Fast Numerical Technique for Simulation of SAW Dispersion in Periodic Gratings and Its Application to Some SAW Materials

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Fast Numerical Technique for Simulation of SAW Dispersion in
Periodic Gratings and Its Application to Some SAW Materials
Natalya Naumenko
Moscow Steel and Alloys Institute
Moscow, Russia
nnaumenko@ieee.org
Benjamin Abbott
TriQuint
Apopka, FL 32703, USA
bpabbott@tqs.com
Abstract —A method for approximation of harmonic admittance
of an infinite periodic grating, which takes into account
interaction between SAW (or LSAW) and BAW, is presented. A
numerical technique based on this method is non-iterative, fast
and accurate and enables extraction of dispersion parameters
from numerical admittance calculated at real-valued spectral
frequencies. Excellent agreement with admittance can be
obtained in a wide range of temporal and spectral frequencies,
including the interval where there is a strong interaction between
SAW and BAW. Application of the method to different types of
surface waves - Rayleigh-type SAW in LiNbO3, SH-type leaky
waves in LiTaO3 and high-velocity LH-type leaky wave in
Li2B4O7, is demonstrated. The extracted dispersion curves are in
a good agreement with the dispersion found by the search of
poles in complex spectral domain.
Keywords - SAW, leaky waves; periodic grating; dispersion;
COM parameters
I. INTRODUCTION
The Coupling-of-Modes (COM) model is an efficient tool
for simulation of SAW devices with resonator-type structures.
The accuracy of simulation depends crucially on COM-
parameters, which characterize the dispersion of SAW in the
analyzed material with a periodic metal grating.
For well behaved Rayleigh-type and SH-type (shear
horizontally polarized) SAW, the wave modes in the grating
satisfy simple analytical equations of ‘classic’ dispersion
employing the Ingebrigtsen’s approximation of surface
impedance [1], and two-parameter COM formalism [2],
respectively. In these two cases, all necessary COM
parameters can be determined if the edges of the Bragg
stopbands are located for open-circuit (OC) and short-circuit
(SC) gratings. In practice, SAW or leaky SAW (LSAW)
propagating in most of material/cut combinations widely used
for SAW devices, is neither Rayleigh-type nor SH-type wave.
For example, leaky waves (LSAW) propagating in rotated
Y-cuts of LiNbO3 or LiTaO3, most widely used in resonator-
type filters, have quasi-SH polarization, which changes with
geometrical parameters of the grating. In wide frequency
interval, the dispersion of these waves varies between two
cases mentioned above. In addition, LSAW propagates with
nonzero loss, which is frequency dependent.
To account for interaction between SAW and bulk waves
(BAW) in these orientations, two-parameter COM model can
be utilized. Many efforts were made to fit this model for leaky
waves in 36°YX and 42°YX cuts of LiTaO3 (for example,
[3,4]). The classic and two-parameter COM models were
combined in the five-parameter approximation suggested by
Abbott and Hashimoto [5]. The main disadvantage of such
approach is that it is too complicated for accurate fitting with
numerical data.
In [6] we suggested an improved method for rational
approximation of the harmonic admittance function, based on
two-parameter COM formalism with frequency-dependent
parameters. Fitting of this approximation with numerical
admittance enables easy and fast calculation of SAW or
LSAW dispersion and can be especially useful when SAW
mode has general type of polarization. In this paper, we
describe the new technique and demonstrate the results of its
application to SAW and leaky waves with different
polarization, propagating in some SAW substrates.
II. DERIVATION OF RATIONAL APPROXIMATION OF
ADMITTANCE
The behavior of surface waves in the periodic grating of
metal electrodes depends on geometry of the grating and the
structure of surface and bulk waves, which is generally
complicated. The concept of harmonic admittance [1,7] is of
great utility for characterization of wave propagation under the
imfinite periodic metal grating. Similar to effective dielectric
permittivity function [8], which characterizes propagation of
surface waves in a semi-infinite substrate, harmonic
admittance is independent on the applied voltage and contains
information about the wave characteristics in the analyzed
substrate. It is a function of temporal and spectral frequencies,
and can be determined as a ratio of the dispersion relations in
the OC and SC gratings,
SC
OC
D
D
jCsfY
0
=
),(
(1)
In classic COM model, which describes interaction between
two counter-propagating modes, at any fixed frequency the
dispersion is assumed to be a linear function of detuning
parameter ?s=0.5-p/?, where p is a period of the grating and ?
is a wavelength. Then, with symmetry of wave propagation
taken into account,
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2
R
2
2
A
2
0
ss
ss
jCsfY
∆−∆
∆−∆
=
),(
(2)
where
in spectral domain, at OC and SC conditions, respectively,
2
A s
∆
and
2
R s
∆
are the roots of the dispersion equations
))(()(
,,,,2211
2
ARARARAR
ffffCfs
−−=∆
(3)
Thus,
and the roots of equation (3) determine the edges of the Bragg
stopbands in OC and SC gratings, (
respectively. Such simple approximation is valid only when
there is no interaction with other waves propagating in the
same substrate. Moreover, even if this approximation is valid,
there always exists an effect of uniform mass load, which is
frequency dependent and affects dispersion in wide interval of
frequencies. Any perturbation of dispersion makes the
frequency dependence (3) non-parabolic. In addition,
interaction with the bulk waves results in branch points, which
are not predicted by the approximation (2).
The effect of interaction between SAW and BAW can be
adequately characterized if two-parameter COM formalism [2]
is applied to dispersion relations in (1),
2
A s
∆
and
2
R s
∆
are parabolic functions of frequency,
)
21AA
ff ,
and ()
21RR
ff ,
,
2
22
A
2
AA OC
sfsfD
ε
ε
η−
η−
η
η−
−∆
∆
+
+
∆
∆
⋅−∆
∆
−
−
∆
∆
=
=
))(())((
(4a)
(4b)
22
RRRSC
sfsfD
−⋅
))(())((
where
normalized frequency
After transformation, we can write the admittance (1) as
5.05.0/ )f(f
−=∆
;
5.05.0/ )s(s
fp
−
.
=
/
∆
, and f means the
BAW
Vf
=

)(
)(
),(
fXX
fXX
jCsfY
RBRB
ABAB
∆−+−
∆−+−
=∆∆
γη
γη
0
(5)
where
order polynomials, with respect to parameter
2
22
/ )(
ηεγ
−=
. Equation (5) is a ratio of two second-
() sfsfXB
∆−∆+∆+∆=
2
1
, (6a)
which can be transformed to
22
sffXB
∆−∆+∆=
(6b)
Then two roots of each polynomial, which characterize the
dispersion in OC or SC gratings, are

CffX
∆−∆±=2
21
η
,
(8)
where

4/ )2(
η −
22
ε=∆Cf
(9)
determines the cut-off frequency of BAW radiation. In terms
of parameters ? and ?, the edges of the stopband are

2
21
/ )(
η ±
,
ε=∆f
(10)
In practical orientations, it is difficult to fit the edges of the
stopband and the cut-off frequency
parameters ? and ? [6]. On the other hand, two-parameter
COM model is able to predict the effects caused by
SAW/BAW interaction. Assuming that the roots of the
dispersion equations
0
=
OC
D
dependent and different from (8), we suggested to use more
general approximation of admittance,
Cf
∆
using the same
and
0
=
SC
D
are frequency-
))((
))((
),(
RBRB
ABAB
XXXX
XXXX
jCsfY
21
21
0
−−
−−
=∆∆
(11)
for fitting it with numerically found harmonic admittance, and
extraction of the frequency-dependent roots of the dispersion
equations. The accuracy of such approximation can be
improved if higher-order polynomials are used in (11),
⋅=
)(
)(
),(
B
n
R
B
n
A
XP
XP
jCsfY
0
(12)
The complicated parameter of approximation, XB(6), is a
function of temporal and spectral frequencies and has branch
points at ?s=±?f, which characterize BAW radiation cut-off at
arbitrary frequency. Without detuning from synchronous
resonance condition, ?s=0, the branch point occurs at ?f=0.
At any branch point, the admittance changes from pure
imaginary to complex value, with real part responsible for
propagation loss. Therefore, approximation (11)-(12), based
on two-parameter COM formalism with frequency dependent
parameters, is able to account for BAW radiation, with
variation of temporal or spectral frequency.
III. NUMERICAL TECHNIQUE
The form of rational approximation (12) lends itself to
linear solution for any specified order of polynomials. Thus, it
is compatible with non-iterative solution. The fitting is
performed exclusively at real-valued spectral frequencies. The
propagation loss results in complex-valued dispersion
parameters extracted from complex harmonic admittance
function.
We found that approximation (11), with second-order
polynomials, enables excellent fit over a broad spectral
frequency range, for most of SAW substrates. To provide an
accurate fit to the harmonic admittance, it is necessary to
bound the range of spectral frequencies where poles for SAW
or LSAW exist. However, when there is a strong SAW/BAW
(or LSAW/BAW) interaction, and the type of the surface wave
differs from SH-wave, approximation obtained in the narrow
interval of spectral frequencies around s=0.5 is preferred to the
approximation in wide spectral domain region. This refers
only to the interval of frequencies close to the BAW cut-off.
Fig.1 illustrates the accuracy of approximation of harmonic
admittance Y(s) at the normalized frequency f=0.497, which
nearly coincides with the BAW cut-off frequency fc, in 42°YX
LT with Al electrode thickness h/p=0.1 and metallization ratio
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a/p=0.5 (this example is also considered in the next section, in
Fig.4 and Fig.5 ). Approximation obtained in the interval of
spectral frequencies between s=0.5 and s=f (Fig. 1a),
provides excellent agreement with numerical admittance and
reveals the existence of a resonance at s=0.4985 or V=4210.3
m/s, though at s>0.52 there is some disagreement with
numerical data. On the contrary, approximation in a wide
range of s (Fig. 1b) is able to characterize correctly the
behavior of admittance at large detuning from synchronism,
but ‘ignores’ the details of its behavior caused by strong
LSAW/BAW interaction around s=0.5. With higher-order
polynomials in (12) (for example, n=3) perfect approximation
can be achieved in wide intervals of s andf.
In the next section, some numerical examples are discussed.
In these examples, the fitting was made with the second-order
polynomial approximation (11). The roots of dispersion
equations obtained in such a way can be associated with two
modes in the grating and enable accurate location of one or
two poles of harmonic admittance, at any frequency.
IV. EXAMPLES OF DISPERSION
a) LSAW in 36°YX cut of LiTaO3 with OC grating. This
example (Fig.2) is most close to SH-wave, because in 36°YX
LT the fast shear bulk wave is nearly pure SH-polarized. In the
OC-grating with thin electrodes (h/p=0.02), LSAW is a
slightly perturbed BAW, and the extracted dispersion of
LSAW looks consistent with two-parameter COM equation
(4a). In this example, the second branch of the dispersion
equation (blue lines) is not required for approximation of
admittance at arbitrary f and s, because it can be predicted
quite accurately using parameters ? and ? determined from the
edges of the Bragg stopbands in OC and SC gratings.
The dispersion extracted from harmonic admittance in the
real-valued spectral domain (red lines) coincides with the
function obtained by numerical search in the complex s-plane
(black lines), though small disagreement occurs at f>0.5,
with constantly growing loss caused by BAW radiation.
Admittance Y(s)
0,4950,505
Spectral frequency s
0,515
Admittance Y(s)
3500
3700
3900
4100
0,465 0,475
Normalized frequency fp/VBAW
Velocity, m/s
0
10
20
30
Fig.3. Velocity dispersion of Rayleigh-type SAW in 128°YX LN
with Al grating (SC), h/p=0.08, a/p=0.5.
Im(V)
Re(V)
slow shear BAW
Fig.1. Numerical (solid lines) and approximated (dashed lines)
admittance Y(s), calculated at the normalized frequency f=0.497,
in 42°YX LT with Al grating, h/p=0.1, a/p=0.5:
a) approximation at s=0.497-0.5,
b) approximation in a wide interval of spectral frequencies. At
the branch points, s=fand s=1-f, the velocity crosses VBAW.
a)
Im(Y)
s=f s=1-f
b)
3900
4100
4300
0,4900,495
Normalized frequency fp/VBAW
0,5000,505
Velocity, m/s
0
10
20
30
Fig.2. Velocity dispersion of quasi-SH type LSAW in 36°YX LT
with Al grating (OC), h/p=0.02, a/p=0.5. Red and blue lines refer to
the solutions of the dispersion equation obtained by rational
approximation. Black lines show dispersion obtained by search in the
complex spectral domain region.
fast shear BAW (SH)
Im(V)
Re(V)
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b) SAW in 128°YX cut of LiNbO3. The next example
(Fig.3) demonstrates that the method is also able to determine
dispersion of Rayleigh-type SAW propagating in this
orientation. Here, the slow shear BAW is SH-polarized and
gives rise to the second SAW branch, which is nearly
uncoupled. The second solution of the dispersion equation,
found due to application of rational approximation (11), refers
to SH-type SAW. Both branches are found simultaneously, in
one non-iterative procedure, while the common method of
finding solutions in the complex spectral frequency domain
would require laborious separate analysis of each mode
propagating in the grating. The second mode does not exactly
coincide with SH-SAW, because of the effect of uniform mass
loading included in the dispersion of this mode.
c) LSAW in 42°YX cut of LiTaO3. In this orientation
(Fig.4), the fast shear BAW includes vertical component of
polarization. Metal electrodes cause additional perturbation of
the wave structure. Though LSAW is still more close to SH-
type than to Rayleigh-type wave, its behavior in the grating is
not adequately characterized by eqn. (4). Therefore, it is
impossible to predict accurately the behavior of admittance
around the branch points using only one mode of the
dispersion equation. Additional mode is essential in this case,
because only the combination of two modes is able to
characterize the effect of LSAW/BAW interaction. This mode
determines additional pole of the dispersion relation at the
frequencies close to
Surface-Bulk Hybrid mode (SBH) in the grating, which arises
from BAW because of its trapping by the counter-propagating
LSAW [9]. With deviation from the frequency interval of
strong LSAW/BAW interaction, the second (blue) mode takes
into account two additional effects: frequency dependent
uniform mass loading of the surface and deviation of the
dispersion from that described by equations (4). In practice, it
is very difficult to separate these effects, and rational
approximation (12) allows determining the combined
influence of few factors on the harmonic admittance. In spite
of some ambiguity of physical interpretation of the second
mode of the dispersion equation, extracted from harmonic
admittance, such approach provides surprisingly high accuracy
of approximation of admittance and enables fast and accurate
calculation of COM parameters necessary for design
simulation.
Fig.6 shows the pole contribution to the harmonic
admittance Yp, which characterizes the coupling of SAW or
LSAW mode in the grating [7] and can be found as
c f =0.497 and can be associated with the
()
()
R s
=
s
SC
ROC
∆
p
sfsD
fsD
jCfY
∂∂
∆
,
=
/
,
)(
(13)
The factor
For any frequency, excluding the interval around fc, the effect
of the second mode gives a constant slowly varying with
frequency. Therefore, one pole (solid lines) characterizes
LSAW mode propagating in the grating, and
(
π
)
RsCjC
ω
sin
0
=
was omitted in calculations.

()
1A OC
XXD
−=
;
()
1RSC
XXD
−=
(14)
In the interval of normalized frequencies between 0.496 and
0.501, the second mode builds physically meaningful solution
– SBH mode, which results in additional pole of harmonic
admittance. If this pole is treated separately from the first one,
its amplitude looks as the dashed lines in Fig.5. Such
approach, which is good for analysis of independent modes,
propagating in the grating, should be modified for the case
when two modes interact and even form additional Bragg
stopband.
Approximation (11) based on two-parameter COM
formalism assumes that each mode is a combination of a
surface and bulk waves. Therefore, commonly used numerical
-0,1
0,0
0,1
0,460,480,50
Normalized frequency fp/VBAW
Amplitude
3000
3500
4000
4500
0,460,48 0,50
Normalized frequency fp/VBAW
Velocity, m/s
0
50
100
150
Fig.4. Dispersion of SH-type LSAW in 42°YX LT with Al grating
(SC), h/p=0.1, a/p=0.5.
fast shear BAW (quasi-SH)
Im(V)
Re(V)
Fig.5. Amplitude of a pole calculated using approximation (11)
with extracted dispersion parameters. Additional pole (dashed lines
on the enlarged fragment) appears at the frequencies where there is a
strong LSAW/BAW interaction.
Re(Yp)
Im(Yp)
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procedure of extraction of SAW pole contribution and
estimation of residual effect of BAW radiation [7] needs
revision. Otherwise, the high accuracy of extracted dispersion
parameters can be degraded in further numerical analysis.
d) Quasi-longitudinal LSAW in LBO. The last example
considers high-velocity LSAW of LH-type. The existence of
low-attenuated quasi-longitudinal leaky waves was predicted
in some orientations of LBO based on exceptional wave
theory [10]. In particular, this theory requires that any
exceptional bulk wave, which can give a rise to low-attenuated
LSAW, must be horizontally polarized. The behavior of
longitudinal horizontally polarized (LH) waves in the grating
is close to that of SH-waves, though LH-waves usually exhibit
higher propagation loss.
Fig.6 shows the dispersion of LH-type LSAW in LBO,
Euler angles (45°,46°,90°), obtained with the method
presented. In this orientation, the propagation loss is nearly
zero near the upper stopband edge, but it grows fast to lower
frequencies. This results in additional asymmetry of
dispersion, compared to that predicted by two-parameter COM
equation. The complicated structure of the wave and large
propagation loss is responsible for some disagreement with
dispersion obtained by analysis in the complex spectral
domain. However, in general, the interaction between LH-type
LSAW and BAW is fairly accurately described by the applied
approximation.
V. CONCLUSIONS
The rational approximation of harmonic admittance, which
was developed for numerical analysis of dispersion,
• can be applied to different types of surface waves;
• includes the effect of the branch points produced by the
neighboring BAW;
• provides fast and high quality fitting over a reasonably
broad spectral range;
• allows non-iterative fit procedure;
• provides high quality fit using real-valued spectral
frequencies;
• is able to reveal the details of SAW/BAW interaction.
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[9] N. F. Naumenko, B. P. Abbott, and S. Malocha, “Effect of
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[10] N. F. Naumenko, “Longitudinal horizontally polarized
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5000
6000
7000
0,44 0,460,480,50
Normalized frequency fp/VBAW
Velocity, m/s
0
500
1000
Fig.6. Velocity dispersion of LH-type LSAW in LBO, Euler
angles (45°,46°,90°), with Al grating (SC), h/p=0.08, a/p=0.9.
longitudinal BAW (LH)
Im(V)
Re(V)
2007 IEEE Ultrasonics Symposium170