Field patterns: A new type of wave with infinitely degenerate band structure

Abstract

Field pattern materials (FP-materials) are space-time composites with PT-symmetry in which the one-dimensionalspatial distribution of the constituents changes in time in such a special manner to give rise to a new type of waves, which we call field pattern waves (FP-waves) [G. W. Milton and O. Mattei, Proc. R. Soc. A 473, 20160819 (2017); O. Mattei and G. W. Milton, https://arxiv.org/abs/1705.00539 (2017)]. Specifically, due to the special periodic space-time geometry of these materials, when an instantaneous disturbance propagates through the system, the branching of the characteristic lines at the space-time interfaces between phases does not lead to a chaotic cascade of disturbances but concentrates on an orderly pattern of disturbances: this is the field pattern. By applying Bloch-Floquet theory we find that the dispersion diagrams associated with these FP-materials are infinitely degenerate: associated with each point on the dispersion diagram is an infinite space of Bloch functions, a basis for which are generalized functions each concentrated on a field pattern, paramaterized by a variable that we call the launch parameter. The dynamics separates into independent dynamics on the different field patterns, each with the same dispersion relation.

Uniform-velocity spacetime crystals

“…The frequency transitions for a backward problem, with the incident wave coming from the right, in medium i, would be found by changing the signs of the superscripts and the sign of v m in (27b). Equations (27) are represented graphically in Fig. 5(b).…”

“…as obtained by rounding the corners of the patches in the figure). Some of the features of such crystals were studied in [23][24][25][26][27] using space-time ray tracing.…”

Spacetime modulation in floating thin elastic plates

“…In fact, time started recently to be considered along with geometry [15], as a design parameter for newly proposed spacetime metamaterials and metasurfaces, with many unprecedented applications [16], such as Fresnel drag [17], signal amplification [18], harmonic generation [19], and photonic circulators [20,21], to name a few. Some earlier studies considered dispersion relations in spacetime periodic media [22,23], and the dynamics * mohamed.farhat@kaust.edu.sa of light propagation in these structures [24,25]. Interestingly, similar effects have been observed in the context of acoustic and mechanical waves in time-modulated structures [26,27].…”

Edge Waves and Transmissions for Temporal Laminates and Imperfect Chiral Interfaces

“…Most notable has been the work of Fink [5], and Bacot et al [6] on image recovery through partial time reversal at temporal interfaces. The recent development in modelling of waves in media with temporal and spatial interfaces was stimulated by the work of Lurie [3], Milton & Mattei [7], and Mattei & Milton [8,9], who introduced the key ideas in this novel subject area.…”

“…With causality in place, a temporal interface still produces wave splitting so that, at a certain time when the elastic or inertial properties of the carrier medium change instantaneously, the wave front will split into two fronts, propagating in different directions. Special combinations of temporal and spatial interfaces can be analysed, where highly non-trivial wave patterns are observed, as discussed in [7,8,9]. These patterns of characteristic lines are known as "field patterns".…”