Tailoring one-dimensional layered metamaterials to achieve unidirectional transmission and reflection

We investigate elastic-wave propagation in a spatially dispersive multilayered, totally passive metamaterial system. At oblique incidence a longitudinal (acoustic) wave can convert to transverse in the solid material comprising the layers, but when the incident wave enters the multilayer from a solid as opposed to a liquid medium, the incident transverse component supported by the solid medium indirectly causes the longitudinal transmission response to be greatly modified, with a similar result for the transverse wave exiting the multilayer into a solid medium in response to an incident longitudinal wave. The conversion between longitudinal and transverse waves is found to lead to the emulation of a characteristic nonreciprocal phenomenon at some frequencies: a directionality in the transmission response, sometimes simultaneously with the reflection response. The directionality can be exploited, for example, in the construction of antiseismic structures or breakwater structures. The inclusion of gain and loss elements can strongly enhance the directionality. Periodicity-breaking defects can cause a great variability in the response, enabling the use of devices based on this phenomenon as sensors.

Figure 1

Metamaterial system under study: oblique wave impinging onto a multilayer stack composed of alternating gain (G) and loss (L) layers, separated by passive material. The materials comprising the layers, A (alumina) and Ep (epoxy thermoset), are discussed in the text. The angle of incidence $\theta$ shown corresponds to an incident longitudinal or transverse wave. The orientation shown corresponds to the forward (F) direction.

Figure 2

Longitudinal-mode response for an A/Ep multilayer system (16 total layers) with no G/L present between two glass half-spaces on which a transverse mode is incident at an angle of $\pi /16$ in either the forward (F) or backward (B) direction.

Figure 3

Longitudinal-mode response for an A/Ep multilayer system (16 total layers) between two glass half-spaces on which a transverse mode is incident at an angle of $\pi /8$ in either the forward (F) or backward (B) direction, (a) with no G/L present and (b) with GL.

Figure 4

Transverse-mode transmission response for an A/Ep multilayer system (16 total layers): (a) without G/L and (b) with G/L between two Ep half-spaces on which a longitudinal mode is incident at an angle of $\pi /16$ in either the forward (F) or backward (B) direction. In (a) both directions give the same result.

Figure 5

Same system and incident conditions as in Fig. 4, but with water replacing the second Ep, as taken from the F direction. Shown are both the transverse-mode reflection and transmission responses.

Figure 6

Transverse-mode transmission response for the same system and conditions as in Fig. 4, but with water or Hg replacing the second Ep, as taken from the F direction.

Figure 7

(a) Transverse-mode and (b) longitudinal-mode transmission response difference between the F and B directions for an A/Ep multilayer system (16 total layers) without G/L between two half-spaces of varying composition. In (a) a longitudinal mode is incident at an angle of $\pi /16$, while in (b) a transverse mode is incident at $\pi /16.$

Figure 8

(a) Transverse-mode and (b) longitudinal-mode transmission response relative directionality between the F and B directions for the same system as depicted in Fig. 7.

Figure 9

Longitudinal-mode transmission response difference between the F and B directions compared to the respective undefected system for a transverse mode incident at an angle of $\pi /16$ on an A/Ep multilayer system (16 total layers) without G/L between two half spaces of glass, HDPE, or polycarbonate.

Figure 10

Longitudinal-mode transmission response for an A/Ep multilayer system (16 total layers) without G/L between Ep/glass half-spaces (referring to the F direction) on which a longitudinal mode is incident at an angle of $\pi /16$ in either the forward (F) or backward (B) direction.

Figure 11

Transverse-mode reflection and transmission responses for an A/Ep multilayer system (16 total layers) without G/L between Ep/glass half-spaces (referring to the F direction) on which a longitudinal mode is incident at an angle of $\pi /16$ in either the forward (F) or backward (B) direction.