Brillouin-zone definition in non-reciprocal Willis monatomic lattices

Exploration of new wave phenomena has been a driving motivation in wave propagation research¹ with a growing interest in elastodynamic non-reciprocity.² Within linear, time-invariant elastic media, a wave emanating from a source towards a receiver must not behave differently if the locations of the source and receiver are interchanged, thanks to elastodynamic reciprocity. Designs of artificial materials have enabled new avenues to manipulate waves beyond what is possible in natural materials, and breakage of wave reciprocity is no exception. Given its potential in enabling novel applications (e.g., one-way vibrational isolators³) researchers have introduced a variety of structural designs tailored for non-reciprocity, such as exploiting nonlinear instability with supra-transmission phenomenon.⁴ Comparably, granular crystals, with their nonlinear and bifurcating-chaotic behavior, have been exploited to asymmetrically transmit energy, with potential application in sensing and energy harvesting.⁵ Another choice to achieve non-reciprocity is by spatiotemporally modulating mechanical properties, with artificial momentum bias forcing waves to propagate differently in the forward and backward directions.^3,6–10

When elastic media defy elastodynamic reciprocity, their dispersion diagram becomes skewed and asymmetric about a zero wavenumber, and conventional Brillouin-zone (BZ) definitions are rendered obsolete. Cassedy and Oliner are amongst the first contributors to study the skewed (non-reciprocal) nature of BZ in an attempt to generalize its definition for wave propagation in space-time modulated electrical circuits.¹¹ Unlike spatial-only modulation with dispersion diagrams perfectly repeating along the wavenumber axis, the presence of time modulation creates dispersion-diagram periodicity along a line with a slope that is a function of such modulation (under small modulation amplitude assumption). However, the adequacy of traditional BZs to fully capture wave non-reciprocity remains an ongoing concern, especially for two-dimensional systems.⁸ 

Examining Eq. (4) and Fig. 1, it is first deduced that the motion of individual masses m is physically coupled through springs k, as in typical MLs.^14,15 Having the momentum bias introduced by the rod's constant speed, the physics of the lattice's motion is influenced in two ways:

1. Besides the spring k, the ith unit-cell displacement is coupled with its next neighbors through negative springs of coupling coefficient $− m v 0 2 / a 2$⁠, which remains negative regardless of the sign of v₀ (or β). Consequently, the effective stiffness (⁠ $k e = k − m v 0 2 / a 2$⁠) of the WML reduces as the rod's speed v₀ increases, jeopardizing dynamical stability when $m v 0 2 / a 2 > k$ (equivalent to $| β | > 1$⁠) as a consequence of its effective stiffness k_e becoming negative.

2. Non-local coupling terms emerge, which are proportional to the velocity of next-neighboring masses of the ith unit cell. Such non-local interactions (related to Willis coupling¹⁶) arise from discretizing the mixed derivative in Eq. (1) and dictate the strength of lattice's non-reciprocity. While the physical motion of the elastic medium enables such non-local coupling, it may be alternatively achieved via feedback control.^17–19 Lastly, it is noteworthy that such non-local (positive-negative) couplings are akin to skew-symmetric gyroscopic coupling studied in literature.^20–25 

Equations (10) and (13) uncover that the BZ in its entirety is shifted and it is within the range $q ∈ [ − π + ϕ , π + ϕ ]$⁠, with its width remaining constant at $2 π$ in analogy to a reciprocal ML. Depending on the sign of β (and subsequently $ϕ$⁠), the BZ moves forward or backward, as seen in the middle row of Fig. 2. Not only that non-reciprocity shifts the BZ, but it also forces the range of the wavenumber corresponding to the forward-going and backward-going waves to be unequal. To maintain a constant BZ width of $2 π$⁠, the amount of shrinkage (or extension) in the forward-going wave wavenumber range (i.e., $q ∈ [ 0 , π + ϕ ]$⁠) is compensated by a larger (smaller) wavenumber range of the backward-going waves (i.e., $q ∈ [ − π + ϕ , 0 ]$⁠), depending on the sign of $ϕ$⁠. This shift $ϕ$ disappears when the relative speed β is zeroed out as expected, which is verifiable from Eq. (13), and the traditional BZ range (⁠ $q ∈ [ − π , π ]$⁠) is recovered.

A few additional observations from the driven-wave dispersion in Fig. 2. (i) The attenuation at a frequency higher than the cutoff frequency $Ω max = 2$ is smaller in WMLs compared to a reciprocal ML. (ii) The real component of the wavenumber with $Ω max > 2$ in WMLs does not have a constant value and varies as the frequency increases, unlike its reciprocal counterpart with a constant real wavenumber of $± π$ within the attenuation zone (⁠ $Ω max > 2$⁠). (iii) The BZ defined in MLs with β = 0 is symmetric about its center, allowing for an irreducible BZ to be defined within the range $q ∈ [ 0 , π ]$⁠. As WMLs have asymmetric dispersion branches, reducing the first BZ may be elusive. A zero wavenumber, however, remains the point that divides the forward- and backward-going waves for both lattices. Finally, the inequivalence of the forward and backward wavenumber ranges is numerically verified via a spatiotemporal fast-Fourier transform (FFT) of the time response of a finite WML (bottom row of Fig. 2), exhibiting excellent agreement with the analytical results of the newly defined BZ definition in the middle row (see supplementary notes 2 for more details).

In a reciprocal square ML, the vertical (horizontal) boundaries of its perfectly square BZ correspond to a vanishing group velocity in the x-direction (y-direction) only. In extension, the same condition shall be imposed on the non-reciprocal WML case to find the horizontal and vertical boundaries of the shifted BZ. By doing so, it is found that the BZ for WML is no longer a square and has irregularly shaped boundaries. Yet, its area remains a constant value of $4 π 2$⁠, identical to its reciprocal case, regardless of the magnitude of β_x and β_y [see Figs. 3(a), 3(b) and supplementary Fig. S3(b)].

Another way to depict the two-dimensional dispersion relation is by using directivity plots and reducing the two variables q_x and q_y into a single (direction) angle equal to $ψ = tan − 1 ( q y / q x )$⁠. In the reciprocal case, all four quadrants of the angle ψ show identical profiles of the dispersion surface, and the same cannot be said when $β x ≠ 0$ and/or $β y ≠ 0$ [Fig. 3(c)]. Complete dispersion surfaces in the directivity plot are generated if the wavenumber values within the non-reciprocal BZ boundaries [shown in Fig. 3(b)] are substituted to the dispersion relation, which is not the case if the traditional BZ values of the wavenumbers q_x and q_y are used instead [see Fig. 3(c) and supplementary Fig. S4]. Finally, a spatiotemporal FFT of the time response of a square WML of finite size is shown in Fig. 3(d) for $β x = β y = 0$ (reciprocal case) and $β x = 0.5$ and $β y = 0.75$ (non-reciprocal case). The FFT contours exhibit a close agreement with their analytical counterparts in Fig. 3(c), substantiating the approach's validity and the completeness of the newly defined BZ (see supplementary note 2 for more details).

In summary, this study demonstrates that the Brillouin zone (BZ) remains constant in width/area with non-reciprocity in a class of Willis monatomic lattices (WMLs), yet its boundaries are shifted in a precisely quantifiable amount. Synthesized from a modified wave equation of longitudinal waves in moving elastic rod,¹² the one-dimensional WMLs is studied analytically and its driven-wave dispersion relation is proven vital for the quantification of the BZ shift $ϕ$⁠. It is shown that the proposed theory perfectly agrees with the numerical simulation, validating BZ shifting, its constant width, and inequivalence of wavenumber ranges occupying the forward-going and backward going waves. It is also established that an increase (decrease) in the forward-going wavenumber range is compensated by a shrinkage (enlargement) in the backward-going one. An extension to a square WML is also established, demonstrating the shifted nature of the BZ and its area being constant and unaffected by the degree of non-reciprocity, dictated by the velocity modulation in the x and y direction. Additionally, it is shown that the two-dimensional BZ is no longer a square, and its deformed vertical (horizontal) boundaries are found by setting the group velocity in the x (y) propagation direction to zero. The established clarification on the definition of BZs is envisioned to be a stepping stone for further investigations in BZ quantification for different types of modulations for wave non-reciprocity, as well as for periodic media with multi-mode dispersion relations.

See supplementary material at https://doi.org/10.1121/10.0022535 for a discussion on group velocity and BZ phase shift, numerical validation procedure, and derivations related to square WMLs (notes 1, 2, and 3, respectively). In addition, supplementary video animations are provided for numerical simulations of one-dimensional WMLs, as well as BZ definition of square WMLs with various values of parameters β_x and β_y.

The author has no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.