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Diaz-Rubio, A.; Tretyakov, S. A.
Acoustic metasurfaces for scattering-free anomalous reflection and refraction
Published in:
Physical Review B
DOI:
10.1103/PhysRevB.96.125409
Published: 07/09/2017
Document Version
Publisher's PDF, also known as Version of record
Please cite the original version:
Diaz-Rubio, A., & Tretyakov, S. A. (2017). Acoustic metasurfaces for scattering-free anomalous reflection and
refraction. Physical Review B, 96(12), [125409]. https://doi.org/10.1103/PhysRevB.96.125409
PHYSICAL REVIEW B 96, 125409 (2017)
Acoustic metasurfaces for scattering-free anomalous reflection and refraction
A. Díaz-Rubio and S. A. Tretyakov
Department of Electronics and Nanoengineering, Aalto University, P. O. Box 15500, FI-00076 Aalto, Finland
(Received 3 May 2017; published 7 September 2017)
Manipulation of acoustic wave fronts by thin and planar devices, known as metasurfaces, has been extensively
studied, in view of many important applications. Reflective and refractive metasurfaces are designed using
the generalized reflection and Snell’s laws, which tell that local phase shifts at the metasurface supply extra
momentum to the wave, presumably allowing arbitrary control of reflected or transmitted waves. However, as has
been recently shown for the electromagnetic counterpart, conventional metasurfaces based on the generalized
laws of reflection and refraction have important drawbacks in terms of power efficiency. This work presents
a new synthesis method of acoustic metasurfaces for anomalous reflection and transmission that overcomes
the fundamental limitations of conventional designs, allowing full control of acoustic energy flow. The results
show that different mechanisms are necessary in the reflection and transmission scenarios for ensuring perfect
performance. Metasurfaces for anomalous reflection require nonlocal response, which allows energy channeling
along the metasurface. On the other hand, for perfect manipulation of anomalously transmitted waves, local and
nonsymmetric response is required. These conclusions are interpreted through appropriate surface impedance
models which are used to find possible physical implementations of perfect metasurfaces in each scenario. We
hope that this advance in the design of acoustic metasurfaces opens new avenues not only for perfect anomalous
reflection and transmission but also for realizing more complex functionalities, such as focusing, self-bending,
or vortex generation.
DOI: 10.1103/PhysRevB.96.125409
I. INTRODUCTION
The interest in quasi two-dimensional devices capable
of manipulating waves revived with the formulation of the
generalized reflection and Snell’s laws [1], which shows a
possibility oftailoring the direction of reflected and transmitted
waves by introducing gradient phase shifts at the interface
between two media. The generalized laws of reflection and
refraction have been applied for controlling the direction
of transmitted and reflected waves in electromagnetism [2]
and acoustics [3–13]. By appropriately varying the phase
shift introduced along the metasurface between 0 and 2π ,
the propagation direction of the reflected/refracted wave can
be controlled. These approaches enable tailoring the energy
propagation direction, but with important restrictions of the
efficiency (the amount of energy that is sent into the desired
direction is smaller than the energy introduced in the system,
even for lossless metasurfaces).
Recently, it has been demonstrated that some additional
considerations about the power conservation can be applied
over the conventional generalized reflection and Snell’s laws
for ensuring full control of electromagnetic energy flow
[14–17]. This advance has attracted much attention due to
the possibility of dramatic improvements of conventional solu-
tions, especially for steep reflection or transmission angles. For
electromagnetic waves, the basis of this “second generation”
of gradient electromagnetic metasurfaces has been established
and numerically verified, and for reflective metasurfaces the
theoretical findings have already been confirmed experimen-
tally [18].
However, it appears that the synthesis tools for acoustic
metasurfaces do not benefit from the new knowledge. In the
acoustic reflection scenario, the generalized reflection law has
been experimentally demonstrated [4–8] using labyrinthine
unit cells which provide a phase-shift profile with the 2π span
in the reflection coefficient phase. However, in view of the
results ofRefs. [14–17], the performance of these designs is not
optimal becausesignificant energy isspread in unwanted direc-
tions. Theoretical studies based on inhomogeneous impedance
along the metasurface [5] show the coexistence of more than
one reflected wave. For perfect control of anomalous reflection
or, in other words, for allowing arbitrary changes of the
direction of reflected plane waves, we need to ensure perfect
steering of all the incident power into the desired direction,
avoiding the generation of parasitic waves propagating in other
directions or losses in the system.
On the other hand, the same approach based on the
generalized Snell’s law has been applied for the design of
refractive acoustic metasurfaces [9–13]. The direction of
the transmitted wave is controlled by linearly modulating
the local phase shift in transmission through the metasur-
face. For the design of unit cells, different topologies have
been used, including space-colling structures [9], slits filled
with different density materials [10], or straight pipes with
Helmholtz resonators in series [11,12].One problem addressed
in the design of the refractive metasurface was the required
impedance matching of each meta-atom in order to obtain
total transmission. In this sense, substantial improvements in
the design of matched unit cells have been achieved by using
tapered labyrinthine units [13]. However, as has been recently
demonstrated for the electromagnetic scenario [14,15,19], by
ensuring perfect matching in the microscopic design of the
metasurface (individual design of each meta-atom) we cannot
obtain the proper macroscopic behavior of the metasurface.
In this paper, we present the foundations for the synthesis
of perfect acoustic metasurfaces, overcoming the fundamental
limitations of conventional designs. The study covers two
different scenarios: anomalous reflection and transmission of
acoustic plane waves. With the purpose of simplifying the
presentation and emphasizing the novelty of this approach, in
2469-9950/2017/96(12)/125409(10) 125409-1 ©2017 American Physical Society
A. DÍAZ-RUBIO AND S. A. TRETYAKOV PHYSICAL REVIEW B 96, 125409 (2017)
FIG. 1. (a) Schematic representations of the desired metasurface
behavior for the anomalous reflection scenario. (b) Equivalent circuit
proposed for the analysis of reflective metasurfaces.
both cases the analysis starts with a comprehensive overview
of the known approaches based on the generalized reflection
and Snell’s laws. After identifying the weaknesses of current
designs, we propose new methods that ensure perfect control
of acoustic energy in reflection and refraction. Finally, we
interpret the theoretical findings in terms of the physical
properties of metasurface unit cells and give examples of
possible realizations.
II. ACOUSTIC METASURFACES FOR REFLECTION
In this section, the reflected wave front manipulation is
studied. Particularly, we focus the study on anomalous reflec-
tion of acoustic plane waves. This fundamental functionality
is the base of many interesting applications such as reflection
lenses, plane wave to surface wave conversion, or acoustic
retro-reflectors.
A. Design based on the generalized reflection law
In order to understand the current status of the synthesis
methods, we start with the analysis of reflective gradient
metasurfaces based on the generalized reflection law (we
use the same short-hand notation, GSL, for both generalized
Snell’s law and the generalized reflection law). If we consider
the scenario illustrated in Fig. 1(a), where the incident
and reflected waves propagate in a homogeneous medium
with density ρ and sound speed c, assuming time-harmonic
dependence e
jωt
, the incident and reflected pressure fields can
be written as
p
i
(x,y) = p
0
e
−jksin θ
i
x
e
jkcos θ
i
y
, (1)
p
r
(x,y) = Ap
0
e
−jksin θ
r
x
e
−jkcos θ
r
y
, (2)
where p
0
is the amplitude of the incident plane wave, k = ω/c
is the wave number in the background medium at the operation
frequency, θ
i
and θ
r
are the incidence and reflection angles, and
A is a constant which relates the amplitudes of the incident
and reflected waves. The velocity vectors associated with these
pressure fields (v =
j
ωρ
∇p) read
v
i
(x,y) =
p
i
(x,y)
Z
0
(sin θ
i
ˆ
x −cos θ
i
ˆ
y), (3)
v
r
(x,y) =
p
r
(x,y)
Z
0
(
sin θ
r
ˆ
x +cos θ
r
ˆ
y
)
, (4)
where Z
0
= cρ is the characteristic impedance of the back-
ground medium.
Assuming that the field beyond the metasurface is zero (an
impenetrable metasurface), the system can be conveniently
modeled by the equivalent circuit shown in Fig. 1(b), where
the impedance Z
s
models the specific impedance of the
metasurface. GSL designs are based on the assumption
that a linear gradient phase shift [
∂
x
∂x
= k(sin θ
r
− sin θ
i
)] is
introduced by the metasurface. In other words, the metasurface
is characterized by the local reflection coefficient with the unit
amplitude, which can be written as
(x) =
e
−jksin θ
r
x
e
−jksin θ
i
x
= e
jk(sin θ
i
−sin θ
r
)x
= e
j
x
, (5)
where the reflection phase is defined as
x
= k(sin θ
r
−
sin θ
i
)x. The reflection coefficient is related with the surface
impedance as =
Z
s
−Z
i
Z
s
+Z
i
, where Z
i
= Z
0
/ cos θ
i
represents
the specific acoustic impedance of the incident wave at the
metasurface. From this expression, the impedance which
models the metasurface can be found as
Z
s
(x) = j
Z
0
cos θ
i
cot(
x
/2). (6)
Figure 2 presents a numerical study of the conventional
designs based on the GSL. The surface impedance modeled
by Eq. (6) is purely imaginary [see Fig. 2(c)], so lossless imple-
mentations are possible for this kind of reflective metasurfaces.
Actual implementations of the desired impedance profiles can
be obtained by using simple rigidly terminated waveguides
with different lengths or, exploiting the longitudinal character
of acoustics waves and so the absence of cutoff frequency,
“space-colling” particles with labyrinth channels [4–8]. In
these cases, each meta-atom has to be carefully tailored for
individually implementing the required surface impedance
profile and producing the required local phase shift. For the
purposes of this study, we assume that the required impedance
profile has been realized and model the metasurface (using
COMSOL software) as an impedance boundary described by
Eq. (6). Figures 2(a) and 2(b) show the results of numerical
simulations when the metasurface is illuminated normally
(θ
i
= 0
◦
) and the reflection angles are θ
r
= 30
◦
and θ
r
= 70
◦
,
respectively. From the comparison of these two examples, it
is easy to see two important issues: First, the “quality” of the
reflected wave decreases when the reflection angle increases,
due to parasitic reflections in other directions; second, the
amplitude of the reflected wave changes with the reflection
angle, although this behavior is not contemplated in the design
statement (A = 1). Clearly, the simple design philosophy
described by Eq. (6) does not ensure the perfect conversion
of energy between the incident and reflected plane waves and
it cannot be considered as an accurate method for the design
of metasurfaces for large values of the reflection angle.
The conclusions extracted from the analysis of the numeri-
cal simulations can be understood as an impedance mismatch
problem. Although the metasurface provides the desired phase
response, the incident and reflected waves have different
specific impedances, so part of the energy cannot be redirected
into the desired direction. Since the metasurface is assumed to
be lossless, part of the incident energy has to be reflected into
other directions (into 0
◦
and −70
◦
in the example of Fig. 2).
125409-2
ACOUSTIC METASURFACES FOR SCATTERING-FREE . . . PHYSICAL REVIEW B 96, 125409 (2017)
FIG. 2. Real part of the scattered pressure field for a metasurface
designed according to Eq. (6) when: (a) θ
i
= 0
◦
and θ
r
= 30
◦
;(b)θ
i
=
0
◦
and θ
r
= 70
◦
. (c) Surface impedance described by Eq. (6)when
θ
i
= 0
◦
and θ
r
= 70
◦
. (d) Efficiency of the GSL gradient metasurfaces
as a function of the reflection angle.
The reflections into parasitic directions can be estimated
introducing reflection coefficient calculated in terms of the
respective impedances:
R =
Z
r
− Z
i
Z
r
+ Z
i
=
cos θ
i
− cos θ
r
cos θ
i
+ cos θ
r
. (7)
Because the metasurface is an impenetrable boundary, the
total pressure of the incident and reflected waves (1 + R)is
equal to the pressure of the wave redirected into the desired di-
rection (A
GSL
), in analogy to transmission of electromagnetic
waves through electric-current sheets [20]:
A
GSL
= 1 + R =
2 cos θ
i
cos θ
i
+ cos θ
r
. (8)
We can now define the efficiency of the metasurface as the
ratio between the incident power and the power reflected in
the desired direction. The acoustic power can be expressed in
terms of the intensity vector
I =
1
2
Re[pv
∗
], (9)
where “∗” represents the complex conjugate. Due to the
periodicity of the system, only the normal component of the
intensity vector will take part in the power balance. The normal
component of the incident power is
P
i
=
ˆ
n ·
I
i
=−
p
2
0
Z
0
cos θ
i
. (10)
The normal component of the power carried in the desired
reflection direction can be calculated as
P
r
=
ˆ
n ·
I
r
= A
2
p
2
0
Z
0
cos θ
r
, (11)
and the efficiency reads
η =
|P
r
|
|P
i
|
= A
2
cos θ
r
cos θ
i
. (12)
Substituting the wave amplitude A from Eq. (8) we can finally
estimate the efficiency of conventional metasurfaces as
η
GSL
=
2 cos θ
i
cos θ
i
+ cos θ
r
2
cos θ
r
cos θ
i
. (13)
Figure 2(d) represents the efficiency estimation given by
Eq. (13) and its comparison with the numerical results. To find
the power efficiency from numerical results, we calculate the
amplitude of the reflected plane wave into θ
r
, A
COMSOL
.This
amplitude can be calculated as
A
COMSOL
=
1
D
D
0
p
r
· e
jksin θ
r
x
dx, (14)
where D is the metasurface period, and the efficiency is
obtained using Eq. (12). It is possible to see how the efficiency
of the generalized reflection law metasurfaces dramatically
decreases when the reflection angle increases.
B. Lossy metasurfaces for anomalous reflection
As we have seen, the amplitude of the reflected wave in the
conventional design is not equal to the amplitude of the inci-
dent plane wave (A
GSL
= 1). If we design a metasurface which
arbitrarily changes the direction of the reflected wave keeping
the amplitude A = 1, the pressure field at metasurface can be
written as
p
tot
(x,0) = p
0
(1 + e
j
x
)e
−jksin θ
i
x
. (15)
The corresponding total velocity at the metasurface reads
v
tot
(x,0) =
p
0
Z
0
(sin θ
i
+ sin θ
r
e
j
x
)e
−jksin θ
i
x
ˆ
x+ (16)
p
0
Z
0
(−cos θ
i
+ cos θ
r
e
j
x
)e
−jksin θ
i
x
ˆ
y. (17)
At this point, we have to satisfy the boundary condition at the
metasurface. We can do that by defining the surface impedance
which models this metasurface as
Z
s
(x) =
p
tot
(x,0)
−
ˆ
n ·v
tot
(x,0)
. (18)
Introducing the expressions for the desired total pressure
(15) and velocity [Eq. (17)] into this equation, we find the
impedance which models such metasurfaces:
Z
s
(x) = Z
0
1 + e
j
x
cos θ
i
− cos θ
r
e
j
x
. (19)
125409-3
A. DÍAZ-RUBIO AND S. A. TRETYAKOV PHYSICAL REVIEW B 96, 125409 (2017)
FIG. 3. Real part of the scattered pressure field for a metasurface
designed according to Eq. (19) when: (a) θ
i
= 0
◦
and θ
r
= 30
◦
and (b) θ
i
= 0
◦
and θ
r
= 70
◦
. (c) Surface impedance described by
Eq. (19)whenθ
i
= 0
◦
and θ
r
= 70
◦
. (d) Efficiency of the gradient
metasurfaces for A = 1 as a function of the reflection angle.
Figure 3 presents the results of a numerical study of acoustic
metasurfaces based on Eq. (19). As in the previous case,
Figs. 3(a) and 3(b) show simulated results for the metasurface
illuminated normally and designed for the reflection angles
θ
r
= 30
◦
and θ
r
= 70
◦
, respectively. The results confirm the
required performance of metasurfaces that anomalously reflect
a perfect plane have with the same amplitude as the incident
wave. The surface impedance given by Eq. (19) is a complex
number as it is shown in Fig. 3(c) for θ
i
= 0
◦
and θ
r
= 70
◦
.
The real part of the impedance takes positive values (modeling
losses) over all the period showing that these metasurfaces
are necessarily lossy, which is a condition for keeping the
amplitude of the reflected wave equal to the incident wave. To
illustrate this behavior, we can analyze the efficiency of the
metasurface found from Eq. (12) when A = 1:
η
A=1
=
cos θ
r
cos θ
i
. (20)
This expression is represented in Fig. 3(d) as a function of
the reflection angle for θ
i
= 0
◦
. Numerical results have been
calculated in the same way as before, using Eq. (14). We
can see that the efficiency dramatically decreases when the
reflection angle increases. When the reflection angle increases,
the power sent into the desired direction decreases and all
FIG. 4. Real part of the scattered pressure field for a metasurface
designed according to Eq. (23) when: (a) θ
i
= 0
◦
and θ
r
= 30
◦
and (b) θ
i
= 0
◦
and θ
r
= 70
◦
. (c) Surface impedance described by
Eq. (23)whenθ
i
= 0
◦
and θ
r
= 70
◦
. (d) Amplitude of the reflected
wave for perfect anomalous reflection (red symbols) compared with
conventional designs based on GSL (blue line).
the remaining energy is absorbed in the metasurface. On the
other hand, if θ
i
>θ
r
, the real part of the surface impedance
becomes negative (gain), meaning that additional energy has
to be introduced in the system in other to obtain the desired
performance.
C. Active-lossy scenario and lossless non local realization
Obviously, the design approach defined by Eq. (19) presents
important drawbacks in terms of power efficiency, although
there are no parasitic reflections into unwanted directions. For
perfect anomalous reflection where all the impinging energy
is sent into the desired direction, we have to ensure η = 1.
From Eq. (12) it is easy to find the amplitude coefficient which
corresponds to the perfect performance: For perfect anomalous
reflection the amplitude of the reflected wave has to be A =
√
cos θ
i
/ cos θ
r
. Figure 4(d) shows a comparison between the
required amplitude for perfect performance and the amplitude
of conventional designs based on GSL when θ
i
= 0
◦
.The
differencebetween both approaches increases with the angle of
reflection, confirming our previous conclusion about the poor
efficiency of conventional design for large differences between
125409-4
