Hybrid Surface-Bulk Mode in Periodic Gratings
Natalya Naumenko1, Benjamin Abbott2
1Moscow Steel and Alloys Institute, 117936, Leninski prosp.,4. Moscow, Russia
2 SAWTEK Inc., 1818 South Hwy 441, Apopka, FL 32703
Abstract – We report on a SAW/BAW hybrid mode (SBH),
which arises from surface skimming bulk wave due to the
trapping of its energy by a counter-propagating surface wave
(SAW) or leaky surface wave (LSAW), in periodic grating
structures. The SBH is the result of an interaction between
surface and bulk waves and exists within limited frequency
region, outside which it is decomposed into counter-
propagating SAW and BAW. Different examples of SBH
mode illustrate interactions of BAW with pure SAW, leaky
and high-velocity leaky waves. The behavior of the SBH mode
was numerically investigated using harmonic admittance
function. Calculation of velocity dispersion revealed the
existence of additional branch of dispersion curve, which
refers to SBH mode. A stopband occurs due to SAW/BAW
interaction and manifests itself by a resonance of harmonic
admittance at its lower edge, which can be considered as the
cut-off frequency of bulk wave radiation.
One of the most important requirements of modern
SAW filters, especially for RF applications, is low
insertion loss, which can be provided by using resonator
structures on piezoelectric substrates with strong
piezoelectric coupling. High propagation velocity of
acoustic wave is also desirable for high frequency devices.
Therefore leaky (or pseudo-surface) acoustic waves
(LSAW), existing in rotated Y-cuts of lithium niobate (LN)
and lithium tantalate (LT)
electromechanical coupling coefficient, are the good
choices of substrate material in low loss filters.
Rotated Y-cuts of LT, X propagation, with rotation
angles from 36° to 42° and rotated Y-cuts of LN with
rotation angles from 41° to 64° are most widely used.
Though any LSAW exhibits nonzero propagation loss
caused by bulk wave radiation, with proper choice of
rotation angle and metal electrode thickness in the grating,
this loss can be minimized. Propagation velocity of LSAW
in the grating also depends on orientation and electrode
thickness and its maximum value tends to that of the fast
shear surface skimming bulk wave with reducing metal
thickness and rotation angle approaching 37° for LT and
41° for LN. In mentioned orientations, fast shear bulk
wave becomes SH-polarized and satisfies stress-free
and having high
mechanical boundary conditions without a metal grating
(In SAW theory such a bulk wave is called exceptional)
. As a result, the bulk wave is strongly generated by the
grating and interacts with the leaky wave, thus giving
parasitic response and degradation of SAW filter
The problem of interaction between LSAW and fast
shear bulk wave in 36YX cut of LT was recently
investigated by various researchers [2-4] using different
numerical techniques. In particular, Fusero et al 
examined harmonic admittance as function of frequency
and wave number, and arrived to the conclusion that
additional “PSAW-like” mode exists in this orientation
when electrode thickness is sufficiently large.
A detailed analysis of the new mode was performed to
verify its existence. As a result, the mechanism of its
forming in the grating was found. The results of a thorough
numerical investigation of the new mode are presented,
and a physical interpretation is suggested. We found that
the SAW/BAW Hybrid mode (SBH) arises from the surface
skimming bulk acoustic wave (BAW) due to the trapping
of its energy by the counter-propagating surface wave. The
mechanism of its forming from the bulk wave is
fundamentally different from that of other known surface
and leaky waves with quasi-bulk structure. SBH mode
exists only in a grating and within limited frequency range.
Outside this range, SBH mode is decomposed into pure
bulk wave and counter-propagating SAW.
Different examples of SBH, which illustrate interaction
of BAW with pure SAW, STW, and leaky waves, (low-
and high-velocity), are described. The lower edge of the
stopband, which occurs due to SAW/BAW interaction, was
found to manifest itself as a resonance of the harmonic
admittance which is indicative of spurious resonances in
SAW filter frequency response.
II. HARMONIC ADMITTANCE AS METHOD OF ANALYSIS
In recent years, rapid progress has been made in the
development of efficient techniques for rigorous analysis
of SAW and LSAW propagation characteristics in grating
structures. In particular, harmonic admittance is very
useful for both the simulation of resonator SAW filters and
Proc. 2001 IEEE Ultrasonics Symposium, Atlanta, USA, Oct.7-10, 2001, pp. 243-248
for extraction of wave characteristics in infinitely periodic
The concept of harmonic admittance was first
introduced by Blotekjaer et al [5,6] in 1973. However,
Blotekjaer used an approximation for the piezoelectric
substrate's effective permittivity. Milsom et al 
introduced the rigorous evaluation of the effective
permittivity for piezoelectric substrates, which Zhang et al
 applied to the evaluation of the harmonic admittance
for periodic massless metal
contributions of Ventura et al  the concept of harmonic
admittance is now applicable to massive metal gratings.
The harmonic admittance represents the admittance of
an individual electrode in a periodic array of electrodes to
which a harmonic excitation is applied. Consider the
periodic array in Fig 1a. The applied voltages, Vn, are
These harmonic excitations vary spatially with the
normalized wavenumber, s=p/V. V is the period of the
harmonic excitation. In Fig.1a, each electrode's current is
proportional to the applied voltage and the harmonic
The harmonic admittance function, Y(f,s), is a powerful
tool for evaluating the spectral and frequency dependences
of SAW interdigital transducers. It may be used to
characterize sequences of electrodes with harmonic
excitations as well as finite length excitations. In
transducers with finite length excitations, the complete
spectrum of spatial harmonics must be considered, 0 s<1.
To evaluate the admittance of a finite length excitation
Fourier analysis is used.
gratings. With the
Consider the electrode sequence shown in Fig 1b, which
is assumed to have a finite excitation length. The spectral
representation of the transducer's voltage is given by the
Fourier transform of the electrode voltages.
The individual electrode voltages are dependent upon the
applied transducer voltage, VT, and each electrodes
The spectral representation of the electrode currents is
given by the product of the spectral voltage and the
The total transducer current, IT, is given by the
summation of the individual electrode currents.
By applying Fourier analysis, the total transducer
current may be represented by the integral of the spectral
representation of the transducer's current.
Given the voltage applied to the transducer is VT, the
transducer's admittance, YT(f), is
Therefore, the admittance of a transducer with a finite
length excitation is dependent upon the complete spectrum
of the harmonic admittance, 0s<1. Thus all resonances
present in the harmonic admittance function manifest
themselves in the finite transducer's admittance. The
resonances may have a negative impact on the performance
of SAW filters. Therefore, it is important to give
consideration to these undesired resonances. Particularly
those in close proximity to the resonances of the desired
surface modes. In the following section the SBH
resonances occurring between surface and bulk modes are
investigated. These hybrid resonances can be particularly
harmful to the performance of SAW filters due to their
close proximity to the resonances of the pure surface
Harmonic admittance can be regarded as an example of
a wavenumber-dependent function, which relates the
electric potential to the normal component of electric
Fig.1. Periodic array of electrodes with a) harmonic
excitation, b) finite excitation
displacement on the surface and contains information
about all acoustic modes in a given piezoelectric substrate.
Other examples of such a function are electric surface
impedance  and effective dielectric permittivity ,
which were first introduced for semi-infinite piezoelectric
medium and appeared to be very efficient approach to the
analysis of acoustic waves in other structures, including
periodic grating. The presence of velocity dispersion
makes analysis of wave propagation more complicated.
However, the basic relations obtained for semi-infinite
medium were successfully applied to the analysis of
grating structures [5,6,10].
In particular, the concept of effective piezoelectric
coupling coefficient, first introduced for non-dispersive
piezoelectric medium , can be generalized for
dispersive wave propagation problems. This coefficient
can be defined in terms of the velocity difference obtained
by changing electrical condition from short-circuited to
open-circuited, and characterizes power transfer from
electromagnetic source to acoustic waves.
For any frequency f, the following approximation is
valid, unless harmonic admittance is perturbed by another
where s0, s and C are functions of frequency. With s=s
being normalized wavelength of acoustic mode in short-
circuited grating, the effect of changing electric boundary
condition upon the solution of dispersive relation can be
assuming C=1 in a narrow frequency interval. With (9)
generalized for complex-valued s, K2(f) also becomes
complex, and s must be determined as an eigenmode of
dispersion relation for short-circuited grating.
Though derived in different way, K2(f) defined by (10)
is analogous to the effective electromechanical coupling
factor of the grating mode introduced by Hashimoto et al
 and shows frequency-dependent efficiency of
transformation of applied electric voltage into acoustic
power radiated in forward and backward directions. As
will be apparent from the following discussion, analysis of
this function helps to understand the effect of SBH mode
on SAW device performance and to see how the energy is
transferred between different modes in the grating.
Numerical technique, which combines matrix formalism
for finding discrete Green functions with BEM analysis of
electrode region, was used to calculate harmonic
admittance and velocity dispersion in the grating.
III. RESULTS AND DISCUSSION
To understand the nature of “PSAW-like” mode found
by Fusero et al , we analyzed the same example, 36YX
cut of LT, with a short-circuited infinite periodic two-
electrode-per wavelength grating structure, with Al
electrodes of thickness h/=0.04, and metallization ratio
w/p=0.5 (where is LSAW wavelength). First, the
harmonic admittance Y(f,s) was calculated as a function of
frequency, at fixed value of normalized wavelength s.
Under the Bragg reflection condition, s=0.5, only one
resonance of Y(f) occurs at the lower edge of the stopband.
Due to the symmetry of analyzed orientation and electrode
structure, there is no resonance at the opposite edge.
However, as was previously reported [2-4], with small
detuning from Bragg condition, Y(f) exhibits second
resonance at frequencies approximately corresponding to
the upper edge of the stopband .
Then, following the method suggested in , we
analyzed harmonic admittance as function of normalized
wavelength, at different frequencies around the upper edge
of LSAW stop band. The results are shown in Fig.2. We
used normalized frequency f’=fp/Vbulk, where p is the
period of the grating and Vbulk=4185,94 m/s is the fast shear
bulk wave velocity (calculations were made with material
constants of LT reported in ). Hence, s=f’ corresponds
to V=Vbulk. Two resonances of Y(s) appear in a narrow
frequency interval between f’=0.4970 and f’=0.4971 and
move with frequency. Both resonances are characterized
by nonzero propagation loss, Re(Y)0. The resonances
merge at f’ about 0.49711. The resulting maximum of
Re(Y) reduces rapidly with further increasing frequency
and vanishes at f’ about 0.49715.
Assuming that Re(Y) is negligible, each resonance in
Fig.2 represents a mode in the grating with velocity
V=f’Vbulk/s. Therefore, the velocity dispersion function can
be derived for each mode from the harmonic admittance
data. Both modes are confined in the velocity interval
(Vmax-LSAW, Vbulk), where Vmax-LSAW is the maximum LSAW
velocity in the short-circuited grating. Furthermore,
according to the behavior of Im(Y), these modes are
Direct calculation of the velocity dispersion function
was performed. The frequency interval around f’=0.497
was carefully examined and revealed that there exists an
additional branch of dispersion curve. In Fig.3, the
calculated velocity dispersion is shown for forward and
backward LSAW modes, with normalized wave numbers
s+ and s-=1-s+, respectively. An additional branch arises
from Vbulk at f’=0.4970 and merges with backward LSAW
at f’=0.49711. These results are consistent with the
behavior of harmonic admittance in Fig.2 and reveal the
nature of the new branch of the dispersion curve as a result
of interaction between the counter-propagating LSAW and
BAW. Hence the new mode can be referred to as
SAW/BAW Hybrid mode (SBH). It exists within limited
frequency interval, outside which it decomposes into pure
BAW and counter propagating LSAW. The stopband
occurs at the frequencies higher than f’=0.4971 due
LSAW/BAW interaction, and Im(V) increases in the
stopband indicating Bragg reflection. Thus the lower edge
of SBH stopband can be considered as the cut-off
frequency of bulk wave radiation. In the absence of SBH
mode, the cut-off frequency is higher and can be
determined as an intersection between the velocity of
backward LSAW and Vbulk .
Fig.4 illustrates how SBH mode changes with electrode
thickness. With thicker electrodes, SBH mode exists in
wider frequency interval. For h/=0.06 the stopband of the
SBH mode merges with the stopband of LSAW. As a
result, Im(V) does not vanish at the upper edge of LSAW
stopband, which makes the rigorous evaluation of this
important characteristic more difficult.
In addition to Re(V) and Im(V), the effective coupling
coefficient K2 of the SBH mode was calculated using (10).
For each electrode thickness analyzed, at the lower edge of
the stopband, which corresponds to SAW/BAW Bragg
reflection, Re(K2) reaches maximum value. Consequently,
SBH mode is expected to manifest itself by spurious
resonance at this frequency. When h/=0.04, Re(K2)
crosses zero value at f’ about 0.49712 and becomes
negative with further increasing frequency. This can be
interpreted as vanishing radiation in forward propagation
direction and further growth of radiation in backward
direction. However, these numerical results should be
treated with caution because approximation (9) used for
definition of coupling coefficient (10) can be invalid here
due to the interaction between LSAW and SBH mode.
Apparently, analytical consideration is required to
understand the mechanisms of energy transfer between
different modes in the grating.
With increasing electrode thickness, the spurious
resonance caused by SBH mode is expected to become
weaker. It is also expected to decrease with increasing
rotation angle of LT orientation. Fig.5 shows calculated
velocity dispersion of LSAW and SBH mode in two
orientations, 36YX and 42YX. With thick electrodes,
h/=0.1, the stopbands of LSAW and SBH mode overlap.
With increasing rotation angle, LSAW/BAW interaction
becomes weaker. If rotation angle exceeds 44, SBH mode
does not occur.
Other examples of the SBH were also found . In
128YX cut of LN, the SBH mode is a result of interaction
between a pure SAW and slow shear bulk wave. Since the
latter becomes exceptional in 131YX cut , it is strongly
radiated in 128YX cut. In this case, SBH is pure surface
mode, having zero propagation loss. In 36Y, X+90 cut of
Fig.2 Harmonic admittance calculated for LSAW in 36YX
LT, with short-circuited Al grating, h/=0.04, w/p=0.5, as
function of normalized wavelength s=p/ for. Normalized
frequency f’=fp/Vbulk changes from 0.49700 to 0.49711.
0.475 0.485 0.495
Fig.3. Velocity dispersion of LSAW (a) in 36YX cut of LT,
h/=0.04, with enlarged fragment (b) showing SBH mode which
appears in close proximity to the upper edge of the stop band.
Dashed lines refer to backward propagating modes.
Normalized frequency fp/Vbulk
quart, SBH mode is a result of interaction between STW
and fast shear bulk wave (Fig.6). In this example, the lower
edge of SBH stopband nearly coincides with the upper
edge of STW stopband and the effective coupling of SBH
mode changes sign at the same frequency, dividing it into
the forward and backward propagating modes and
indicating that there is a transfer of energy between two
modes, STW and SBH, with frequency. More detailed
analysis of this and other examples, including analytical
approach, could give a further insight of the nature of SBH
mode and its role in the transformation of the wave
structure from SAW to LSAW (or from LSAW to high-
velocity LSAW) when the propagation velocity crosses
that of the bulk wave.
An example of SBH having HVPSAW nature, which
arises from quasi-longitudinal bulk wave, has been found
in LBO cut with Euler angles (45,46,90) (Fig.7).
All types of SBH modes considered above have some
typical common features. The fundamental feature, which
distinguishes SBH mode from other known quasi-bulk
waves, is that it exists only in a grating and in limited
frequency interval of SAW/BAW interaction. At low
frequencies, the velocity of SBH mode approaches to that
of the bulk wave. The bulk wave, which gives rise to SBH
mode, must be close to exceptional one. Therefore,
analysis of exceptional wave lines  can help predict the
Fig.6. Velocity dispersion and Re(K2) in 36˚Y,X+90˚ cut of
quartz, including STW and SBH mode, h/Λ=0.04, w/p=0.9.
Fig.4. Velocity dispersion and effective coupling coefficient
near the upper edge of LSAW stopband in 36YX cut of LT, with
Al electrodes of different thickness. SBH velocity and backward
LSAW velocity are shown with solid and dashed lines,
Fig. 5. Velocity dispersion of LSAW propagating in two cuts of
LT with Al grating, h/=0.1, w/p=0.5 (a).
Normalized frequency fp/Vbulk
Normalized frequency fp/Vbulk
Normalized frequency fp/Vbulk
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existence of SBH mode and, if necessary, change crystal
orientation to reduce spurious resonance caused by this
Rigorous numerical analysis of “PSAW-like” mode,
recently found to exist in 36YX cut of LT with
sufficiently thick periodic structure, was performed and
revealed that this SAW/BAW hybrid mode is a result of
trapping of surface skimming BAW by counter-
propagating LSAW. Different examples of the SBH were
found and investigated. The typical features of the SBH
behavior can be generalized as follows.
1) SBH occurs in certain crystal orientations providing
that, in the absence of a grating, one of surface skimming
bulk waves is nearly exceptional. With deviation from this
selected orientation, the SBH mode gradually disappears.
2) The SBH mode can be pure SAW, LSAW or high
velocity LSAW, dependent on whether the generating
BAW is slow quasi-shear, fast quasi-shear or quasi-
3) In contrast to other quasi-bulk waves, the SBH mode
exists only in grating structures, within limited frequency
range, outside which it decomposes into the pure bulk
wave and counter-propagating SAW or LSAW.
4) Due to Bragg reflection condition fulfilled for
SAW/BAW interaction, a stopband occurs for the SBH
mode. The lower edge of this stopband can be regarded as
the cut-off frequency for bulk wave radiation and manifests
itself by a resonance of harmonic admittance.
5) Finally, the existence of the SBH mode can play an
important role in transformation of the wave structure from
pure SAW to LSAW and from LSAW to high-velocity
LSAW, with increasing frequency. Analysis of energy
transfer between SAW (or LSAW) and SBH mode could
reveal the mechanism of such transformation.
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Fig.7. Real part of velocity versus normalized frequency,
calculated for LBO cut with Euler angles (45, 46, 90), including
HVPSAW and SBH modes; h/Λ=0.04,w/p=0.9.
Normalized frequency fp/Vbulk