Flat lens for pulse focusing of elastic waves in thin plates
Supplementary materials
Large Angle Negative Refraction
(a) 5 kHz
Γ
(b) 10 kHz
M
X
Γ
(c) 10 kHz with double step
M
X
Γ
M
X
The above three scenarios illustrate the anisotropic effect on an incident wave in the ΓM direction.
(a) At 5 kHz the isofrequency contour still circles the Γ-point and the outgoing wave remains
divergent. Blue arrow: Incident free-space k-vector; orange arrow: Crystal k-vector; red arrow:
Group velocity pointing outward. (b) At 10 kHz the isofrequency contour within the crystal circles
the M-point and the direction of energy transport points inward (red arrow), leading to partial
focusing of the outgoing wave. However, the free-space isofrequency contour (blue contour) is
much larger than the one in the crystal (red contour), hindering the focus resolution. (c) By doubling
the thickness of the plate (2mm-thick) on each side of the crystal (1mm-thick), “free-space”
velocity is increased resulting in a better match of the two contours. Diffraction-limited
subwavelength focusing is then possible since large angles contribute to negative refraction.
Analytical model
In what follows, we present the analytical model used in the accompanying article to describe the
band diagram and isofrequency for a square array of square holes, which served as a basis for the
design of the platonic lens. Vibrations of densely packed arrays of square perforations in thin plates
can be modeled stricto sensu as vibrations of intersecting beams (Fig. S1). The advantage of such
an asymptotic model is two-fold. It captures the essence of the physics and shows its remarkably
simple features. This also facilitates the analysis of the band spectrum, as its structure can be read
at a glance thanks to Eq. 1 reproduced here:
sin(𝛽𝑙) [cos(𝑘1 𝑙) + cos(𝑘2 𝑙) − 2 cos(𝛽𝑙)][cos(𝑘1 𝑙) + cos(𝑘2 𝑙) − 2 cos h(𝛽𝑙)] = 0
(S1)
The first factor explains occurrence of flat bands (slow waves), the second describes dispersion of
Floquet-Bloch waves (including frequency intervals with anomalous dispersion and stop bands),
and the third factor corresponds to evanescent Lamb waves.
The way the asymptotic approach unfolds is fairly intuitive: Consider the thin interstitial space
between square holes (see Fig. S1). These are homogeneous beams within which bending waves
propagate according to the Bernoulli–Euler equation
𝑑4 𝑢𝑛
𝑑𝑥 4
𝜔2
− Ξ4 𝑢𝑛 = 0,
(S2)
where un is the out-of-plane displacement of beam n,  is the frequency and Ξ4 = 𝐸𝐼 ⁄𝜌 𝐴 is a
normalization parameter. Here, 𝐼 = 𝑏 ℎ3 ⁄12, 𝐴 = 𝑏ℎ (b is the beam width), respectively denote
the area moment of inertia and the area of the beam cross section, ρ stands for its density and E its
Young modulus. The general solution in each beam is
𝑢𝑛 (𝑡) = 𝑎𝑛 sin (
√𝜔
𝑥)
Ξ
√𝜔
𝑥)
Ξ
+ 𝑏𝑛 cos (
√𝜔
√𝜔
+ 𝑐𝑛 sinh ( Ξ 𝑥) + 𝑑𝑛 cosh ( 𝛯 𝑥)
(S3)
We therefore have 16 constants to evaluate, thus require 16 boundary conditions.
We now set up the boundary value problem for four beams within the elementary cell shown in
Fig.S1, using Floquet–Bloch boundary conditions:
(𝑗)
(𝑗)
(𝑗)
(𝑗)
𝑢1 (𝑙) = 𝑒 2𝑖𝑘1 𝑙 𝑢3 (𝑙), 𝑢2 (𝑙) = 𝑒 2𝑖𝑘2 𝑙 𝑢4 (𝑙),
(S4)
(𝑗)
(0)
with 𝑢𝑖 (𝑥) = 𝑢𝑖 and 𝑢𝑖 (𝑥) = 𝑑 𝑗 𝑢𝑖 ⁄𝑑𝑥 𝑗 , where k  (k1 , k2 ) is the Bloch vector, and i =
1,2,3,4.
At the central node where the functions ui meet, we expect continuity and differentiability (the
elastic solid corresponding to the intersection of all four beams in the unit cell “bends smoothly”)
as follows:
u1( j ) (0)  u3( j ) (0) , u2( j ) (0)  u4( j ) (0)
(S5)
where j = 0,1. Moreover u1(j)(0) = -u4(j)(0) ensures that beams meet at right angle and (S5) is fulfilled
for j = 2 if we assume the rotational inertia vanishes at the central node from the conservation of
the angular momentum. Moreover, u1(3)(0)=u3(3)(0) from the conservation of linear momentum if
we assume that the mass of the central node is zero. This leads to dispersion relation in Eq. 1 of the
main article.
There is a nearly perfect match between the finite element computations and the asymptotic
dispersion curve in the acoustic branch. The discrepancy between the continuous and dashed curves
can be further reduced by considering a system of masses at the junction between the beams: unlike
for a square array of square perforations whereby junctions play a secondary role, here one should
take into account the moment of inertia of interstitial material between circular perforations
(roughly shaped as a diamond). Nevertheless, the lattice beam approximation substantially
degrades above the acoustic curve as seen in Fig. 1b: asymptotic formula (Eq. 1) breaks down
above the acoustic band when it is used as an approximation for non-square perforations.
Node
U
U
3
2
b U
1
U
l
4
(a)
Figure S1 : (a) Diagrammatic view of a unit cell consisting of four thin beams and their associated flexural displacements
u1, u2, u3, u4. (b) Isofrequency contours of the acoustic curve obtained from Eq. 1 when the Bloch vector varies within the
first Brillouin zone ΓXM. We note the strong similarity of this plot with Fig. 1a of the accompanying article.
Movie
Time evolution of the amplitude-squared, |A(r,t)|2 (envelope) of the spatial distribution of the
velocity. Incident pulse has central frequency f0 = 10 kHz and spectral bandwidth \sigma = 2.3
kHz. Focusing is best seen at t = 833 µs.