surface acoustic waves and saw materials
TRANSCRIPT
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8/9/2019 Surface Acoustic Waves and SAW Materials
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8/9/2019 Surface Acoustic Waves and SAW Materials
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5 8 2
PROCEEDINGS
OF THE IEEE, MAY 1976
1
auk
lineartrain-mechanical dis-
ski =-
-x, :
placement elations (2 1
aD,
axi
_ -
-0
derived from Maxwell 's equations
acp underheuasi-staticssumption3)
E,. =--
ax i
q = p '
s
- '
vkllniiEn
lineariezoelectricon- (4 )
D~ = e;kl skl+ ,$ En
stitutive elations
where
T
is the stress,
p
the mass densi ty, u he m echanical dis-
placement,
S
thestrain,
D
the electricdisplacement, E the
electric field, and cp the electric potentia l. The primed quanti-
t ies, tha t
is
the elastic constants
(c j jk l ) ,
the piezoelectric con-
stants
(e ;k ) ,
and the dielectric constants
( ~ k ) ,
efer to a ro-
ta ted oordinate ystemhroughheEulerransformation
mat r ix [6 ] inwhich wave propa gation will alwaysbealong
the 1 di rec t ion . No te tha t the summ at ion convent ion (over
1,
2, 3) for repeated indices is employed.
By subst i tut ion, (1) through (4) can be reduced o
Ckkll(k,li
+
e i i iP ,k i
=
pu i ,
j
=
1, , 3
( 5 )
e:kluk,li- Ei'kV,ki = 0.
(6 )
Thedotnotation refers to differentia t ionwi th espect to
t ime, while an ndexprecededbyacommadenotesdiffer-
entia t ion with respect to a space coordinate .
Equa t ions (1) through (6) a re , of course, valid only within
the crystalline substrate, i .e., fo r x3 > 0 as defined in Fig. 3.
This figure also defines the geometry under considerat ion and
illustrates the meaning of
o = 0,
and
wh =
corresponding
to a horted urface nd ree urface, respectively. Fo r
- h Q x3
4
3
-
o L i N b O ,
I
B i , , G c 0 2 0
:
:
500
lo00 1500 2000
3
u)
F R E Q U E N C Y ( M H + )
Fig. 11. SAW attenuation due t o air loading as a function of frequency
results
for LiNbO, and Bi,, GeO,, .
or materials listed in Fig. 10. It is interesting to note nearly identical
so-called air loading can be eliminated by vacuum encapsula-
tion or minimized by the use of a light
gas.
Propagation ossescanbedeterminedbydirectlyprobing
the acoust icenergywitha aser
[
261.In hismethod, he
surface wave deflectsasmall raction of the ncident ight,
which is detected with a photomult iplier ube and measured
with a lock-in amplifier. The deflected light is directly propor-
t iona l to the acoustic power of th e surface wave.
Air
loadingcanbe determinedbyplacingdelay lines ina
vacuum system and reducing th e pressure below 1 torr while
mon itoring the change n nsert ion loss. Vacuu m attenuation
is, of course,. the difference between th e total propa gation loss
in
air and the air loading component .
Frequency dependence of vacuum attenu ation and air load-
ing for three of the m ost popular SAW substrates [261, [291,
[
301 are illustrated in Figs. 10 and
11 .
Note the approximate
f 2 dependence of the former and the linear dependence of the
latter.This allows n mpirical xpression forpropagation
loss to be derived from the data .
Propagation
Loss
(dB//&) = (VAC) F 2 (AIR )
F
(26 )
where F is in GHz. The coefficients VAC and AIR are tabu-
lated orpopularsubstratesat heend of thispaper.Equa-
t ion
( 2 6 )
would be used , for example, when designing filters
having particular bandpass characteristics.
DIFFRACTION ND
BEAM
STEERING
Parabolic Diffraction Theory
Diffract ion of surface waves is aphysicalconsequence of
their propagation and can vary considerably depending upon
the anisotropy of the substrate chosen. In fact , i t is the slope
of the power flow angle which determin es the extent of bo t h
diffract ion nd beam teering [3 1] . There is annherent
tradeoff between these two impo rtant sources f loss.
A useful heory for calculating diffraction fields when the
velocity anisotropy near pure-mode axes can be approxim ated
by a parabola has been developed by Cohen [3 2 ]. By using a
small angle appro ximatio n, he show ed hat for certain cases,
the higher orders of t he expressio n for the velocity could be
neglected past the second order . That is,
where y = &$/a6 and
Bo
is the angular orientat ion of the pure-
mode axis. By comparing hesespproximations to an exact
solut ion orelectromagneticdiffract ion nuniaxially aniso-
tropicmedia,Cohenshowed that he diffraction ntegral re-
duces to Fresnel's integral with the follow ing change
z^ =z^ll
+ y l .
28)
Szabo and Slobodnik
[
3 1 ] in t roduced the absolute magni tude
signs to account for hose materia ls having
y