# Surface corrugated laminates as elastic grating couplers: splitting of SV-
and P-waves by selective diffraction

G. J. Chaplain  Department of Mathematics, Imperial College London, London SW7
2AZ, UK<EMAIL_ADDRESS>R. V. Craster  Department of
Mathematics, Imperial College London, London SW7 2AZ, UK  Department of
Mechanical Engineering, Imperial College London, London SW7 2AZ, UK  UMI 2004
Abraham de Moivre-CNRS, Imperial College London, London SW7 2AZ, UK

###### Abstract

The phenomenon of selective diffraction is extended to in-plane elastic waves
and we design surface corrugated periodic laminates that incorporate crystal
momentum transfer which, due to the rich physics embedded within the vector
elastic system, results in frequency, angle and wave-type selective
diffraction. The resulting devices are elastic grating couplers, with
additional capabilities as compared to analogous scalar electromagnetic
couplers, in that the elastic couplers possess the ability to split and
independently redirect, through selective negative refraction, the two body
waves present in the vector elastic system: compressional (P) and shear-
vertical (SV) elastic waves. The design paradigm, and interpretation, is aided
by obtaining isofrequency contours via a non-dimensionalised transfer matrix
method.

††preprint: AIP/123-QED

##  I  Introduction

Numerous parallels exist between the in-plane vector elastic wave system and
the scalar wave systems of polarised electromagnetism, acoustics, flexural
waves, linear water waves and shear-horizontal polarised elasticity.
Transposing physics into the vectorial elastic system is often nuanced and not
straightforward with different shear and compressional wavespeeds in
elasticity, and coupling between shear and compression at interfaces, that can
lead to additional effects. An example being array-guided surface waves
confined to elastic diffraction gratings, so-called Rayleigh-Bloch waves
Colquitt  _et al._ (2015), that are elastic analogues to electromagnetic spoof
surface plasmon polaritonsPendry, Martin-Moreno, and Garcia-Vidal (2004);
Hibbins, Evans, and Sambles (2005), which solely owe their existence to
periodic structuring. The nature of the vector elastic system dictates that
these modes exist for traction-free boundary conditions, rather than simple
Neumann conditions as in, for example, electromagnetic gratings Wilcox (1984).
In two-dimensions the nuances are driven in the vector elastic system by the
presence of two body waves, shear-vertical polarised (SV) and compressional
(P), that co-exist and travel with different wavespeeds $c_{s}$ and $c_{p}$
respectively, such that $c_{s}<c_{p}$. Additionally these wave types couple to
each other (and can scatter into surface Rayleigh waves if defects are
present) at traction-free interfacesGraff (1975), and therefore they are
inherently coupled and resist independent manipulation. Since the advent of
metamaterials there has been considerable effort to manipulate these distinct
elastic wave types, drawing on techniques developed in optics and
electromagnetism, particularly those associated with the abrupt phase changes
endowed to wavefronts which traverse a particular metasurfaceYu and Capasso
(2014). Anomalous reflection, refraction, transmission and wave splitting
effects have been proposed, by designing metamaterials and metasurfaces based
on phased gratings; spatially dependent structurings give the required
tailored phase profile, allowing generalised Snell’s laws to be observedSu and
Norris (2016); Su, Lu, and Norris (2018); Ahn  _et al._ (2019).

Figure 1: Splitting of SV- and P- elastic waves by the selective diffraction
created from surface corrugation decorating an elastic bilayer laminate: (a)
Compressional elastic potential field, $\phi$. (b) Shear-vertical potential
field, $\bm{\psi}$. Each are excited by separate line forcings (green line)
which match the relative preferential particle motion direction of each wave
type (half-headed arrows). The ray path is shown by the black arrows. The two
wave types are spatially separated upon transmission through the laminate;
only the compressional P-waves receive a momentum "kick" due to the surface
corrugation, resulting in selective negative refraction. Geometry detailed in
Appendix A.

We extend conventional grating coupler concepts from electromagnetism and
optics using periodic decorations attached to an elastic laminate, see Fig. 1,
by leveraging momentum transfer to generate a selective momentum "kick". In
optics the generation of a momentum "kick" has been used to create negative
refraction Lu  _et al._ (2007), however due to the scalar nature of polarised
electromagnetism there is no possibility for the analogues of compressional
and shear wave splitting.

By using isofrequency contours as a design tool these surface corrugations on
the outer interfaces of an elastic laminate, composed of periodic bilayers
analogous to one-dimensional dielectric photonic crystalsJoannopoulos  _et
al._ (2008), we are able to give the "kick" preferentially to one wave type
and create selective negative refraction (in Fig. 1 the kick is gifted to the
compressional wave); the presence of the two distinct elastic body wave types
allows these extended coupler concepts to provide flexible control over P and
SV waves independently. Figure 1 motivates these additional capabilities of
the elastic system showing the compressional and shear elastic potentials
$\phi$ and $\bm{\psi}$; the finite element method is used in COMSOL
Multiphysics to perform the simulations. To emphasise the different effects we
perform multiple scattering simulations, with alternate line forcings, to
preferentially excite either P or SV-waves; dominant compressional motion is
achieved when the forcing direction is perpendicular to the line source,
whilst shear motion is dominant when the forcing direction is parallel to it.
This enhances visualisation of the effects by distinguishing between the
mixture of the two wave types which result from a general source (Appendix C
highlights an example where the total solid displacement field is visualised).
The forcing directions are highlighted by the green half-headed arrows in
Figure 1, which follow the particle motion.

Alongside the additional physics of wave splitting for elasticity, more
conventional grating coupler effects, analogous to those from
electromagnetism, are also recoverable. Indeed the term grating coupler is
almost exclusively used to describe electromagnetic devices, particularly
prevalent in integrated silicon photonic devices and fibre optics Taillaert
_et al._ (2002); Chuang (2012); Marchetti  _et al._ (2017, 2019). The desired
outcome in these systems is to couple incident light efficiently into a
waveguide, and ensure its confinement. We show, as an additional example in
Appendix C, that this is entirely possible for elastic laminate waveguides by
suitable design of the surface corrugation.

The structure design rests on obtaining isofrequency contours for the elastic
laminate corresponding to purely P or SV-waves propagating through the
laminate. We obtain those through employing the Transfer Matrix Method (TMM),
as outlined in Section II. Advantages of this methodology arise through its
simplicity, permitting the design of structures with a tailored response over
a wide range of frequencies and angles of incidence. Extending the concepts of
periodic surface corrugation to elastic grating couplers thus allows the
decoupling of elastic waves at an interface, by a simple alteration of the
surface structure. The resulting control gained over each independent wave-
type has practical implications in non-destructive evaluation and vibration
isolationChimenti (1997).

##  II  Periodic Bilayer Laminates and the Transfer Matrix Method

Elastic waves in isotropic, homogeneous layers are governed by the elastic
wave equationGraff (1975)

$\rho\bm{\ddot{u}}=(\lambda+2\mu)\nabla(\nabla\cdotp\bm{u})-\mu\nabla\times\nabla\times\bm{u},$
(1)

where, in Cartesian coordinates, $\bm{u}=(u,v,w)$ is the displacement vector.
The acceleration is given by the double time derivative, $\bm{\ddot{u}}$, with
$\rho$, $\lambda$ and $\mu$ being the material density and Lamé’s first and
second parameters respectively. Through Helmholtz decomposition, the
displacement vector $\bm{u}$ can be written in terms of the divergence-less
shear potential, $\bm{\psi}$, and the curl-less compressional potential,
$\phi$, such that

$\bm{u}=\nabla\times\bm{\psi}+\nabla\phi.$ (2)

Time harmonic behaviour, of form $\exp({-i\omega t})$, is henceforth assumed.
Following this, equation (1) can be formulated such that each potential
satisfies the wave equations

$\displaystyle(\nabla^{2}+k_{p}^{2})\phi=0,\qquad(\nabla^{2}+k_{s}^{2})\bm{\psi}=0,$
(3)

with $k_{p}$ and $k_{s}$ the compressional and shear wavenumbers related to
the respective wavespeeds, $c_{p},c_{s}$, through
$k_{\delta}=\omega/c_{\delta}$, with $\delta=p,s$, such that
$c_{p}=\sqrt{\frac{\lambda+2\mu}{\rho}}$ and $c_{s}=\sqrt{\frac{\mu}{\rho}}$.
The elastic potentials can then be written in terms of the displacement field
through

$\displaystyle\phi=-\frac{(\nabla\cdotp\bm{u})}{k^{2}_{p}}=-\frac{\text{Tr}(\bm{\varepsilon})}{k^{2}_{p}},\qquad\bm{\psi}=\frac{(\nabla\times\bm{u})}{k^{2}_{s}},$
(4)

where $\varepsilon$ is the strain tensor. The potentials provide a useful
approach for separately visualising the compressional or shear components of
an elastic wavefield.

We first focus on a one-dimensional periodic bilayer elastic laminate
structure, as shown in Figure 2(a), that is of infinite extent and take
advantage of periodicity to extract Floquet-Bloch dispersion curves that
relate the Bloch wavenumber to frequency. The periodicity means that we can
take a unit cell, and apply Bloch conditions; the unit strip is of width $l$,
comprising two layers of linear elastic media, such that $l=a_{1}+a_{2}$,
where $a_{1},a_{2}$ are the widths of the first and second layers
respectively. Following conventions in optics we define the direction of
periodicity to be along the $z$-direction, with the strips assumed to be of
infinite extent in the orthogonal $x$-direction. Each material has density
$\rho$ along with Lamé’s first and second parameters $\lambda$, $\mu$ and when
required, the elastic quantities and parameters in each layer are
distinguished with a subscript $i=1,2$ for each layer.

To design finite gratings capable of performing selective diffraction, the
isofrequency contours for a periodic array of the bilayer cell must be
evaluated. Given the assumed time harmonic behaviour, the potential fields in
each layer are written as a superposition of left- and right-propagating plane
waves. To determine the structure’s dispersive properties and band structure
we employ the TMM, that relates the potential field amplitudes on either side
of the unit cell and this is achieved by splitting the problem into two parts.
The first requires the continuous physical quantities at an interface to be
determined. In the elastic case considered here these are the normal and
tangential displacements to the interface $(w,u)$, and the corresponding
stresses $(\sigma_{zz},\sigma_{xz})$. As detailed in Appendix B, the vector of
continuous quantities $\bm{W}_{i}=(w,u,\sigma_{zz},\sigma_{xz})^{\text{T}}$
across the interface within the $i^{th}$ layer is constructed through

$\displaystyle\bm{W}_{i}=§\begin{pmatrix}\phi_{,z}-\bm{\psi}_{,x}\\\ \\\
\phi_{,x}+\bm{\psi}_{,z}\\\ \\\
(\lambda_{i}+2\mu_{i})\nabla^{2}\phi+2\mu_{i}(\phi_{,z}-\bm{\psi}_{,xz}-\nabla^{2}\phi)\\\
\\\ \mu_{i}(2\phi_{,xz}+\bm{\psi}_{,zz}-\bm{\psi}_{,xx})\end{pmatrix},$ (5)

where it is implicitly assumed that the potential fields correspond to those
in the $i^{th}$ layer.

The second step relates the potential field amplitudes due to propagation
through a layer, until the next interface is encountered, and therefore
incorporating the phase change across the finite layer. By combining the two
steps (using straightforward matrix multiplication) the amplitudes on one side
of the unit strip are related to those on the other, through a matrix
equation.

Figure 2: (a) Infinitely periodic bilayer elastic laminate: Periodic unit
strips, of length $l=a_{1}+a_{2}$ where $a_{1}$ and $a_{2}$ are the
thicknesses of elastic media labelled 1 and 2 respectively, comprise the
laminate. The TMM is applied, along with Bloch conditions, across one unit
cell, highlighted in blue. (b) Finite corrugated periodic bilayer elastic
laminate: Surface corrugation takes place on the outermost layers of a finite
laminate composed of the bilayers in (a). The vertical periodicity of the
corrugations are defined by $a_{c1}$, $a_{c2}$ on the left and right sides of
the laminate respectively, in the direction orthogonal to the laminate axis.
In general $a_{c1}\neq a_{c2}$. Provided these periodicities are designed
appropriately, for a given incident plane wave (P or SV) at angle $\theta$ to
the laminate axis with wavevector $\bm{k}_{p(s)}$, the SV and P potential
fields ($\bm{\psi}$ and $\phi$) can be separated by incorporating the momentum
transfer by the grating. Half-headed arrows show particle motion. In each case
the laminate is assumed to be infinite in the direction orthogonal to the
periodicity.

Due to its relative simplicity, TMMs are often utilised to obtain transmission
and reflection coefficients from finite structures, periodic or otherwise, (in
any number of dimensions); for $N$ identical layers the resultant transfer
matrix is simply raised to the $N^{th}$ power. As such this method has a long
history, with much success in its applications to layered structures in
electromagnetism Markos and Soukoulis (2008), solid-fluid interactions
Schoenberg (1984), disordered systems Pernas-Salomón  _et al._ (2015) and
notably elasticityThomson (1950); Haskell (1953); Huang, Wang, and Rokhlin
(1996); Fomenko  _et al._ (2014). By additionally invoking the quasi-periodic
Floquet-Bloch conditions, the resultant matrix equation can be re-written as
an eigenvalue problem which allows the dispersion relation to be obtained,
with the Bloch wavenumber the eigenvalue for a given (non-dimensional)
frequency $\Omega$, incidence angle and wave type. In Appendix B we detail
this methodology, adopting a non-dimensional formalism; the matrices are
composed of quantities expressed entirely in terms of the ratios of parameters
of the two materials comprising the bilayer e.g. the ratios of the speeds in
each material. The advantages of this are that we retain generality whilst
mitigating numerical instabilities which arise through the required matrix
inversion; the TMM is susceptible to instabilities in the so-called large f-d
problemDunkin (1965). Alternatives to the TMM are those based on the global
matrix formulationKnopoff (1964) or spectral collocationAdamou and Craster
(2004).

The method rapidly generates dispersion curves, and subsequently the
isofrequency contours associated with the compressional or shear components of
the incident wave. Two independent calculations are required depending on the
component of the incident wave under consideration; the angles of reflection
and refraction are expressed in terms of the incident angle of either P- or
SV-waves through Snell’s law for in-plane elasticityAchenbach (2012)

$\frac{\sin\theta_{1p}}{c_{1p}}=\frac{\sin\theta_{1s}}{c_{1s}}=\frac{\sin\theta_{2p}}{c_{2p}}=\frac{\sin\theta_{2s}}{c_{2s}},$
(6)

where $\theta_{i\delta}$ and $c_{i\delta}$ give the angle and wavespeed
corresponding to the P- and SV ($\delta=p,s$) waves in layers $i=1,2$. A
schematic of this coupling for an incident compressional wave, i.e. through
$\phi$, is shown Figure 3.

Figure 3: Snell’s law for in-plane elasticity: An incident compressional wave
in layer 1, described by the potential $\phi^{(i)}_{1p}$, couples to reflected
and transmitted compressional waves $\phi^{(r)}_{1p}$ and $\phi^{(t)}_{2p}$ in
layers 1 and 2 with wavespeeds $c_{1p}$ and $c_{2p}$ respectively.
Additionally the incident compressional wave couples to reflected and
transmitted shear-vertical potential fields $\bm{\psi}^{(r)}_{1s}$ and
$\bm{\psi}^{(t)}_{2s}$ with wavespeeds $c_{1s}$ and $c_{2s}$ respectively. The
angles of reflection and transmission are related to the incident angle
through (6).

Once the isofrequency contours of the infinitely periodic structure are
obtained, these can be used to infer the behaviour of a finite laminate, given
a large enough number of bilayers are takenJoannopoulos  _et al._ (2008). The
design of the corrugation for a particular effect then rests on interpreting
these contours, as we do below.

##  III  Selective Diffraction through Surface Corrugation

Figure 4: Splitting of SV- and P-waves through selective diffraction by
surface corrugation: (a) Total compressional potential field, $\phi$.
Compressional waves are preferentially excited by forcing the line source
perpendicular to its direction, at a non-dimensional frequency $\Omega=24$.
The line forcing is oriented at $51^{\circ}$ to the laminate axis, and marked
by a green line, with the particle motion shown by the half-headed arrow. Here
no surface corrugation is present. The laminate consists of 10 bilayers such
that $a_{1}=a_{2}=5$\mathrm{cm}$.$ (b) Similar to (a) but with corrugation
present on both sides of laminate in the form of extrusions of layer 1, with
dimensions $a_{c}\times 0.25a_{c}$, where $a_{c}=5.7$\mathrm{cm}$$ is the
periodicity of the corrugation, shown in inset. (c) Isofrequency contours of
the laminate (green) with isocircle of incident P-wave in material 2
surrounding the laminate shown in red. The construction line (tangential
component of $\bm{k}_{p}$ shown in dashed-red) intersects the contours and so
conventional refraction occurs at an angle defined by the normal to the
intersection. The reflected, refracted and transmission directions are shown
by the arrows marked with $r$, $R$ and $t$ respectively, corresponding to
those shown in (a-b). (d) Similar to (a) but for SV excitation, which are
preferentially excited by applying a forcing parallel to the line direction.
At this frequency and angle there exists a bandgap (yellow regions); the
construction line in (f) of $\bm{k}_{s}$ (dashed blue line) does not intersect
any of the laminates contours and so only reflection occurs, shown by $r_{1}$.
(e) Similar to (b) but for SV excitation. Negative refraction and higher order
reflection occurs due to the intersection of the translated construction line
via momentum transfer of $\bm{k}_{s}-\bm{G}_{c}$, as predicted by the
isofrequency contours shown in (f), which has the same vertical scale as (c).
Here the isocircle of the incident SV-wave in material 2 is shown by the blue
circle, with the directions of reflection, negative refraction and, by
reciprocity, transmission shown by the arrows labelled $r_{1}$, $r_{2}$, $NR$,
$t_{1}$ and $t_{2}$ respectively. The red arrow highlighted in (f) shows a
non-causal wavevector; the energy flow within the crystal is towards the
source and so cannot be excitedFoteinopoulou and Soukoulis (2005). Low
reflecting boundary conditions are applied on the edges of the domain in
(a,b,d,e).

Inspired by the ability to achieve negative refraction through surface
corrugation in optical systemsLu  _et al._ (2007), we extend this to
elasticity showing that the resulting negative refraction is not only
selective by angle, as in the optical case, but also by wave type, due to the
propagation of the two elastic body waves at different speeds. The idea rests
upon momentum transfer due to the periodic nature of the surface corrugation,
on account of wavevectors in a periodic structure being defined up to a
reciprocal lattice vector $\bm{G}_{c}$. This concept is commonly exploited in
grating couplers Chuang (2012), 2D photonic crystalsFoteinopoulou and
Soukoulis (2005) and recently for manipulating surface elastic Rayleigh
wavesChaplain  _et al._ (2020).

The design of the surface corrugation relies upon interpreting the
isofrequency contours gained from the TMM method. Given an incident P(SV)-wave
of wavevector of $\bm{k}_{p(s)}$ at an angle $\theta_{1p(s)}$ to the laminate
axis (Figure 2(b)), there is then a Bloch wave component
${k}_{z}=\cos\theta_{1p(s)}|\bm{k}_{p(s)}|$ in the $z$-direction of
periodicity, along with a vertical component parallel to the laminate
interface ${k}_{x}=\sin\theta_{1p(s)}|\bm{k}_{p(s)}|$. As such, for each
frequency a contour is defined over the angles of incidence. Example
isofrequency contours at the non-dimensional frequency $\Omega=24$ are shown
in Figure 4(c,f) for the cases of incident P- and SV-waves respectively. The
$z$-direction is periodic such that $\Gamma\equiv\bm{k}_{z}=0$ and $\pm
X\equiv\bm{k}_{z}=\pm\frac{\pi}{(a_{1}+a_{2})}$.

Figure 4 shows the effect of selective diffraction by angle, frequency and
wave-type for elastic materials with arbitrary properties such that
$c_{2p}/c_{1p}\sim 4.3$ and $c_{2s}/c_{1s}\sim 2.8$, similar to the ratios
achievable between some polymersMason (1958). We stress here that given the
non-dimensional approach taken in constructing the TMM, the dispersion
relations for laminates composed of arbitrary materials can quickly be
characterised. This highlights an additional benefit of the adopted
methodology, since comparisons between materials with unusual elastic
properties (e.g. metamaterials with effective propertiesKadic  _et al._
(2012)) can be considered. The laminate is composed of 10 unit bilayer strips
such that $a_{1}=a_{2}=5$\mathrm{cm}$$, embedded in a material with the same
properties as material 2, as shown in Figure 2(b). As such, additional to the
contours shown in Figure 4(c,f) we show the isotropic contours of P and SV
waves in material 2, shown in red and blue respectively. Determining the
design of the corrugation for splitting SV- and P-waves then rests on
interpreting the intersections of the tangential component of the incident
wave at the interface (sometimes referred to as the construction
lineFoteinopoulou and Soukoulis (2005)) with the contours of the periodic
structure.

In the presence of a vertical periodic corrugation, the vertical wavevector
component, $k_{x}$, of the incident wave is then also defined up to a
reciprocal lattice vector of the vertical grating such that
$\bm{G}_{c}=[0,2\pi/a_{c}]$ with $a_{c}$ the vertical period (recall the
$(z,x)$ coordinate system). As such, translations of the construction line, by
integer multiples of $\bm{G}_{c}$, result. The desired wave splitting is then
achieved by ensuring this induced momentum "kick" results in selective
negative refraction for only one wave type. This is predicted from
intersections of the translated construction lines with the isofrequency
contours of each wave type, and the resulting directions of energy flow
(normal to this intersection).

In Figures 4(a,b) we show the compressional potential for an incident wave,
predominantly of compressional motion due to the perpendicular forcing of the
line source, at $51^{\circ}$ to the laminate axis. Figure 4(a) shows the case
where the laminate has no corrugation on the outer interfaces, whilst Figure
4(b) shows the effect of adding a surface corrugation of periodicity
$a_{c}=5.7$\mathrm{cm}$$ on both sides of the laminate. The corrugation takes
the form of a periodic array of extrusions of the first material, i.e. blocks
of dimension $a_{1}\times 0.25a_{1}$, shown by the inset in Figure 4(b). We
note that the exact form of the grating, and therefore the guided modes it
supports, is somewhat arbitrary. The selective diffraction mechanism only
rests on the periodicity of the structure. As such the corrugation could
easily be, for example, a gentle sinusoidal variation in thickness; the
coupling efficiency will however be effected by the depth of the grating, just
as in its optical counterpartsMarchetti  _et al._ (2019). Interpreting the
contours in Figure 4(c) shows that the similarities in (a,b) are expected; at
this angle the construction line intersects the a contour of the laminate,
with or without the presence of the grating. The angle of refraction is then
determined by the normal to this intersection point. As such the P-waves
remain unaffected by the surface corrugation.

In stark contrast to this, are Figures 4(d,e), which show the effect without
and with the corrugation on the shear potential. Shear waves are predominantly
excited by forcing the line source in a direction parallel to it. Figure 4(d)
shows the shear potential at the same frequency and angle as before, with no
corrugation present. There is no intersection with the construction line and
the contours within the laminate, shown in Figure 4(f), and as such total
reflection occurs at the interface. However when the corrugation is included,
as in Figure 4(e), we see that incorporating momentum transfer by the grating
by the subtraction of a reciprocal lattice vector $G_{c}=[0,2\pi/a_{c}]$
(often attributed to the Umklapp mechanismFoteinopoulou and Soukoulis (2005);
Chaplain and Craster (2020)) from the incident $\bm{k}_{s}$ results in an
intersection of the translated construction line with the laminate contours.
This is shown in Figure 4(f). Determining the normal to this intersection then
gives the angle of negative refraction for the incident SV-wave, which is
clearly visible in Figure 4(e). Additionally to this there is a higher order
reflection at the interface, also shown, again due to momentum transfer via
the corrugation. The angles of transmission on the other side are shown,
arising due to reciprocity. Hence by incorporating the grating the transmitted
SV- and P-potential fields are split.

##  IV Conclusions

We have theoretically shown that it is possible to separate the shear-vertical
and compressional elastic wave components through selective diffraction, by
incorporating surface corrugation on the outer interfaces of a periodic
bilayered elastic laminate. The design rests on obtaining its corresponding
isofrequency contours, and then incorporating the desired momentum transfer
from the vertical periodic stratification. To demonstrate this effect we used
block extrusions of the first medium in the bilayer to form a grating on
either side of the laminate. We note that the form of the grating is somewhat
arbitrary, all that is required is there is some periodicity which provides
the relevant momentum "kick" to the incident wave. Indeed the grating need not
be symmetric on either side of the laminate, providing a further avenue for
the manipulation of elastic waves. The simplicity of this then has advantages
over say, tailored phased metasurfaces, as all that is required to predict the
effects are the isofrequency contours of the layered structure, which we
obtain through a non-dimensionalised version of the TMM. The demonstrated
separation of the elastic potential fields was realised for two gratings which
enabled selective negative diffraction for only P-waves (Fig. 1) and only for
SV-waves (Fig. 4). Additionally, for certain frequency ranges and angles of
incidence, both wave types will be affected and experience different angles of
refraction, depending on the band structure of the distinct wave types. As
such the presented surface corrugated laminate structures form a class of
elastic grating couplers capable of physical effects further than their scalar
wave counterparts.

###### Acknowledgements.

G.J.C gratefully acknowledges financial support from the EPSRC in the form of
a Doctoral Prize Fellowship. The support of the UK EPSRC through grant
EP/T002654/1 is acknowledged as is that of the ERC H2020 FETOpen project
BOHEME under grant agreement No. 863179.

## Data availability

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

##  Appendix A  Parameters for Figure 1.

The elastic parameters of the bilayers and surrounding elastic material
defined in Figure 1 are the same as those utilised in Figure 4. All that
differs are the dimensions of the corrugation; the periodicity of which is
defined as $a_{c}=12$\mathrm{cm}$$, with width $a_{c}/2$ and height $a_{c}/4$.
The non-dimensional frequency of excitation in this case is $\Omega=40$ at an
angle of $30^{\circ}$ with respect to the laminate normal.

##  Appendix B  Transfer Matrix Method

Here we detail the TMM by firstly nondimensionalising the governing equations,
in order to circumvent numerical instabilities which arise in the TMM.
Particularly, this is unstable at high frequencies and layer thicknesses (the
large f-d problemDunkin (1965)). We reformulate the governing wave equations
in terms of the shear wave speed in the first layer, $c_{1s}$. This allows us
to express the TMM formalism entirely in terms of ratios of the speeds and
material properties in each layer. We introduce the length scale $l$ such that
$Z=zl$ and $X=lx$. Therefore the equation for the shear potential in the first
layer reads

$\nabla_{Z}^{2}\bm{\psi}_{1}+\Omega_{1}^{2}\bm{\psi}_{1}=0,$ (7)

where
$\nabla_{Z}^{2}=\frac{1}{l^{2}}\left(\partial_{Z}^{2}+\partial_{X}^{2}\right)$
and $\Omega_{1}=\frac{l\omega}{c_{1s}}$. Letting subscripts denote the layer
order, and implicitly incorporating the re-scaling, the rest of the potentials
are then governed by

$\displaystyle\begin{split}\nabla^{2}\phi_{1}+(\Omega_{1}\xi_{1p})^{2}\phi_{1}=0,\\\
\nabla^{2}\bm{\psi}_{2}+(\Omega_{1}\xi_{2s})^{2}\bm{\psi}_{2}=0,\\\
\nabla^{2}\phi_{2}+(\Omega_{1}\xi_{2p})^{2}\phi_{2}=0,\end{split}$ (8)

where $\xi_{i\delta}^{2}=\frac{c_{1s}^{2}}{c_{i\delta}^{2}}$, with $i=1,2$ and
$\delta=p,s$. Two separate versions of the TMM then arise, one for incident P
waves, and one for incident SV waves. Solutions then arise in the form of a
superposition of left and right propagating waves in each layer. Considering
the case of a purely incident P wave, the angles can all be expressed in terms
of the incident angle $\theta_{1p}$ from (6). As such the solutions can be
written

$\displaystyle\begin{split}\bm{\psi}_{1}&=(A_{1s}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{i\Omega_{1}\beta_{1}Z}+A_{2s}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{-i\Omega_{1}\beta_{1}Z})\bm{\hat{\textbf{j}}}\\\
\phi_{1}&=A_{1p}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{i\xi_{1p}\Omega_{1}\beta_{0}Z}+A_{1p}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{-i\xi_{1p}\Omega_{1}\beta_{0}Z}\\\
\bm{\psi}_{2}&=(B_{1s}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{i\xi_{2s}\Omega_{1}\gamma_{1}Z}+B_{2s}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{-i\xi_{2s}\Omega_{1}\gamma_{1}Z})\bm{\hat{\textbf{j}}}\\\
\phi_{2}&=B_{1p}e^{i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{i\xi_{2p}\Omega_{1}\gamma_{2}Z}+B_{2p}e^{-i\xi_{1p}\Omega_{1}\alpha_{0}X}e^{i\xi_{2p}\Omega_{1}\gamma_{2}Z},\end{split}$
(9)

where $\alpha_{0}=\sin\theta_{1p}$, $\beta_{0}=\cos\theta_{1p}$,
$\beta_{1}=\cos(\sin^{-1}(\xi_{1p}\alpha_{0}))$,
$\gamma_{1}=\cos(\sin^{-1}(\zeta_{2s}\alpha_{0}))$,
$\gamma_{2}=\cos(\sin^{-1}(\zeta_{2p}\alpha_{0}))$ where
$\zeta_{i\delta}=\frac{c_{i\delta}}{c_{1p}}$. The amplitudes $A_{1(2)}$
correspond to the right(left) propagating waves in the first layer, with
$B_{1(2)}$ to those in the second. In order to maintain the non-dimensional
formulation we then construct the vector
$\bm{W}^{\prime}_{i}=(\frac{w}{i\Omega_{1}},\frac{u}{i\Omega_{1}},\frac{\sigma_{xz}}{\Omega_{1}^{2}c_{1p}^{2}\rho_{1}},\frac{\sigma_{xz}}{\Omega_{1}^{2}c_{1p}^{2}\rho_{1}})^{\text{T}}$.
Factorising the $e^{i\xi_{1p}\Omega_{1}\alpha_{0}x}$ terms allows us to write

$\bm{W}^{\prime}_{1}=C_{1}D_{1}\bm{A},$ (10)

where $\bm{A}=(A_{1p},A_{2p},A_{1s},A_{2s})^{\text{T}}$, and $D_{1}$ is the
propagator matrix for layer 1 such that
$D_{1}=\text{diag}(\exp{\big{[}i\xi_{1p}\Omega_{1}\beta_{0}Z},{-i\xi_{1p}\Omega_{1}\beta_{0}Z},{i\Omega_{1}\beta_{1}Z},{-i\Omega_{1}\beta_{1}Z}\big{]})$.
Similarly in the second layer

$\bm{W}^{\prime}_{2}=C_{2}D_{2}\bm{B},$ (11)

where $\bm{B}=(B_{1p},B_{2p},B_{1s},B_{2s})^{\text{T}}$, and $D_{2}$ is the
propagator matrix for layer 2 such that
$D_{2}=\text{diag}(\exp{\big{[}i\xi_{2p}\Omega_{1}\gamma_{2}Z},{-i\xi_{2p}\Omega_{1}\gamma_{2}Z},{i\xi_{2s}\Omega_{1}\gamma_{1}Z},{-i\xi_{2s}\Omega_{1}\gamma_{1}Z}\big{]})$,
with the forms of $C_{1}$ and $C_{2}$ shown below. We focus on a unit strip
consisting of two layers, centred at $Z=0$ with the first layer of width
$a_{1}$ and the second of $a_{2}$, such that $l=a_{1}+a_{2}$. Without loss of
generality we make the coordinate shift $Z\rightarrow Z+a_{1}$, and as such at
the edge of layer 1

$\displaystyle\begin{split}\bm{W}^{\prime}_{1}(-a_{1})=C_{1}D_{1}(-a_{1})\bm{A}\\\
\implies\bm{A}=C_{1}^{-1}\bm{W}^{\prime}_{1}(-a_{1}).\end{split}$ (12)

Similarly at the opposite edge of layer 2,

$\displaystyle\begin{split}\bm{W}^{\prime}_{2}(a_{2})=C_{2}D_{2}(a_{2})\bm{B}\\\
\implies\bm{B}=[C_{2}D_{2}(a_{2})]^{-1}\bm{W}^{\prime}_{2}(a_{2}).\end{split}$
(13)

Evaluating (10) and (11) at the centre $Z=0$ i.e. the interface between layers
1 and 2 gives

$\displaystyle\begin{split}\bm{W}^{\prime}_{1}(0)&=P_{1}\bm{W}^{\prime}_{1}(-a_{1}),\\\
\bm{W}^{\prime}_{2}(0)&=P_{2}\bm{W}^{\prime}_{2}(a_{2}),\end{split}$ (14)

where $P_{1}=C_{1}D_{1}(0)C_{1}^{-1}$ and
$P_{2}=C_{2}D_{2}(0)D_{2}^{-1}(a_{2})C_{2}^{-1}$. Enforcing the continuity
equations requires that

$\displaystyle\begin{split}\bm{W}^{\prime}_{1}(0)&=\bm{W}^{\prime}_{2}(0)\\\
\implies\bm{W}^{\prime}_{2}(a_{2})&=P_{2}^{-1}\mathds{1}P_{1}\bm{W}^{\prime}_{2}(-a_{1}).\end{split}$
(15)

Enforcing the Bloch condition, given the structure is periodic in $Z$, then
gives

$\displaystyle\begin{split}\bm{W}^{\prime}_{2}(a_{2})=e^{i\xi_{1p}\Omega_{1}\beta_{0}}\bm{W}^{\prime}_{1}(-a_{1}).\end{split}$
(16)

As such the dispersion relation arises from solving the eigenvalue problem

$(Q-e^{i\xi_{1p}\Omega_{1}\beta_{0}}\mathds{1})\bm{W}^{\prime}_{1}(-a_{1})=0,$
(17)

which is satisfied when
$\det[Q-e^{i\xi_{1p}\Omega_{1}\beta_{0}}\mathds{1}]=0$, with
$Q=P_{2}^{-1}\mathds{1}P_{1}$. A similar approach is adopted for an incident
SV-wave, with each angle being written in terms of $\theta_{1s}$. We note that
despite the advantages of nondimensionalisation, for large incident angles the
matrices can become close to singular due to small entries within $P_{2}$.
Inaccuracies surrounding this can be alleviated by using the pseudoinverse.

###  B.1 $C_{i}$ Matrices

The forms of $C_{i}$ are given below, in terms of the non-dimensional ratios
of speeds, $\xi$ and $\zeta$ defined above and in terms of the non-dimensional
density $\Upsilon=\frac{\rho_{2}}{\rho_{1}}$. Where the matrix components are
too lengthy they are shown separately in the form of $C^{(pq)}_{i}$ with $p,q$
denoting the row and column, in the $i^{th}=1,2$ corresponding elastic layer.

$\displaystyle C_{1}$
$\displaystyle=\begin{pmatrix}\xi_{1p}\beta_{0}&-\xi_{1p}\beta_{0}&-\xi_{1p}\alpha_{0}&-\xi_{1p}\alpha_{0}\\\
\\\ \xi_{1p}\alpha_{0}&\xi_{1p}\alpha_{0}&\beta_{1}&-\beta_{1}\\\ \\\
C^{(31)}_{1}&C^{(32)}_{1}&2\xi_{1p}^{3}\alpha_{0}\beta_{1}&-2\xi_{1p}^{3}\alpha_{0}\beta_{1}\\\
\\\
-2\xi_{1p}^{4}\alpha_{0}\beta_{0}&2\xi_{1p}^{4}\alpha_{0}\beta_{0}&\xi_{1p}^{2}\left(\left(\xi_{1p}\alpha_{0}\right)^{2}-\beta_{1}^{2}\right)&\xi_{1p}^{2}\left(\left(\xi_{1p}\alpha_{0}\right)^{2}-\beta_{1}^{2}\right)\end{pmatrix},$
(18) $\displaystyle C_{2}$
$\displaystyle=\begin{pmatrix}\xi_{2p}\gamma_{2}&-\xi_{2p}\gamma_{2}&-\xi_{1p}\alpha_{0}&-\xi_{1p}\alpha_{0}\\\
\\\
\xi_{1p}\alpha_{0}&\xi_{1p}\alpha_{0}&\xi_{2s}\gamma_{1}&-\xi_{2s}\gamma_{1}\\\
\\\
C^{(31)}_{2}&C^{(32)}_{2}&2\zeta_{2s}^{2}\Upsilon\xi_{1p}\xi_{2s}\alpha_{0}\gamma_{1}&-2\zeta_{2s}^{2}\Upsilon\xi_{1p}\xi_{2s}\alpha_{0}\gamma_{1}\\\
\\\
-2\zeta_{2s}^{2}\Upsilon\xi_{1p}\xi_{2p}\alpha_{0}\gamma_{2}&2\zeta_{2s}^{2}\Upsilon\xi_{1p}\xi_{2p}\alpha_{0}\gamma_{2}&\zeta_{2s}^{2}\Upsilon\left(\left(\xi_{1p}\alpha_{0}\right)^{2}-\left(\xi_{2s}\gamma_{1}\right)^{2}\right)&\zeta_{2s}^{2}\Upsilon\left(\left(\xi_{1p}\alpha_{0}\right)^{2}-\left(\xi_{2s}\gamma_{1}\right)^{2}\right)\end{pmatrix},$
(21)
$\displaystyle\begin{split}C^{(31)}_{1}&=2\xi_{1p}^{4}+\frac{2i\xi_{1p}^{3}\beta_{0}}{\Omega_{1}}-\xi_{1p}^{2},\\\
C^{(32)}_{1}&=2\xi_{1p}^{4}-\frac{2i\xi_{1p}^{3}\beta_{0}}{\Omega_{1}}-\xi_{1p}^{2},\\\
C^{(31)}_{2}&=\Upsilon\xi_{1p}^{2}\left(\alpha_{0}^{2}(2\zeta_{2s}^{2}-\zeta_{2p}^{2})-\gamma_{2}^{2}\right)+2\zeta_{2s}^{2}\Upsilon\left(\frac{i\xi_{2p}\gamma_{2}}{\Omega_{1}}+\xi_{2p}^{2}\gamma_{2}^{2}\right),\\\
C^{(32)}_{2}&=\Upsilon\xi_{1p}^{2}\left(\alpha_{0}^{2}(2\zeta_{2s}^{2}-\zeta_{2p}^{2})-\gamma_{2}^{2}\right)+2\zeta_{2s}^{2}\Upsilon\left(\xi_{2p}^{2}\gamma_{2}^{2}-\frac{i\xi_{2p}\gamma_{2}}{\Omega_{1}}\right).\end{split}$
(24)

##  Appendix C  Elastic Grating Coupler

As elucidated, the term grating coupler is almost exclusively used to describe
electromagnetic devices, particularly prevalent in integrated silicon photonic
devices and fibre optics Taillaert  _et al._ (2002); Chuang (2012); Marchetti
_et al._ (2017, 2019). In elasticity, coupling achieved by periodic gratings
typically describes excitation of surface acoustic waves utilising
piezoelectric coupling White and Voltmer (1965); Zhang, Desbois, and Boyer
(1993); Golan  _et al._ (1991). Certain elastic devices have utilised
diffractive effects from non-uniform gratings to introduce frequency dependent
time delays for shear wavesCoquin and Tsu (1965). We show here, as an
additional example, that conventional grating coupler modalities can be
achieved in the elastic setting. The result is that, due to the presence of
designed grating in a local region along the laminate, wave energy can enter
and be trapped within the laminate due to subsequent total internal
reflections.

Figure 5 shows a direct comparison to optical devices, whereby the presence of
a periodic surface structuring enables incident radiation to couple to a
waveguide. The structure is similar to that considered in Figure 4, all that
has altered is the grating; the periodicity of which is defined as
$a_{c}=9.5$\mathrm{cm}$$, with width $a_{c}/2$ and height $a_{c}/4$. There are
40 cells present in the simulation of Figure 5(b), only present on the left
interface of the structure. Figure 5(a) shows the total solid displacement
field which results given a perpendicular forcing of the line source, oriented
at $9.8^{\circ}$ to the laminate normal; for the chosen frequency of
excitation there is no supported mode within the waveguide and as such
reflection from the interface occurs. Introducing a periodic grating in the
form of a surface corrugation on the outer interface of the laminate allows,
via the transfer of crystal momentum, excitation of a diffractive order
capable of coupling with a waveguide mode. This is shown in Figure 5(b).
Furthermore the grating is only introduced in the region desired to couple
incident radiation; no grating is present anywhere else along the array and as
such, by total internal reflection the waveguided mode is now trapped within
the laminate.

Figure 5: Elastic Grating Coupler: (a) Total solid displacement field
($|\bm{u}|$) excited by perpendicular line forcing (green line, similar to
Fig. 4(a)) at $9.8^{\circ}$ to the laminate axis. The non-dimensional
frequency of excitation is $\Omega=50$. There is no surface grating present on
the laminate structure and as such no elastic energy penetrates into the
laminate, a schematic of which is shown in Figure 2. (b) Total solid
displacement field for the same scenario as in (a), except there is a grating
present on the outer interface of the laminate (the internal structure is
drawn in light grey as to not obscure the field). The presence of the grating
incorporates a transfer of crystal momentum related to the periodicity of the
grating (shown in inset). The wave is subsequently trapped in the laminate
waveguide by total internal reflections. The laminates are composed of bilayer
strips of elastic materials labelled 1,2 (coloured gold and silver
respectively in the inset), similar to the schematic shown in Figure 2.

## References

  * Colquitt  _et al._ (2015) D. J. Colquitt, R. V. Craster, T. Antonakakis, and S. Guenneau, “Rayleigh–Bloch waves along elastic diffraction gratings,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, 20140465 (2015).
  * Pendry, Martin-Moreno, and Garcia-Vidal (2004) J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847–848 (2004).
  * Hibbins, Evans, and Sambles (2005) A. P. Hibbins, B. R. Evans, and R. J. Sambles, “Experimental verification of designer surface plasmons,” Science 308, 670–672 (2005).
  * Wilcox (1984) C. H. Wilcox, “Scattering theory for diffraction gratings,” Mathematical Methods in the Applied Sciences 6, 158–158 (1984).
  * Graff (1975) K. Graff,  _Wave motion in elastic solids_ (Oxford University Press, 1975).
  * Yu and Capasso (2014) N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature materials 13, 139–150 (2014).
  * Su and Norris (2016) X. Su and A. N. Norris, “Focusing, refraction, and asymmetric transmission of elastic waves in solid metamaterials with aligned parallel gaps,” The Journal of the Acoustical Society of America 139, 3386–3394 (2016).
  * Su, Lu, and Norris (2018) X. Su, Z. Lu, and A. N. Norris, “Elastic metasurfaces for splitting sv-and p-waves in elastic solids,” Journal of Applied Physics 123, 091701 (2018).
  * Ahn  _et al._ (2019) B. Ahn, H. Lee, J. S. Lee, and Y. Y. Kim, “Topology optimization of metasurfaces for anomalous reflection of longitudinal elastic waves,” Computer Methods in Applied Mechanics and Engineering 357, 112582 (2019).
  * Lu  _et al._ (2007) W. T. Lu, Y. J. Huang, P. Vodo, R. K. Banyal, C. H. Perry, and S. Sridhar, “A new mechanism for negative refraction and focusing using selective diffraction from surface corrugation,” Optics Express 15, 9166 (2007).
  * Joannopoulos  _et al._ (2008) J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade,  _Photonic Crystals, Molding the Flow of Light_ , 2nd ed. (Princeton University Press, Princeton, 2008).
  * Taillaert  _et al._ (2002) D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. Van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE Journal of Quantum Electronics 38, 949–955 (2002).
  * Chuang (2012) S. Chuang,  _Physics of photonic devices_ , Vol. 80 (John Wiley & Sons, 2012).
  * Marchetti  _et al._ (2017) R. Marchetti, C. Lacava, A. Khokhar, X. Chen, I. Cristiani, D. J. Richardson, G. T. Reed, P. Petropoulos, and P. Minzioni, “High-efficiency grating-couplers: demonstration of a new design strategy,” Scientific reports 7, 1–8 (2017).
  * Marchetti  _et al._ (2019) R. Marchetti, C. Lacava, L. Carroll, K. Gradkowski, and P. Minzioni, “Coupling strategies for silicon photonics integrated chips,” Photonics Research 7, 201–239 (2019).
  * Chimenti (1997) D. E. Chimenti, “Guided Waves in Plates and Their Use in Materials Characterization,” Applied Mechanics Reviews 50, 247–284 (1997).
  * Markos and Soukoulis (2008) P. Markos and C. M. Soukoulis,  _Wave propagation: from electrons to photonic crystals and left-handed materials_ (Princeton University Press, 2008).
  * Schoenberg (1984) M. Schoenberg, “Wave propagation in alternating solid and fluid layers,” Wave motion 6, 303–320 (1984).
  * Pernas-Salomón  _et al._ (2015) R. Pernas-Salomón, R. Pérez-Álvarez, Z. Lazcano, and J. Arriaga, “The scattering matrix approach: a study of elastic waves propagation in one-dimensional disordered phononic crystals,” Journal of Applied Physics 118, 234302 (2015).
  * Thomson (1950) W. T. Thomson, “Transmission of elastic waves through a stratified solid medium,” Journal of applied Physics 21, 89–93 (1950).
  * Haskell (1953) N. A. Haskell, “The dispersion of surface waves on multilayered media,” Bulletin of the seismological Society of America 43, 17–34 (1953).
  * Huang, Wang, and Rokhlin (1996) W. Huang, Y. Wang, and S. Rokhlin, “Oblique scattering of an elastic wave from a multilayered cylinder in a solid. transfer matrix approach,” The journal of the acoustical society of america 99, 2742–2754 (1996).
  * Fomenko  _et al._ (2014) S. Fomenko, M. Golub, C. Zhang, T. Bui, and Y.-S. Wang, “In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals,” International Journal of Solids and Structures 51, 2491–2503 (2014).
  * Dunkin (1965) J. W. Dunkin, “Computation of modal solutions in layered, elastic media at high frequencies,” Bulletin of the Seismological Society of America 55, 335–358 (1965).
  * Knopoff (1964) L. Knopoff, “A matrix method for elastic wave problems,” Bulletin of the Seismological Society of America 54, 431–438 (1964).
  * Adamou and Craster (2004) A. Adamou and R. Craster, “Spectral methods for modelling guided waves in elastic media,” The Journal of the Acoustical Society of America 116, 1524–1535 (2004).
  * Achenbach (2012) J. Achenbach,  _Wave propagation in elastic solids_ , Vol. 16 (Elsevier, 2012).
  * Foteinopoulou and Soukoulis (2005) S. Foteinopoulou and C. M. Soukoulis, “Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects,” Physical Review B 72, 165112 (2005).
  * Chaplain  _et al._ (2020) G. J. Chaplain, J. M. De Ponti, A. Colombi, R. Fuentes-Dominguez, P. Dryburgh, D. Pieris, R. J. Smith, A. Clare, M. Clark, and R. V. Craster, “Tailored elastic surface to body wave umklapp conversion,” Nature communications 11, 1–6 (2020).
  * Mason (1958) W. Mason,  _Physical Acoustics and the Properties of Solids_ , Bell Telephone Laboratories series (Van Nostrand, 1958).
  * Kadic  _et al._ (2012) M. Kadic, T. Bückmann, N. Stenger, M. Thiel, and M. Wegener, “On the practicability of pentamode mechanical metamaterials,” Applied Physics Letters 100, 191901 (2012).
  * Chaplain and Craster (2020) G. J. Chaplain and R. V. Craster, “Ultrathin entirely flat umklapp lenses,” Physical Review B 101, 155430 (2020).
  * White and Voltmer (1965) R. M. White and F. W. Voltmer, “Direct piezoelectric coupling to surface elastic waves,” Applied physics letters 7, 314–316 (1965).
  * Zhang, Desbois, and Boyer (1993) Y. Zhang, J. Desbois, and L. Boyer, “Characteristic parameters of surface acoustic waves in a periodic metal grating on a piezoelectric substrate,” IEEE transactions on ultrasonics, ferroelectrics, and frequency control 40, 183–192 (1993).
  * Golan  _et al._ (1991) G. Golan, G. Griffel, A. Seidman, and N. Croitoru, “Grating-assisted surface acoustic wave directional couplers,” Journal of applied physics 70, 46–52 (1991).
  * Coquin and Tsu (1965) G. Coquin and R. Tsu, “Theory and performance of perpendicular diffraction delay lines,” Proceedings of the IEEE 53, 581–591 (1965).