1. Introduction

Ever since Franken et. al. first demonstrated optical harmonic generation in a nonlinear crystal [1], marking the beginning of the field of nonlinear optics, there have been steady advances in the efficiencies of the various wave-mixing processes. In broad terms, there are two main principles to supporting highly efficient wave-mixing processes: first, one should make use of the largest available nonlinear coefficient; second, the particular configuration must avoid the destructive effects of phase mismatch. The former has led to the characterization of numerous nonlinear crystals and polymers in the search for ever-larger second- and third-order nonlinear coefficients [2, 3]. The latter, on the other hand, requires implementation of one method or another to coax the interacting waves to travel with the same phase [4, 5]. Many such ‘phase matching’ schemes have been proposed and employed, generally relying either on the properties of anisotropic materials, or on layered or otherwise graded mediums [6]. While these methods have seen success, there still exist many advantageous configurations and processes that are not easily phase matched due to the modest range of linear and nonlinear properties supported by natural materials. Configurations in which one or more interacting waves propagate counter to the others, in particular, have been the subject of numerous theoretical studies [7–11], with experimental realizations few and limited [12, 13].

In recent years, however, a variety of novel and anomalous properties have been demonstrated in a new class of artificial materials comprising periodic arrangements of subwavelength, polarizable elements. These artificial materials, known collectively as ‘metamaterials,’ have led to the realization of such exotic phenomena as negative refraction [14] and electromagnetic cloaking [15]. In addition, the inclusion of nonlinear elements has caused metamaterials research to spill over into the field of nonlinear optics [16], with demonstrations including frequency generation [17–19], parametric amplification [20], self-phase modulation, and bistability [21, 22]. Through careful design of their constituent elements, nonlinear metamaterials are capable of supporting an unprecedented range of linear and nonlinear properties. Considering again the two main guiding principles in constructing highly efficient nonlinear processes, nonlinear metamaterials offer unique advantages to both. While the enhancement of nonlinear coefficients has been studied previously [23], we concentrate here on the various phase matching schemes supported by nonlinear metamaterials.

We begin the analysis with the general expressions for phase matching, discussing the conventional techniques and their usages and limitations. Next, we define five broad categories of phase matching techniques in metamaterials: anomalous dispersion phase matching, birefringence phase matching, quasi-phase matching, negative-index phase matching, and index-near-zero phase matching, which make up Sections 3 through 7. While Sections 3 through 5 discuss the application of conventional techniques to metamaterials, Sections 6 and 7 outline unique phase matching configurations that have only been achieved in metamaterials. In each case, numerical and/or experimental examples are given to demonstrate the various forms of phase matching. Finally, we conclude our paper in Section 8, summarizing our results and offering an outlook for the potential applications of nonlinear metamaterials.

2. Wave-mixing and phase mismatch: an overview

Wave-mixing refers broadly to processes involving the interaction between multiple photons, mediated by a nonlinear medium. As such, these processes are subject to two major constraints: conservation of photon energy and conservation of photon momentum. For simplicity, we will focus on three-wave mixing in a medium with a bulk second-order nonlinear susceptibility, noting that the conclusions of this paper can be extended to higher-order wave-mixing processes. For three-wave mixing of plane waves in a homogeneous medium, energy and momentum conservation can be written as

Since the wave-mixing interaction requires spatio-temporal overlap between the interacting waves, most systems are restricted to the case where the wavevectors are co-linear. Taking propagation to be along the z-axis, we can write ${\overrightarrow{k}}_i=\pm \frac{n_i{\omega }_i}{c}\widehat{z}$, where the ± corresponds to propagation in the positive/negative z-direction. We summarize the four possible co-linear phase matching configurations in Table 1. In labeling these configurations, we borrow the conventional notation of birefringent phase matching, such that Type I refers to like-propagating lower-frequency waves, and Type II refers to unlike-propagating lower-frequency waves. While most nonlinear devices to date are based on the parallel-I configuration, the anti-parallel configurations are crucial to several highly intriguing and advantageous devices, such as the mirror-less optical parametric oscillator (MOPO) [7–10] and the nonlinear optical mirror [27]. However, since all natural materials are constrained by n > 0, the anti-parallel configurations require rather extreme techniques to overcome phase mismatch [25].

Table 1. Directions of Propagation and Corresponding Phase Matching Condition for the Four Co-linear Three-Wave Mixing Configurations

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Among artificial materials, metamaterials retain the advantage of being described by homogenized, constitutive parameters. Moreover, the characterization and language of nonlinear metamaterials has recently been extended to the standard formalism of nonlinear optics [26]. As such, the general equations and physics of wave mixing processes and phase matching used in the analysis of natural nonlinear materials are equally applicable to metamaterials. The advantage of metamaterials then, in this context, is that they vastly extend the achievable range of linear and nonlinear properties. These novel and often extreme properties, such as negative refraction, can offer a variety of solutions to phase matching in wave-mixing processes.

3. Anomalous dispersion phase matching

The parallel-I configuration is the simplest and most common wave-mixing setup, in which all three waves propagate in the same direction. It is easily shown that, considering waves of the same polarization and frequencies far from any absorption features, the refractive index of a medium increases monotonically with frequency and thus satisfaction of the phase matching condition given in Table 1 is impossible [6]. However, by carefully choosing these frequencies to be around a medium’s absorption feature, the anomalous dispersion in this frequency range can, in principle, exactly counteract the normal dispersion. This technique is known as anomalous dispersion phase matching, and has been achieved, for example, in polymer waveguides through the incoroporation of nonlinear chromophores with an absorption maximum between the lower-frequency and high-frequency waves [28].

Many metamaterials are purposefully designed around an analogous absorption feature, originating from a resonance in its constituent elements. As an example, consider the split-ring resonator (SRR) medium, composed of a periodic array of planar metallic rings with a capacitive gap [29]. The SRR displays an LC circuit resonance, and has been shown to be characterized, in a range of frequencies around its resonance, by a permeability [26]

4. Birefringence phase matching

Alternatively, one can consider mixing between waves of differing polarizations, mediated by the appropriate off-diagonal elements of the nonlinear susceptibility tensor. Thus, in an anisotropic medium, where the refractive index is dependent on a wave’s direction of propagation and polarization, the phase matching conditions in Table 1 can potentially be satisfied by selecting the polarizations of the interacting waves. The principal configurations for this technique, known as birefringent phase matching, are generally divided into Types I and II, where Type-I involves lower-frequency waves of the same polarization, and Type-II involves lower-frequency waves with orthogonal polarizations. In a uniaxial crystal, propagation of ordinary and extraordinary polarized waves is governed by the ordinary index $n_i^o$ and the extraordinary index $n_i^e(\theta )$ in the material, where

The anti-parallel configurations, on the other hand, require more extreme material properties. For example, let us consider Type II-(eoo) birefringent phase matching in the anti-parallel-IIa configuration for the nearly degenerate case, ω₁ ≈ ω₂, where wave 1 is anti-parallel to waves 2 and 3. Invoking (eoo) polarizations in the refractive indices of Table 1, phase matching in this case requires a birefringence of $n_1^e-n_1^o\ge 2n_3^o$. Taking $n_3^o\approx n_1^o$, this constraint implies an extraordinary index that is roughly three times the ordinary index, something no natural material has been found to support. If instead ω₁ << ω₂, this constraint is relaxed, leading to speculation that birefringent phase matching for backward wave oscillation will likely only be achievable with a signal frequency in the mid- or far-infrared spectrum, while the idler and pump are near-infrared or higher frequency waves [7].

In addition, birefringent phase matching requires a sufficiently strong cross-term in the nonlinear tensor, corresponding to the direction of propagation and polarizations required by phase matching considerations. The strength of the various cross-term nonlinear coefficients vary by material and are often identically zero due to crystal symmetries [2]. These constraints on both the linear and nonlinear tensors limit the desirable wave-mixing configurations, as well as the useable nonlinear materials.

Metamaterials, meanwhile, have greatly extended the achievable types and magnitudes of material anisotropy, introducing, for example, indefinite media [30]. Anisotropic metamaterials with extraordinary indices of refraction in excess of five times the ordinary indices of refraction have been demonstrated at terahertz frequencies [31]. Such large and configurable anisotropies can lift many of the conventional limitations in applying birefringence phase matching to natural materials. In addition, engineering of the nonlinear tensor in metamaterials can ensure that the particular cross-term involved is both non-zero and sufficiently large.

As a simple and demonstrative example, we consider a metamaterial composed of overlapping silver bars arranged in a cubic lattice with a lattice constant of 1 μm, immersed in a hypothetical nonlinear dielectric with ɛ_d = 2ɛ₀ and d₁₁ = 10 pm/V. Such a structure is easily fabricated by existing techniques, with a frequency range of operation that includes the far-infrared wavelength of 10.6 μm, corresponding to the output of a CO₂ laser. Labeling the crystal axes as in Fig. 1(b), this metamaterial supports coupling between the structure and the incident fields only for electric fields polarized in the Z-direction. Furthermore, the fields coupled into the structure naturally localize in the capactive gaps between overlapping bars, with a dominant electric field component in the X-direction. The symmetry of the metamaterial prevents linear coupling between these polarizations, and thus the metamaterial is strongly biaxial with effective linear susceptibility tensors that are diagonal in the XYZ basis. The resulting nonlinear tensors, however, are not diagonal. Considering Type-I(eeo) and propagation in the YZ-plane, there is significant overlap of the X-components of the electric fields of the three modes in the capacitive gaps, and the metamaterial thus supports a non-zero d₃₅ nonlinear coefficient. This artificially engineered nonlinearity, combined with the massive anisotropy, can support birefringence phase matched oscillations without a mirror, with distinct benefits compared to alternate implementations.

 

Fig. 1 A nonlinear metamaterial consisting of overlapping silver bars embedded in a nonlinear dielectric, designed to operate as a birefringence phase matched MOPO. (a) Schematic of a single layer of the metamaterial, showing the propagation direction and angle relative to the crystal axes. (b) The basic unit-cell of the metamaterial. (c) The retrieved extraordinary and ordinary indices of refraction. (d) Plot of the phase matched signal and idler frequencies as a function of angle. (e) Real and imaginary parts of the retrieved nonlinear coefficient.

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To verify the linear and nonlinear properties, we implement full-wave, finite element simulations on a single unit-cell of the nonlinear metamaterial using COMSOL Multiphysics. We employ periodic conditions in the transverse directions and implement a Drude model for the dielectric function of silver [33], using a plasma frequency of 2179 THz and a collision frequency of 4.35 THz. We then retrieve the linear properties via standard effective parameter retrieval techniques [32]. Constraining propagation to the YZ plane, the metamaterial is positive uniaxial to within simulation error, with extraordinary and ordinary waves corresponding to TM and TE polarizations, respectively. The principal values of the extraordinary and ordinary indices are shown in Fig. 1(c), displaying anomalously large birefringence, as expected. Furthermore, we perform Type-I(eeo) difference frequency generation simulations using the techniques outlined in Ref. [23], using an ordinary-polarized 10.6 μm pump wave as wave 3, and sweeping the frequencies of waves 1 and 2. We use the results of these simulations in the transfer matrix-based nonlinear retrieval method to determine the d₃₅ nonlinear coefficient [34], plotted in Fig. 1(e). Although the cross-terms of the local nonlinear tensor in the dielectric are zero everywhere, the metamaterial supports an effective d₃₅ coefficient that is both non-zero and several times larger than the d₁₁ in the background dielectric, owing to the field localization enhancement effect [23]. Renaming waves 1, 2 and 3 to signal, idler, and pump, these results are immediately applicable to the nonlinear process of optical parametric oscillation. Indeed, the anisotropy is large enough to support birefringent phase matching of a counter-propagating signal wave, corresponding to the anti-parallel-IIa configuration. By rotating the direction of propagation z relative to the optical axis Z, depicted in Fig. 1(a), the anti-parallel-IIa phase matching condition can be satisfied for signal and idler frequency pairs over a wide frequency range, as shown in Fig. 1(d). Thus, for pumping with a CO₂ laser, this metamaterial can be expected to support mirror-less oscillations, generating a tunable signal wave with a frequency ranging from 1 to 8 THz. Moreover, the natural material chosen as the embedding dielectric does not require significant birefringence nor nonlinear cross-terms, allowing for high flexibility in the choice of embedding material, and, consequently, the potential achievement of competitively-low oscillation thresholds. We note that the large anisotropy, however, is accompanied by proportionately large walk-off, except near θ = 90°, corresponding to ω₁ ≈ 2π × 8.18 THz and ω₂ ≈ 2π × 20.10 THz. Similarly, walk-off can be eliminated at any single signal-idler frequency pair by redesigning the anisotropy of this metamaterial to ensure that the principal values of the refractive indices satisfy the phase matching condition.

5. Quasi-phase matching

If instead we allow for an inhomogeneous medium, a third phase matching technique becomes available. This technique, known as quasi-phase matching (QPM), relies on the introduction of some sort of periodicity in the direction of propagation [4, 5]. Thus, the wavevectors can be conserved up to a multiple of the reciprocal lattice vector $\overrightarrow{G}=\frac{2\pi }{\Lambda }\widehat{z}$, where Λ is the length of one period. Typically, the inhomogeneity employed is a periodic modulation of the sign of the nonlinear susceptibility, called periodic poling, but it can be much more general [35]. Including the contribution from the periodicity in the above phase relations, QPM is achieved for Δk = mG⃗, or equivalently Λ = mL_coh, where m is any integer. As such, the possible frequencies for phase matching are strictly limited by the choice of poling period, with small associated tuning bandwidths [35]. The effective nonlinear coefficient of the wave-mixing process, however, is necessarily reduced compared to the bulk value, with a reduction factor of 2/π in the case of first-order QPM [6].

QPM in natural nonlinear crystals has been achieved for second-harmonic generation in the anti-parallel-I configuration [12] and for MOPO in the anti-parallel-IIa configuration [13]. Due to the massive phase mismatch introduced by the backward-propagating signal, the poling period required in the MOPO was a remarkable 800 nm. In principle, the other anti-parallel configurations presented in Table 1 can be achieved in a similar fashion, though the fabrication constraints of such small poling periods will invariably limit the useable materials, frequencies, and interaction lengths.

QPM extends naturally to metamaterials, which have proven viable options in various laminar and gradient-style devices [15, 36]. At microwave frequencies, periodically poled QPM has been demonstrated by the precise orientation of the nonlinear inclusion itself (see Fig. 2(a)) [37]. At infrared and visible frequencies, however, the application of periodically poled QPM will depend on the metamaterial design. In nonlinear metamaterials composed of periodic metallic structures embedded in a nonlinear dielectric [23], for example, the conventional techniques of ferroelectric domain engineering can be applied directly to the embedding dielectric. Assuming it represents the dominant source of nonlinear activity, the reversal of the phase of the embedding dielectric’s nonlinear susceptibility will necessarily cause an inversion of the sign of the effective bulk nonlinearity. In any case, precise control of the metamaterial unit-cell can allow for complex spatial distributions of the nonlinear parameters for the purpose of QPM.

 

Fig. 2 (a) Diagram of the periodic poling technique employed in Ref. [37] for the varactor-loaded split-ring resonator medium, whereby the phase of the nonlinear coefficient is periodically flipped by reversing the orientation of the nonlinear element in each individual unit-cell. (b) Schematic of tunable parallel-I QPM difference frequency generation in an active metamaterial Bragg cell. An external stimulus is used to produce a periodic variation in the linear properties of the metamaterial with a tunable period Λ.

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The generalized form of QPM, which utilizes a periodic variation in any of the electromagnetic material properties, is particularly attractive in the special class of active metamaterials. This stems from the fact that active metamaterials have been shown to support dynamically tunable linear properties, using active mechanisms such as photoconductivity, electrical bias, and temperature tuning [38–41]. In such a system, one can envision introducing a periodic grating with a tunable periodicity, analogous to an acoustically-induced Bragg cell but with a modulation depth several orders of magnitude larger, depicted in Fig. 2(b). Thus, it was demonstrated theoretically that a wave-mixing process taking place in an active metamaterial Bragg cell can support QPM over a widely tunable frequency range [42]. However, active metamaterials of this kind can be difficult to realize for frequencies above the THz regime. Furthermore, this form of tunable QPM comes at the price of a largely reduced effective nonlinear coefficient.

6. Negative-index phase matching

One of the first and most exciting applications of metamaterials was the achievement of a negative index of refraction through the engineering of a composite with both Re[ɛ] < 0 and Re[μ] < 0 in a certain frequency band [14, 43]. This, in turn, spurred several theoretical studies of the nonlinear phenomena supported by nonlinear negative-index mediums [27, 44, 45]. These studies showed that the phase matching conditions of the anti-parallel wave-mixing configurations could be satisfied if the backward-propagating wave was in a negative-index band. This arises from the fact that the wavevector and Poynting vector are oppositely directed in a negative-index medium, such that a wave propagating in the negative z-direction supports a positvely-directed wavevector. In particular, there were two processes that proved highly intriguing from a theoretical viewpoint: mirror-less optical parametric amplification and oscillation, and the nonlinear optical mirror effect.

The so-called nonlinear optical mirror effect is the name given to a device that both converts the frequency of an incoming wave and redirects it backwards. As originally conceived, the device operates via the degenerate case of the anti-parallel-I configuration, in which the system is excited by a forward-propagating fundamental wave, generating a backward-propagating, or ‘reflected’, second-harmonic [27]. Thus, in a homogeneous medium, the phase matching condition is given by n₁ = −n₃ and requires either the fundamental or second-harmonic wave to propagate in a negative-index band and the other in a positive-index band. Such a device has been demonstrated at microwave frequencies, shown schematically in Fig. 3(b) [37]. The dispersion relation of the nonlinear metamaterial was found to be

 

Fig. 3 (a) Plot of the calculated coherence lengths for parallel-I (dashed purple) and anti-parallel-I (solid orange) second-harmonic generation in a negative-index waveguide. The index of refraction at the fundamental frequency is included (dotted black), showing the negative-index and index-near-zero regimes (vertical dashed lines). (b) Schematic of the nonlinear optical mirror effect in a negative-index metamaterial. (c) Schematic of simultaneous QPM in a periodically poled index-near-zero metamaterial. S_i refers to the direction of energy flow of the i^th wave.

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The mirror-less parametric processes could be achieved through either of the anti-parallel-II configurations, employing wave 3 as a pump. The backward-propagating wave provides an automatic feedback mechanism, removing the need for a mirror and the associated calibrations and complexities. While the dynamics of a negative-index MOPO are analogous to the previously discussed quasi-phase matched MOPO, the materials involved and the nature of the phase matching are entirely different. However, the losses associated with negative-index mediums will likely limit the overall efficiencies of wave-mixing processes. Negative-index phase matching should thus have its most meaningful applications in those devices specifically based on negative refraction [46, 47].

7. Index-near-zero phase matching

Furthermore, there exists an interesting class of both natural materials and metamaterials whose real parts of the refractive index vanish at some frequency, with a number of intriguing characteristics as a consequence [48, 49]. In general, this material property can be found near the plasma frequencies of various metals, or in any material with a sufficiently strong Lorentzian contribution to its material properties [50]. If any of the waves involved in a wave-mixing process should propagate in such an index-near-zero band, it can be seen from Table 1 that the phase matching conditions of multiple unique configurations become degenerate. For example, if n₃ = 0, the anti-parallel-IIa and anti-parallel-IIb configurations are both satisfied by n₁ω₁ = n₂ω₂. This characteristic of index-near-zero materials can be used to achieve simultaneous phase or QPM of two or more configurations, opening avenues to a wide range of simultaneous and/or cascaded nonlinear processes.

Indeed, simultaneous QPM of degenerate (ω₁ = ω₂) parallel-I and anti-parallel-I configurations has been achieved at microwave frequencies, using the same metamaterial described above by Eqs. (7) and (8), but in the region where n₁ ≈ 0 [37]. Figure 3(a) demonstrates that, at the index-near-zero frequency given by the second dashed vertical line, the coherence lengths of the parallel-I and anti-parallel-I processes are equal. Thus, periodic poling was employed with a poling period of $\Lambda \approx \frac{2\pi }{\Delta k}$, where Δk ≈ |k₃| for both configurations, so that an incident fundamental wave was able to simultaneously generate second-harmonic waves in the positive and negative z-directions, shown schematically in Fig. 3(c).

8. Conclusion

We have studied the potential phase matching solutions for nonlinear metamaterials in the context of co-linear three-wave-mixing. We find that metamaterials support multiple paths to overcoming phase mismatch, including alternate and even novel methods compared to those used in natural materials. These results validate the potential application of metamaterials to nonlinear devices by demonstrating the feasibility of phase matching in such structures. Moreover, we have shown that the novel phase matching configurations provide advantages of their own, especially with regard to exotic devices such as MOPOs and nonlinear optical mirrors, in which one of the interacting waves propagates opposite to the others. Birefringence phase matching in metamaterials deserves particular attention due to the unprecedented level of engineering that metamaterials offer in both the linear and nonlinear tensors. Through this engeineering, we have demonstrated numerically a relatively simple and flexible metamaterial design for a MOPO that can produce widely tunable THz radiation from the output of a CO₂ laser.

However, there remain several important limitations to the application of metamaterial-based nonlinear devices. First, fabrication considerations will inevitably limit the complexity and scale of the desired bulk metamaterial mediums. One potential solution is the implementation of thin films or channels of metamaterials, effectively changing the problem to that of wave-mixing in a dielectric waveguide. Though not considered here, a similar analysis could be carried out to include waveguide dispersion with analogous conclusions. More fundamental is the subject of losses, which are invariably introduced by the inclusion of metal structures. While it has been shown that non-resonant nonlinear metamaterials can potentially support only modest absorption [23], metamaterials based on metallic components will tend to be limited to operation in the THz and far- and mid-infrared frequency ranges, where metals are less lossy. Otherwise, the bulk metamaterial must be sufficiently compact compared to an absorption length, with a consequent reduction in total efficiency. These constraints are partially responsible for the fact that current experimental demonstrations of phase matching in metamaterials are limited to microwave frequencies. However, the potentialities of metamaterials, both in terms of enhanced nonlinearities and phase matching, are great enough that these limitations should not be considered insurmountable. We expect this work to provide a crucial step in connecting current research in the design of metamaterials with outstanding linear and nonlinear properties, to the functional goal of highly efficient and intriguing nonlinear devices.