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