are the frequencies and angles of propagation (with respective to normal to the temporal interface) for the incident, time-refracted and time-reflected beams, respectively at any time t [Figs. 21(c) and 21(d)].291 This geometric approximation holds for an ideal scenario of a nondispersive medium where the index variation can be instantaneous (e.g., the index boundary has infinite speed). However, causality and the non-instantaneous response of physical materials cause the index change in the media to happen at a “fast” but finite speed (i.e., index boundary moves at vg ≈ c). From a relativistic perspective, in this case, a more general space-time refraction effect occurs where both the frequency and momentum of propagating light must change to conserve energy and momentum [Figs. 21(e) and 21(f)]. Such relativistic events are often explained from various perspectives including photon acceleration,322,328,329 photon energy lifting,330 and self-phase modulation331 where the frequency of the light is gradually modified, or “adiabatically shifted” from one frequency to another due to the interaction with the moving boundary, very similar to an acoustic Doppler shift.

FIG. 21.

Illustration of the generalized concept of refraction and reflection for (a) and (b) space (dn/dt = 0) (c) and (d) time ( d n d t ) and (e) and (f) space-time (dn/dt ≈ finite) boundaries. The lower panels (b), (d), and (f) depicts the corresponding change in wavevector and/or frequency in the phase space diagram depending on the type of interface encountered during propagation. When the change from n1 to n2 happens over a time longer than the period of the input wave, adiabatic changes occur to both wavelength and frequency as shown in (e) and (f). Figures (a)–(d) are adapted from Galiffi et al., Adv. Photonics 4(1), 014002 (2022). Copyright 2022 Author(s). Licensed under a Creative Commons Attribution License.

FIG. 21.

Illustration of the generalized concept of refraction and reflection for (a) and (b) space (dn/dt = 0) (c) and (d) time ( d n d t ) and (e) and (f) space-time (dn/dt ≈ finite) boundaries. The lower panels (b), (d), and (f) depicts the corresponding change in wavevector and/or frequency in the phase space diagram depending on the type of interface encountered during propagation. When the change from n1 to n2 happens over a time longer than the period of the input wave, adiabatic changes occur to both wavelength and frequency as shown in (e) and (f). Figures (a)–(d) are adapted from Galiffi et al., Adv. Photonics 4(1), 014002 (2022). Copyright 2022 Author(s). Licensed under a Creative Commons Attribution License.

Close modal

Frequency conversion through TVMs is important because unlike nonlinear wave mixing processes, this process is not restricted by conditions like phase matching or photon energy conservation and can have 100% internal conversion efficiency for all photons within the TVM, leading to a shift of entire spectrum. Furthermore, the final frequency shift can be tuned by adjusting the strength of the index modulation.

The basic expression for TVM induced frequency shift can be simply derived by taking the time derivative of momentum conservation condition k(ω)c = ωn(ω), given as
d d t ( ω n ) = ω d n d t + n d ω d t = d d t ( k c ) = 0 ; d ω ω = d n n .
(19)
The above equation shows the direct relationship between frequency shift and index modulation. Note here, the expression for frequency shift in Eq. (19) is not exact and only valid for qualitative analysis, one must resort to a full derivative approach to account for dispersion and other relevant complexities in a quantitative analysis.132 Along with index modulation, another important factor is how fast the change occurs. In a TVM, modulation occurring on a timescale shorter than the cycle time of the signal wave effectively creates a temporal boundary [Figs. 21(c) and 21(d)] whereas longer yet comparable modulation time leads to the space-time boundary scenario [Figs. 21(e) and 21(f)]. The magnitude of frequency shift occurring from the interaction with any of these two boundaries therefore depends primarily on two factors—(a) modulation strength (larger the dn, more shift), and (b) modulation duration (the time, Δtchange it takes for the index to change by dnmax) in comparison to the cycle of interacting wave.

To realize temporal boundaries in the optical regime (≈100s of Terahertz scale) the modulation period (Δtchange) needs to be less than or approximately on the order of 1–10 fs scale. Given that, achieving femtosecond-scale modulation of permittivity is most likely feasible through optical methods such as nonlinear interactions via electron polarization or free carrier absorption.48 In the ideal scenario, when considering instantaneous index changes in a nondispersive TVM, the temporal slope of index change, dn dt ( Δ n Δ t change ) tends to infinity as Δtchange→0 implying the relative frequency shift (Δω/ω) to be proportional only to the relative index change (Δn/n) for EM field of any frequency [Eq. (19)].

Complexities arise when considering real dispersive materials where the index modulation is limited by strength vs speed trade-off between various nonlinear approaches [refer to Fig. 1(b) and discussion in Sec. IV on types of Drude nonlinearities]. In conventional dielectrics, only the bound electronic polarization nonlinearities can induce the change in susceptibilities on the desired femtosecond timescale but lack in modulation strength (although recent advances in free carrier effects are enabling fast rise times as well, further details see Sec. IV). For example, with reference to Kerr nonlinearity, the maximum achievable index change (Δn) is on the order of 0.04 for Si and 0.006 for Si3N4 in the near IR spectrum even at 1 TW/cm2 excitation.332 This is neither sufficient nor practical to achieve any appreciable frequency shift. Another approach of optical frequency shift has been reported with RF signal driven electro-optic modulators by changing the modulating frequency but maximum achievable shift through such process is limited (only a few gigahertz level) and inefficient due to sideband generation.333,334 This dilemma explains the inherent challenges in developing an optical TVM based frequency converter compared to early experimental approaches employed in various systems such as mechanical,335 optomechanical,336 acousto-optic,337 and electro-optic338 ranging from kilohertz to gigahertz operational regime. However, space-time refraction relaxes the modulation frequency limit and allows for adiabatic frequency conversion (AFC) with modulation time slightly longer than the optical cycle given that the modulation strength is large. This fits quite well with the strengths of free carrier nonlinearities and has led to several recent studies of AFC using semiconductor, ENZ, and cavity resonance approaches. More on these approaches will be discussed shortly.

AFC in recent approaches can be classified into two types depending on whether the conversion happens to photons residing in a cavity/resonator or while propagating through a medium (film/waveguide). Regardless, the duration or length of interaction plays an important role in improving the efficiency of the adiabatic frequency shift. The modulation duration (Δtchange) ideally needs to occur on a timescale less than or equal to the propagation time (tprop) of the signal through the system (Δtchange ≤ tprop). In the context of resonator/cavity medium, this condition can be interpreted in that the modulation should be much faster than photon lifetime (τp) in the cavity while being slower than the cavity round trip time (τrt) for gradual change to occur (τrt < Δtchange ≪ τp).132 

To further elucidate the interplay between the modulation period and interaction limit, we first need to understand the role of excitation pulse characteristics in the process of index modulation across the TVM. When considering an ideal time boundary in the optical regime, it can be understood as an optically modulated index front moving with an infinite speed. Therefore, all photons within the medium can uniformly and simultaneously experience the induced change. In this case, an implicit assumption for the optical excitation that would induce such instantaneous change in index is to be an impulse or Dirac-delta function of + δ ( t ) d t = 1 (implying it is a spike of infinite amplitude with no temporal/spatial width). This assumption is based on the fact that the excitation pulse's envelope shapes the spatiotemporal profile of modulated index. However, real-life experiments mostly rely on Gaussian pulses of finite duration and spatial width and the modulated refractive index profile (n + Δn) in TVM will closely follow the Gaussian envelope of pump.

In general, the spatial (Δp) and temporal pulse width (tp) of a Gaussian pulse are linked with the group velocity (vg,exc) via the relation Δp = vg tp. For a fast nonlinearity induced index change, we can assume the propagation velocity of the modulation front to be the same as the group velocity of the excitation pulse (vchange = vg,exc). As the modulation closely follows the temporal slope of the pulse front, and hence the modulation duration can be approximated as Δtchange ≈ tp/2 [Fig. 22(a)].

FIG. 22.

Illustrations of the frequency shift dynamics depending on spatiotemporal characteristics of optical pump-induced index modulation. (a) Schematic depicting the temporal dynamics of the pump pulse, shown in red and the induced index change, shown in black. As an example, a typical index change profile, of widely explored free carrier nonlinearities involving real transitions, is shown. Since, the rise in the index change is nearly instantaneous and it follows the pump's temporal response, we can assume the propagation velocity of the modulation front to be the same as the group velocity of the excitation pulse (vchange = vg,exc). Furthermore, since the frequency shift (Δω) is proportional to the rate of change of index (dn/dt), (b) depicts the rate of change of index in a black curve, and underlaid on the figure is the magnitude and direction of frequency shift of the signal photons, which are assumed to be stationary, for simplicity. The shift direction (blue for ω2 > ω1, red for ω2 < ω1) is usually opposite to the change of index experienced by the signal photons according to the time refraction relation given as ω2 = ω1 n1/n2. While both, red and blue shift of the frequency are possible, the largest frequency shift is observed at the rising edge (hence also labeled the modulation front). (c) When considering a finite size medium of length L, and signal photon propagating with a finite velocity v (black line shows the position of signal photon over time, t, the background color corresponds to the index, which varies linearly between n1 and n2, and the dashed gray lines correspond to the bounds of index changed experiences by the photon), there exists a bound on Δtchange, if the signal photons were to experience the full extent of index change (n1 → n2). The condition is that Δtchange should be faster than the propagation time, τprop, and the interplay is shown in (c). In other words, this implies pulses with small temporal widths are preferred. At the same time, (d) highlights the importance of a large vchange. A plot of the percentage shift calculated from Eq. (21) as a function of modulated index, n2 and modulation velocity vchange for photon of base wavelength 1550 nm normally incident upon a nondispersive medium with initial index, n1 = 1.5, is shown. The region to the right of the asymptote (white dashed line) shows that faster modulation velocity achieves large shifts for even smaller index changes. Near the lower extreme of vchange the shift becomes negligible. The equation used for the calculation is based on a simplistic model of relativistic photon propagation and therefore shift magnitudes/signs calculated from it are not exact but highlight the need of a large vchange. With vchange (or vg,exc) linked to Δtchange, by the relation Δp = vg,exc 2Δtchange, there exists a trade-off, since small temporal widths (Δp), small spatial widths and large group velocities, are not all simultaneously achievable, especially for fs pulses. Positive (pos.) and Negative (neg.). Panel (c) is adapted from Ref. 132.

FIG. 22.

Illustrations of the frequency shift dynamics depending on spatiotemporal characteristics of optical pump-induced index modulation. (a) Schematic depicting the temporal dynamics of the pump pulse, shown in red and the induced index change, shown in black. As an example, a typical index change profile, of widely explored free carrier nonlinearities involving real transitions, is shown. Since, the rise in the index change is nearly instantaneous and it follows the pump's temporal response, we can assume the propagation velocity of the modulation front to be the same as the group velocity of the excitation pulse (vchange = vg,exc). Furthermore, since the frequency shift (Δω) is proportional to the rate of change of index (dn/dt), (b) depicts the rate of change of index in a black curve, and underlaid on the figure is the magnitude and direction of frequency shift of the signal photons, which are assumed to be stationary, for simplicity. The shift direction (blue for ω2 > ω1, red for ω2 < ω1) is usually opposite to the change of index experienced by the signal photons according to the time refraction relation given as ω2 = ω1 n1/n2. While both, red and blue shift of the frequency are possible, the largest frequency shift is observed at the rising edge (hence also labeled the modulation front). (c) When considering a finite size medium of length L, and signal photon propagating with a finite velocity v (black line shows the position of signal photon over time, t, the background color corresponds to the index, which varies linearly between n1 and n2, and the dashed gray lines correspond to the bounds of index changed experiences by the photon), there exists a bound on Δtchange, if the signal photons were to experience the full extent of index change (n1 → n2). The condition is that Δtchange should be faster than the propagation time, τprop, and the interplay is shown in (c). In other words, this implies pulses with small temporal widths are preferred. At the same time, (d) highlights the importance of a large vchange. A plot of the percentage shift calculated from Eq. (21) as a function of modulated index, n2 and modulation velocity vchange for photon of base wavelength 1550 nm normally incident upon a nondispersive medium with initial index, n1 = 1.5, is shown. The region to the right of the asymptote (white dashed line) shows that faster modulation velocity achieves large shifts for even smaller index changes. Near the lower extreme of vchange the shift becomes negligible. The equation used for the calculation is based on a simplistic model of relativistic photon propagation and therefore shift magnitudes/signs calculated from it are not exact but highlight the need of a large vchange. With vchange (or vg,exc) linked to Δtchange, by the relation Δp = vg,exc 2Δtchange, there exists a trade-off, since small temporal widths (Δp), small spatial widths and large group velocities, are not all simultaneously achievable, especially for fs pulses. Positive (pos.) and Negative (neg.). Panel (c) is adapted from Ref. 132.

Close modal
Because of the finite speed and pulse width of the excitation, it also creates a spatial index gradient along its trail. Because of the gradient, the index modulation of the medium is nonuniform and the index contrast felt by the photons at any instant ( d n d t) becomes dependent on the slope of the spatial index profile and its velocity [Fig. 22(b)], which can be expressed as
d n d t = Δ n Δ t c h a n g e 2 Δ n t p = 2 Δ n Δ p . v change .
(20)
Therefore, maximizing the index contrast through a traveling modulation front necessitates a spatially and temporally compressed excitation pulse (small Δp,tp), characterized by its short duration. This preference holds true irrespective of the dispersion of the material. Hence, a short pulse with a low group velocity condition is favored for the small spatial width of the excitation pulse. However, it introduces a trade-off between requiring a short pulse and ne