ARTICLE https://doi.org/10.1038/s42005-020-0352-0 OPEN Mimicking bio-mechanical principles in photonic metamaterials for giant broadband nonlinearity ✉ Taeyong Chang 1, Suwan Jeon 1, Minsung Heo1 & Jonghwa Shin 1 1234567890():,; Microscopic structuring can change the effective properties of a material by several orders of magnitude. An example of this is animal bone, which has an effective elastic modulus that is more than 1,000 times larger than that of the constituent proteins. Here, we propose a broadband-enhancement principle of photonic nonlinearity that has a similar mathematical origin as the bone example. The proposed staggered array metamaterials violate the standard Miller’s rule in nonlinear optics and can enhance the third-order nonlinearity by more than a thousand to a billion times, depending on target operation frequencies. This metamaterial principle also enables manipulation of the individual components of the linear and nonlinear susceptibility tensors. Our biomimetic approach overcomes the fundamental speed-efficiency trade-off in current resonant enhancement schemes, making faster and more efficient all- optical devices possible for 1.55 μm wavelength. The principle is also applicable to ionic diffusion, heat conduction, or other transport problems. ✉ 1 Department of Materials Science and Engineering, KAIST, Daejeon 34141, Republic of Korea. email: [email protected] COMMUNICATIONS PHYSICS | (2020) 3:79 | https://doi.org/10.1038/s42005-020-0352-0 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0352-0 onlinear photonic processes, such as harmonic genera- mechanical and electromagnetic properties is presented in the Ntion, phase modulation, and multi-photon absorption, discussion section. have diverse applications including laser sources and From the point of view of materials, SAMMs can be considered optical communications1–3, three-dimensional lithography4, bio- homogeneous, crystalline photonic materials because their imaging5, and many others6–10. However, light–matter interac- periods are much smaller than the wavelength—as those of tion is predominantly linear for common transparent photonic natural crystals—and their effective material properties, such as materials, thus necessitating the injection of high-intensity arti- the linear and nonlinear susceptibilities and refractive indices, are ficial light sources along the optimal crystallographic directions11. well defined and remain constant regardless of the thickness of Therefore, numerous studies have been conducted with the aim to the sample22–24. For these metamaterials, their linear (n = 1) and attain photonic nonlinearity enhancement. Most of these studies χ~ðnÞ nonlinear (n > 1) homogenized electric susceptibilities eff can be have focused on the introduction of tailored electromagnetic reso- 13,19,25 – calculated based on the reciprocity theorem using the nances near the operation wavelength to boost light matter inter- χ~ðnÞð Þ 12–19 electric susceptibility tensors, d r , of the constituent materials actions . Both quantum-electrodynamic approaches involving ðωÞ 12,13 fi ð Þ¼ the engineering of intersubband transition energy levels ,and themselves and point-wise eld-enhancement factors, Fαi r classical electrodynamic enhancement methods—mostly utilizing α Á ðωÞð Þ=j ðωÞj ðωÞjj fi – Eloc r Eave (with Eave i), which quanti es how strong the photonic cavities or resonant structures13 19—have been proposed. ðωÞ α-directional components of the local electric fields (E ðrÞ) are However, these resonant-enhancement schemes exhibit an inherent loc fi ðωÞ trade-off relationship between the degree of enhancement and the when the homogenized, macroscopic electric eld (Eave ) is along frequency bandwidth, thereby hindering their practical applications. the i-direction. α and i are the unit vectors in the α- and i- Even for applications that do not require large bandwidths, large directions, respectively (see Methods and Supplementary Note 1 resonance-induced energy losses and a narrow fabrication error for detailed derivation). margin represent substantial obstacles. Moreover, only a few studies Using this approach, one can analytically derive the homo- have investigated the controllability of the tensor form of nonlinear genized nonlinear susceptibility tensor of the proposed structure susceptibilities10,13,18, despite the fact that it may have important for second-harmonic generation (SHG). In this section, we practical consequences. assume for the sake of simplicity, the use of a metal with perfect Here, we achieve both extreme enhancement and tensor form electric conductivity and an insulator with frequency- controllability of the linear and nonlinear susceptibilities of a independent nonlinear coefficients, thus resulting in the Klein- nonlinear staggered array metamaterial (SAMM) over a broad man symmetry condition11,26 formulated according to ðωÞ ðnωÞ bandwidth. The SAMM represents a periodic composite system Fβγ ¼ Fβγ ¼ Fβγ. These restrictions are eliminated in the of nonlinear materials and conductors with a deep subwavelength rigorous analysis of the actual enhancements and numerical period. We analytically and numerically verify that the third- demonstrations presented in the following sections. For the 9 order nonlinear susceptibility can be enhanced by more than 10 proposed structure, the local electric field is mainly oriented in 4 times at terahertz frequencies and by 10 times at optical tele- the z-direction inside the dielectric layers owing to the parallel- communication frequencies. In addition, we show that 18 dif- plate capacitor-like geometry of the metal plates above and below ferent crystal classes with second-order nonlinearity can be the dielectric. Subsequently, the relevant field-enhancement realized based on simple variations of the spatial configuration of factors are only Fzx, Fzy, and Fzz, and other factors are being a given nonlinear material and a conductor. The proposed negligibly small. The z-directional local electric field- broadband-enhancement principle is mathematically analogous enhancement factors under an x-directional macroscopic electric to that of naturally occurring mechanical SAMMs, such as animal fi fi = = eld, Fzx, of a speci c structure (ax 4800 nm, ay 800 nm, bones and bone-like materials (e.g., nacre and sea shell), which = = = – gx,y 0.2ax,y, and hd hm 20 nm) is plotted in Fig. 1e f (see have enhanced elastic modulus by more than three orders of Methods). The actual profiles of F and F are very simple with a 20,21 zx zy magnitude compared to their constituent soft proteins . large and almost spatially uniform value (around 120 in the aforementioned case) throughout the space between the top and bottom metal plates, and with a negligible value elsewhere. These Results fi = maximum eld-enhancement values are approximately |Fzx | Bespoke nonlinear susceptibility tensor of the proposed = 27 0.5ax/hd and |Fzy | 0.5ay/hd . For the nonlinear dielectric, we SAMM. The proposed SAMM structure is composed of alter- assume that the microscopic pole directions are either the +z-or nating metal and nonlinear dielectric layers (Fig. 1a). The metal −z-direction or unpoled, thus resulting in the intrinsic second- layers are composed of insulated metal patches in a planar lattice, ð Þ ð Þ order nonlinear susceptibility of the form χ 2 ðrÞ¼χ 2 pðrÞ with and the nearest metal layers are staggered with half-unit-cell 33 33 r =+ − shifts in both the x- and y-directions (denoted as “A” and “B” p( ) 1, 1, or 0, respectively. As a representative example, we fi fi r = layers in Fig. 1a). The metal and nonlinear dielectric layers (‘I’) consider a speci c spatial pole con guration of p( ) sign r – are stacked in the sequence A–I–B–I–. The lateral unit-cell [Fzy( )]. Figure 1g h show z-directional local second-order dimensions in the x- and y-directions, the lateral gap sizes in the nonlinear enhancement factor under an x-directional macro- fi χð2Þ 2 =jχð2Þj same directions, and the thicknesses of the metal and dielectric scopic electric eld, 33 Fzx 33 , of this example. The nonlinear layers are denoted as ax and ay, gx and gy, and hm and hd, enhancement factor is very large (around 14,000) and spatially respectively. Figure 1b depicts one unit cell of the SAMM. nearly uniform throughout the aforementioned high-field region The proposed microscopic (subwavelength) structure closely between plates. The contracted notation11,26,28 of the effective resembles the mechanical microscopic structure of mineralized nonlinear susceptibility simplifies to collagen fibrils in animal bone and other hard biological materials 0 1 a 2 ay 20,21 fi fi 000001 ð x Þ ð Þ (Fig. 1c) . In a simpli ed model, the brils in the bones are an B 16 hd hd C ð Þ ð Þ B a a a C array of isolated hard mineral plates embedded in a soft protein 2 2 1 ðax Þ2ð y Þðy Þ3 ð y Þ χ~ ¼ χ ; B 00 0 C matrix, and the mineral plates are in a staggered formation as in eff d 33@ 16 hd hd hd hd A Fig. 1a. Figure 1d schematically shows the microscopic deforma- ðay Þ 000h 00 tion of the mechanical SAMM under macroscopic tensile strain. d ð Þ The mathematical parallelism between the enhancement of 1 2 COMMUNICATIONS PHYSICS | (2020) 3:79 | https://doi.org/10.1038/s42005-020-0352-0 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0352-0 ARTICLE Fig. 1 Staggered-array metamaterial (SAMM) with extremely enhanced electromagnetic nonlinearity. a A schematic of the proposed SAMM structure. The structure is infinitely periodic in all three Cartesian directions. b A single unit cell of the crystal and the positions of representative cross-sections. The cross-sections normal to the x-, y-, and z-axes are denoted as ‘csx’, ‘csy’, and ‘csz’, respectively. c A macroscopic image and a schematic of the microscopic structure of the bone (adapted by permission from Springer Nature Customer Service Centre GmbH: Springer, Int. J. Fract., Gao21). d Microscopic structure of bone fiber with and without macroscopic tensile strain.