Brillouin-zone (BZ) definition in a class of non-reciprocal Willis monatomic lattices (WMLs) is analytically quantified. It is shown that BZ boundaries only shift in response to non-reciprocity in one-dimensional WMLs, implying a constant BZ width, with asymmetric dispersion diagrams exhibiting unequal wavenumber ranges for forward and backward going waves. An extension to square WMLs is briefly discussed, analogously demonstrating the emergence of shifted and irregularly shaped BZs, which maintain constant areas regardless of non-reciprocity strength.

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 

In this effort, BZ definition is unraveled for a class of non-reciprocal monatomic lattices (MLs), synthesized from a non-reciprocal wave equation governing longitudinal waves in an axially moving elastic (continuum) rod as follows:12,
ρ 2 u t 2 + 2 ρ v 0 2 u x t + ( ρ v 0 2 E ) 2 u x 2 = 0 ,
(1)
where ρ, E, and u(x,t) denote the density, modulus of elasticity, and the longitudinal displacement as a function of space x and time t of the rod, respectively. The rod's constant speed is defined as v 0 = β c, where c = E / ρ is the elastic-medium's sonic speed and β [ 1 , 1 ] [+(–) sign indicates forward (backward) motion] is the non-dimensional relative moving velocity.12 As a matter of fact, the nature of non-reciprocity of Eq. (1) is intertwined with that of elastic rods with spatiotemporally modulated properties. At the low-frequency limit and slow modulation, the macroscopic behavior of an elastic rod with a spatiotemporal wave-like modulation is governed by a Willis-type equation,7 which is identical in form to that of Eq. (1) for moving rods.12 Not only that Eq. (1) is of Willis type, but it is also gyroscopic as it can be re-casted as13 
M 2 u t 2 + G u t + K u = 0 ,
(2)
where M = ρ , G = 2 ρ v 0 x, and K = ( ρ v 0 2 E ) x 2 are the mass, gyral, and stiffness operators. Whether subjected to spatiotemporal modulation or motion at a constant speed, elastic rods obeying Eq. (1) exhibit linear momentum bias (inducing non-reciprocity), as evident from the mixed derivative term. The ramifications of the induced non-reciprocity on BZ shall be first established for the lattice version of Eq. (1) and then extended to a two-dimensional counterpart.
To find an equivalent ML to that of the moving rod, henceforth referred to as Willis monatomic lattices (WMLs), the spatial derivatives in Eq. (1) are discretized in the spatial domain x using the central differencing scheme, resulting in
ρ u ¨ i + ρ v 0 ( u ̇ i + 1 u ̇ i 1 a ) + ( ρ v 0 2 E ) ( u i + 1 2 u i + u i 1 a 2 ) = 0 ,
(3)
where a is the finite difference spacing and constitutes the lattice's constant (Fig. 1). The spring constant and mass of WML are defined as the effective stiffness k = E A / a and mass m = ρ A a of a rod segment of length a and area A. Using the introduced parameters, a few mathematical manipulations result in the equation of motion for the ith unit cell of WML,
m u ¨ i + m v 0 a ( u ̇ i + 1 u ̇ i 1 ) + ( k m v 0 2 a 2 ) ( 2 u i u i + 1 u i 1 ) = 0 .
(4)
Fig. 1.

Schematic of a non-reciprocal Willis monatomic lattice (WML), synthesized from discretizing Eq. (1). The lattice's masses m are evenly spaced with a lattice constant a and the displacement of the ith mass is donated as ui. As seen from the figure, two types of springs exist: (i) a conventional spring k arising from the elasticity of the rod and (ii) a negative spring m ( v 0 / a ) 2 induced as a consequence of the rod's motion. Non-local coupling parameters that are proportional to the velocity of the next neighbors of an ith unit cell break the lattice's reciprocity (shown for the ith unit cell only).

Fig. 1.

Schematic of a non-reciprocal Willis monatomic lattice (WML), synthesized from discretizing Eq. (1). The lattice's masses m are evenly spaced with a lattice constant a and the displacement of the ith mass is donated as ui. As seen from the figure, two types of springs exist: (i) a conventional spring k arising from the elasticity of the rod and (ii) a negative spring m ( v 0 / a ) 2 induced as a consequence of the rod's motion. Non-local coupling parameters that are proportional to the velocity of the next neighbors of an ith unit cell break the lattice's reciprocity (shown for the ith unit cell only).

Close modal

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 ke 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 

Assuming harmonic motion u i = u ̂ i e i ω t and applying Bloch boundary conditions u i ± 1 = e ± i q u i, where q is the non-dimensional wavenumber, ω is the excitation frequency, and i is the imaginary unit, a non-dimensional dispersion relation is derived from Eq. (4),
Ω 2 + 2 Ω β sin ( q ) 4 ( 1 β 2 ) sin 2 ( q 2 ) = 0 .
(5)
Here, the definition of the non-dimensional frequency is Ω = ω / ω 0, where ω 0 = k / m. Note that v 0 = β c a β ω 0. The dispersion relation in Eq. (5) has a single (positive-frequency) dispersion branch, which reads
Ω = 2 | sin ( q 2 ) | 1 β 2 sin 2 ( q 2 ) β sin ( q ) .
(6)
The term sin ( q ) is what makes the dispersion relation in Eq. (6) non-reciprocal as its sign changes simultaneously with q. Corroborating our earlier observation regarding the lattice's dynamical stability with the speed v0, Eq. (6) indicates that | β | > 1 yields complex values of the frequency Ω, signaling dynamical instability.
Conventionally, the dispersion relation is plotted by sweeping the non-dimensional wavenumber within the first BZ of the range q [ π , π ] and solve for Ω in Eq. (6) for each value of q, a method known as free-wave dispersion relation. The results of this process are shown in the top row of Fig. 2 for different values of β, where β 0 induces dispersion asymmetry about zero wavenumber (i.e., q = 0). Note that β = 0 recovers the dispersion diagram of a classical (and reciprocal) ML15 and results in a perfectly symmetric dispersion about the center of BZ (q = 0). However, the conventional BZ does not accurately capture the range of wavenumbers spanning the forward-going and backward-going waves for β 0. To show that, Eq. (5) can be reformulated to solve for the wavenumber q by having the frequency Ω as an input, resulting in the driven-wave dispersion relation. After rewriting the trigonometric terms in their exponential form, one can show that
( 1 β 2 i β Ω ) e i q + ( Ω 2 2 ( 1 β 2 ) ) + ( 1 β 2 + i β Ω ) e i q = 0 .
(7)
Fig. 2.

Dispersion diagram of WMLs, depicted using free-wave (top) and driven-wave (middle) methodologies. A positive (negative) β results in a positive (negative) shift ϕ, as seen from the bias towards the forward-going (backward-going) waves, while β = 0 recovers the dispersion diagram of a reciprocal ML. However, the free-wave dispersion does not capture the shifted nature of BZ, showing the driven-wave approach significance in unraveling the BZ shift ϕ. Observe that the attenuation (represented by the imaginary component of the wavenumber in the driven-wave dispersion) is smaller in WMLs relative to its reciprocal ML counterpart (β = 0) when Ω > 2, i.e., a frequency higher than the lattice's cutoff frequency Ω max = 2. Also, Ω > 2 results in a non-constant real component of the wavenumber in WML cases only. Note that, for β = ± 0.5 shown here, the corresponding BZ shift is ϕ 0.3 π. (Bottom) Spatiotemporal FFT for the time response of WML for an impulse excitation in the left and right ends, quantifying forward- and backward-going wavenumbers, respectively, and demonstrating excellent agreement with the new BZ definition in the middle panel.

Fig. 2.

Dispersion diagram of WMLs, depicted using free-wave (top) and driven-wave (middle) methodologies. A positive (negative) β results in a positive (negative) shift ϕ, as seen from the bias towards the forward-going (backward-going) waves, while β = 0 recovers the dispersion diagram of a reciprocal ML. However, the free-wave dispersion does not capture the shifted nature of BZ, showing the driven-wave approach significance in unraveling the BZ shift ϕ. Observe that the attenuation (represented by the imaginary component of the wavenumber in the driven-wave dispersion) is smaller in WMLs relative to its reciprocal ML counterpart (β = 0) when Ω > 2, i.e., a frequency higher than the lattice's cutoff frequency Ω max = 2. Also, Ω > 2 results in a non-constant real component of the wavenumber in WML cases only. Note that, for β = ± 0.5 shown here, the corresponding BZ shift is ϕ 0.3 π. (Bottom) Spatiotemporal FFT for the time response of WML for an impulse excitation in the left and right ends, quantifying forward- and backward-going wavenumbers, respectively, and demonstrating excellent agreement with the new BZ definition in the middle panel.

Close modal
Observe that the first and last terms in Eq. (7) are complex conjugates, which can be rewritten as | z | e ± i ( q q s ), where
| z | = β 2 Ω 2 + ( 1 β 2 ) 2 ; q s = tan 1 ( Ω β 1 β 2 ) .
(8)
The emergent phase shift qs is a function of both Ω and β and shall dictate the new definition of BZ. Following the parametrization in Eq. (8), the dispersion relation in Eq. (7) is recast to
cos ( q q s ) = 2 ( 1 β 2 ) Ω 2 2 β 2 Ω 2 + ( 1 β 2 ) 2
(9)
and the driven-wave formulation of the dispersion relation is obtained by solving for q,
q = ± cos 1 ( 2 ( 1 β 2 ) Ω 2 2 β 2 Ω 2 + ( 1 β 2 ) 2 ) + q s .
(10)
For β 0, Eq. (10) output is no longer bounded by the traditional BZ boundaries q [ π , π ], evincing the shifted nature of BZ due to the phase shift qs. Owing to the group velocity vanishing at BZ boundaries for periodic media,26 WML's group velocity must be first derived to precisely pinpoint its BZ boundaries. Then, the BZ shift shall be obtained by evaluating qs at the frequency corresponding to a zero group velocity. To do so, consider the (non-dimensional) group velocity cg for WML, derived by taking the derivative of Eq. (6) with respect to the wavenumber q (see supplementary note 1 for more discussion on group velocity and phase shift),
c g ( q ) = Ω q = sin ( q ) ( 1 2 β 2 sin 2 ( q 2 ) ) 2 | sin ( q 2 ) | 1 β 2 sin 2 ( q 2 ) β cos ( q ) .
(11)
By equating the group velocity in Eq. (11) to zero, the following wavenumber roots are found:
q max = 4 tan 1 ( β ± 1 + β 2 ) .
(12)
Substituting Eq. (12) into Eq. (6) reveals that the frequency at a zero group velocity is Ω max = 2, which is constant, independent of β, and equivalent to the cutoff frequency of a reciprocal ML. Finally, the BZ shift, denoted as ϕ and constitutes the key result in this study, is derived by evaluating qs at Ω max = 2,
ϕ = tan 1 ( 2 β 1 β 2 ) .
(13)

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).

Next, the analysis for the one-dimensional WML is extended to a two-dimensional counterpart, using an elastic membrane with velocity modulations vx and vy in the x and y directions, respectively, and a proposed governing equation of
2 u t 2 + v x 2 u x t + v y 2 u y t + 1 2 ( ( v x 2 c 2 ) 2 u x 2 + ( v y 2 c 2 ) 2 u y 2 ) = 0 ,
(14)
where u(x,y,t) symbolizes the membrane’s transverse displacement. After discretizing Eq. (14) as a square grid, it can be shown that the dispersion relation for the resulting square WML is (see supplementary note 3)
Ω 2 + Ω ( β x sin ( q x ) + β y sin ( q y ) ) 2 ( ( 1 β x 2 ) sin 2 ( q x 2 ) + ( 1 β y 2 ) sin 2 ( q y 2 ) ) = 0 ,
(15)
where q x , y and β x , y = v x , y / c are the wavenumber and relative velocity in the x (or y) direction, respectively. Note that β x , y [ 1 , 1 ] and the dispersion relation of a traditional square ML is recovered if β x = β y = 0, as expected. The dispersion surface Ω of the square WML in Eq. (15) is found by solving the quadratic equation for values of qx and qy within a newly defined (and shifted) BZ. To reveal such a BZ, an identical procedure to the one used in deriving Eq. (9) is followed here, yielding an alternative form of Eq. (15),
Ω 2 ( 2 β x 2 β y 2 ) + | z x | cos ( q x q s x ) + | z y | cos ( q y q s y ) = 0 ,
(16)
where
| z x , y | = β x , y 2 Ω 2 + ( 1 β x , y 2 ) 2 ; q s x , y = tan 1 ( Ω β x , y 1 β x , y 2 ) .
(17)
Numerically, it can be shown that the dispersion relation of the square WML has a constant cutoff frequency at Ω max = 2, regardless of βx and βy [see supplementary Fig. S3(a)]. Analogous to Eq. (13), and after plugging in Ω max = 2 back into q s x , y in Eq. (17), it is concluded that the two-dimensional BZ corners are shifted with x and y direction shifts of
ϕ x , y = tan 1 ( 2 β x , y 1 β x , y 2 ) .
(18)

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 βx and βy [see Figs. 3(a), 3(b) and supplementary Fig. S3(b)].

Fig. 3.

(a) Dispersion contours of a reciprocal square ML ( β x = β y = 0) and a non-reciprocal square WML ( β x = + 0.5 and β y = + 0.75), highlighting the definition of BZ in each case. For the WML case, the BZ area, while irregular in shape, is identical to that of its reciprocal counterpart. (b) A close-up of BZ for square WMLs, illustrating the shift in the x and y propagation directions (i.e., ϕ x and ϕ y, respectively), as well as the vanishing group velocity in the x-direction (y-direction) for the deformed vertical (horizontal) BZ boundaries. (c) Directivity plots for a variety of combinations of βx and βy, using the proposed definition of BZ for square WMLs. The depicted white lines correspond to the directions at which the cutoff frequency occurs. (d) Numerically constructed directivity plots for β x = β y = 0 (reciprocal case) and β x = + 0.5 and β y = + 0.75 (non-reciprocal case), achieved via spatiotemporal FFT, showing excellent agreement to their theoretical counterpart in sub-figure (c).

Fig. 3.

(a) Dispersion contours of a reciprocal square ML ( β x = β y = 0) and a non-reciprocal square WML ( β x = + 0.5 and β y = + 0.75), highlighting the definition of BZ in each case. For the WML case, the BZ area, while irregular in shape, is identical to that of its reciprocal counterpart. (b) A close-up of BZ for square WMLs, illustrating the shift in the x and y propagation directions (i.e., ϕ x and ϕ y, respectively), as well as the vanishing group velocity in the x-direction (y-direction) for the deformed vertical (horizontal) BZ boundaries. (c) Directivity plots for a variety of combinations of βx and βy, using the proposed definition of BZ for square WMLs. The depicted white lines correspond to the directions at which the cutoff frequency occurs. (d) Numerically constructed directivity plots for β x = β y = 0 (reciprocal case) and β x = + 0.5 and β y = + 0.75 (non-reciprocal case), achieved via spatiotemporal FFT, showing excellent agreement to their theoretical counterpart in sub-figure (c).

Close modal

Another way to depict the two-dimensional dispersion relation is by using directivity plots and reducing the two variables qx and qy 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 qx and qy 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 βx and βy.

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.

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Supplementary Material