MetaMAT Weekly Seminars

In 2001, the word “metamaterial” was coined by R. M. Walser [WALSER 01 ], who gave the following definition: macroscopic composite having a manmade, three-dimensional, periodic cellular architecture designed to produce an optimized combination, not available in nature, of two or more responses to specific excitation. Since then, the definition of metamaterials has been polished to encompass all kinds of waves and a generally accepted definition is that proposed by M. Wegener and coauthors for which metamaterials are rationally designed composites made of tailored building blocks, which are composed of one or more constituent bulk materials, leading to effective medium properties beyond those of their ingredients [KADIC 19 ].

The concept of metamaterial is rooted in the nano-scale world and electromagnetism, with the seminal works of V. Veselago [VESELAGO 68 ] and J. Pendry [PENDRY 99 ] [PENDRY 00 ] that revolutionized the way researchers model light propagation in complex media, where the refraction index can take any value on the real line, or actually in the complex plane. Indeed metamaterials can have refraction index with a positive or negative real part, and same for their imaginary part, thanks to the concepts of passive and active media, in accordance with the Kramers-Kronig relations [GRALAK 10 ].

The refraction index need not be a complex number, it can also be a complex valued tensor describing anisotropic heterogeneous media with sign-shifting phases [MILTON 02 ] [SMITH 04 ] [CAI 10 ] [ZHELUDEV 12 ]. The fine structure of negatively refracting media usually consists of periodic arrangements of elements with size much smaller than the considered wavelength (typically hundreds of nanometers) that acquire effective properties of materials with negative optical index [PENDRY 00 ], or highly anisotropic materials such as hyperbolic metamaterials [IOR 13 ] [PODDUBNY 13 ] or invisibility cloaking devices [PENDRY 06 ], which are based on the form invariance of Maxwell’s equations that behave nicely under coordinate changes [NICOLET 94 ], unlike the Navier equations [MILTON 06a ], which are transformed into so-called Willi’s equations [WILLIS 81 ], which go beyond Newton’s second law [MILTON 07 ]. Two other cloaking techniques, via scattering cancellation [ALU 05 ] and anomalous resonances [MILTON 06b ] have also generated a huge interest in the applied mathematics community. The transition from the electromagnetic to acoustic metamaterials was made in particular possible thanks to phononic crystals, which are artificial handcrafted structures. They range from a few meters down to hundreds of nanometers or less. At this scale, matter appears as continuous and the laws of classical mechanics can be applied. The search for structures with complete phononic band gaps began in 1992 with work by Sigalas and Economou [ECONOMOU 93 ]. Interestingly, as unveiled in [OBRIEN 02 ] high-permittivity dielectric rods display stop bands induced by low frequency localized modes, which are associated with artificial magnetism.

In a similar way, high-density rods and spheres display unique features upon resonance, such as a negative effective density [LIU 00 ]. Just like the refractive index, the effective density need not be a real number, but is, in general, a complex valued matrix, as can be seen in mechanical metamaterials [CHRISTENSEN 15 ] [ACHAOUI 16 ] [MINIACI 16 ] [BERTOLDI 17 ] [KADIC 19 ]. Perhaps more surprisingly, the analogies drawn between electromagnetic, acoustic and mechanical waves propagating in complex media, have been further stretched to encompass heat, mass and light diffusion phenomena [GUENNEAU 13 ] [SCHITTNY 13 ] [SCHITTNY 14 ].