CONTINUOUSLY TUNABLE ACOUSTIC METASURFACE. . . PHYS. REV. APPLIED 10, 054066 (2018)
microstructures with a graded variation of geometry to
reshape the reflected or transmitted phase profile. Among
various metasurface design strategies, the most represen-
tative material microstructure is the labyrinth type, which
has been proposed by Liang and Li [11], see Fig. 1(b).
The space-coiling design will force the acoustic waves to
propagate through the labyrinth passages, which are much
longer than their external dimension. As shown in the
example of Fig. 1(b), a large phase delay, ϕ = k
0
H =
n
eff
k
0
L, is introduced within a small space, where k
0
is the
wave number in the background fluid, H is the channel
length of the “sound path,” and n
eff
is the effective refrac-
tive index, see Fig. 1(c). Then units with designed gradient
channel lengths constitute a metasurface, and the chan-
nel length variation should be predefined to match the
needed phase profile. As a result, the transmitted phase
profile is reshaped [Fig. 1(b)]. However, once manufac-
tured, the traditional space-coiled metasurfaces cannot be
tuned again to meet different phase modulations. There-
fore, a continuously tunable channel length and a flexible
phase regulation spurs designers to obtain a continuously
tunable metasurface. To this end, one should construct
reconfigurable phase delays to realize a broadband tunable
phase profile by reconstructing the geometrical gradient.
The present design can fully achieve this purpose, and
its basic idea will be introduced and described in the
following.
Based on a screw-and-nut mechanism, we propose
a continuously tunable metasurface composed of spiral
channel units as shown in Fig. 1(d). When the screwed
depth (L
) is tuned, the spin-number as well as the trans-
mitted phase profile changes to meet various metasurface
functions. Similar to the labyrinth passage, the spiral chan-
nel will delay the phase as ϕ = k
0
H = n
eff
k
0
L where the
channel length H, which is tunable with the screwed depth
L
,isH = 2π R
e
L
/W + L − L
with R
e
the effective radius
of the spiral channel and W the thread lead [Fig. 1(e)].
In this way, the present flexible tunable units constitute
a metasurface which can arbitrarily reshape the transmit-
ted wave front through a change of the screwed depth.
The designed helical unit components can be flexibly
assembled into a two-dimensional (2D) metasurface [see
Fig. 2(a)] or even a curved metasurface. Such a tunability
mechanism is a significant step to promote metasurfaces
for practical applications.
The detail of the tunable metasurface design is illus-
trated schematically in Fig. 2. Figure 2(a) shows the 2D
metasurface with distributed helical cylinders screwed into
a plate. The tunability of the metasurface is based on a
screw-and-nut mechanism with a tunable screwed depth,
as shown in Fig. 2(b). The plate is perforated with circular
holes, and the inside wall of each hole is cut into a left-
handed screw with shallow grooves [see Fig. 2(d)]. The
helical cylinder is partially screwed into the plate, and a tail
with the length of L − L
may remain outside. Hereafter,
(a) (b) (c)
(d)
FIG. 2. The detail of the tunable metasurface design based
on the screw-and-nut mechanism. (a) The anomalous refraction
phenomenon on the tunable metasurface is governed by the gen-
eralized Snell’s law. The gradient phase is tuned by the screwed
depth of the helical cylinders. (b) The helical cylinder with two
spiraling blades connected through a central slender column. The
cylinder is screwed into a block and then a tunable unit com-
ponent is combined. (c) The geometric details of the helical
cylinder. (d) The drilled hole with the inside wall cut into shallow
grooves.
this external part is called the “helix’s tail.” After partially
screwing the helical cylinder [Fig. 2(c)] into the hole, a spi-
ral channel is formed between the inside wall of the hole
and the helical cylinder. The channel length and thus the
transmitted phase can be continuously tuned by adjusting
the screwed depth. Thus a new degree of freedom for con-
trolling the wavefronts can be attained by introducing an
arbitrary form of the phase shift along the metasurface. The
wavefront modulation is based on Snell’s law. For the sake
of simplicity, we assume a one-way phase gradient along
the x direction as shown in Fig. 2(a). For an incident wave
in the x-z plane, the refraction angle θ
t
is determined by
Eq. (1), see also the schematic diagram of Fig. 2(a).
One of the key steps in the design of the tunable meta-
surface is to obtain the relationship between the phase
changes and the screwed depth of the helical cylinders at
different frequencies with consideration for a high trans-
mission. Here, we would like to mention that a nontrans-
mitting material would be a better choice for the substrate
material used in the present design. As we have stated
before, the phase reshaping of the present metasurface
relies on the precisely designed spiral air channels. The
sound only propagates through the channels, and the grad-
ual variation of the microstructural geometry generates the
phase gradient and thus controls the transmitted sound [see
Figs. 1(b) and 1(c)]. The units of the spiral air channels
are distributed dispersively, and some parts of the metasur-
face consist of only substrate material without air channels
[see Fig. 1(c)]. If transmitting material is used, no phase
gradient can be generated when sound passes through the
substrate part without air channels. Therefore, a transmit-
ting substrate is not a good choice for the present design.
In fact, a high-energy transmittance should be guaranteed
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