Download Surface excitation of hypersound in piezoelectric crystals by

Z. Physik B 21, 1 - 10 (1975)
© by Springer-Verlag 1975
Surface Excitation of Hypersound in Piezoelectric Crystals
by Plane Electromagnetic Waves*
O. Weis
Institut fiir Angewandte Physik, Universit~it Heidelberg
Received January 30, accepted February 14, 1975
Abstract. The generation of hypersound at a free surface of a piezoelectric crystal by means
of an incident plane electromagnetic wave is considered and the corresponding boundary
problem is discussed in detail. The formula developed in this paper are quite general and
can be applied to any piezoelectric crystal and any face orientation. As an important
example, the excitation of sound waves at several quartz faces is treated numerically and
the results are presented in diagrams showing directly the power conversion from the
plane incident electromagnetic wave into the sound waves as function of the angle of
incidence and of polarization directions.
1. Introduction
Langevin [-1] first exploited the electromechanical
coupling due to the piezoeffect in order to generate
and detect ultrasound electrically. He used a X-cut
quartz plate with flat electrodes at both sides as a
sound transmitter and receiver (Fig. 1). If the electrodes
are connected to a if-voltage, the alternating electric
displacement excites the quartz to forced mechanical
thickness vibrations which will become especially large
in the odd harmonics of mechanical thickness resonances. On the other hand a mechanical vibration in
these odd harmonics produces an electric displacement
of the same frequency across the crystal and hence
gives rise to a signal voltage between the electrodes
which can be used for the detection of sound. Since
quartz transducers can generate and detect sound
waves, they are reversible electromechanical transducers. They are a standard tool in ultrasonic research.
Experiments with sound waves in the hypersound
region, i.e. at frequencies above 1 GHz, became feasible
due to the work of Baranskii [2] and B6mmel and
Dransfeld [3]. Baranski found that extremely high
mechanical harmonics can be excited at room temperature by a cw-method even at about 1 GHz in quartz
* Supported by the Deutsche Forschungsgemeinschaft.
plates of 12 mm thickness. He used Bragg reflection
of light as a method to detect the generated sound
waves. He put the crystal in the concentrated electric
displacement field of a re-entrant type of coaxial
resonator in order to get a high electric displacement
at the position of the quartz plate. At hypersound
frequencies sound absorption becomes important, increasing at room temperature proportional to the
square of frequency. B6mmel and Dransfeld suppressed
this interaction with thermal phonons by cooling the
quartz crystal to temperatures of liquid helium. Moreover, the cw-excitation was replaced by short if-pulses
of about 1 gs duration producing hypersound pulses
of the same duration. Quartz crystals in the shape
of rods were used plunging only with one end into
the concentrated dielectric field of the re-entrant cavity
(Fig. 1). The excited sound pulses propagate down the
rod and can be detected in a similar re-entrant cavity
arranged at the other end of the quartz rod, or it
can be detected after reflection in the same cavity
where they were excited some microseconds before.
Since only a small fraction of sound energy is transferred to electromagnetic energy, the sound pulses are
reflected many times, and these reflections produce the
wellknown echo pattern of this pulse-echo method.
2
Z. Physik B 21 (1975)
THICKNESS
RESONANCE~
QE QUARTZPLATES
MECHANICAL ELECTRICAL
STRUCTURE: STRUCTURE:
TRANSMITFERA~
REC~ R
OF ELECTRIC
RF-PULSES
QU
RESONATOR
BROADBAND
REC~ERTRANSMr[TER~
SURFACE EXCITATION___
OF QUARTZRODS
IN A RE-ENTRAN~
OF ELECTRIC
CAVITY
RF-PULSES
~
SURFACE EXCITATION
OF QUARTZUSING A
FOCUSSED IR-LABER
LASER BEAM
BROADBAND
RESONATOR
BROADBAND
SUPERCONDUCTING
BOLOMETER
AS DETECTOR
BROADBAND
1
HYPERSOUND
Fig. 1. Types of experimental arrangements for conversion of
electromagnetic power into sound waves by means of piezoelectric
coupling at a quartz surface
As Jacobsen [4] pointed out, the sources of sound
are the gradients in the piezoelectric stresses. Since
at the free surface of a piezocrystal the piezoelectric
constants and consequently the piezoelectric stresses
make a jump from zero to a finite value, these surfaces
are the most effective sources for sound waves in
piezocrystals.
In the d~vice of Langevin, sound waves are excited
simultaneously at both crystal faces leading to constructive interference in odd harmonics and to destructive interference in even harmonics.
In the arrangement of B6mmel-Dransfeld sound is
generated only at one end-surface of the quartz rod.
Since, for example, at a frequency of 10GHz the
wavelength of longitudinal sound in X-cut quartz is
about 6000,~ and since the surface excitation takes
place with equal phase over the whole end-surface of
the quartz rod (diameter 2 ... 3 mm, typically) a high
collimated beam of hypersound respectively of coherent phonons is emitted having a beam divergence equal
to that of a 6 328 A HeNe laser. In pulse-echo experiments the detection is coherent, since, for an optimal
detection signal, incident sound waves must have a
wave front parallel within a part of wavelength to
the detecting surface in order to get no phase cancellation in the mean value of the produced electric field.
Parallelism of wave front and crystal surface is destroyed by material inhomogenities and insufficient
parallel and flat end faces of the crystal rod. These
requirements limit the usefulness of coherent detection
to frequencies below 100 GHz. In fact, hitherto pulse
echo experiments have been performed only up to
l l 4 G H z (Ilukor and Jacobsen [5]). At these frequencies a re-entrant cavity becomes too small and
a rectangular microwave cavity made from a metal
coated quartz quader had been used. All these experiments were done with quartz crystals.
Experiments with coherent phonon beams at still
higher frequencies require a new technique (Ilukor
and Jacobsen [6]). First of all, the coherent detection
must be replaced by incoherent phonon detectors like
superconducting tunnel junctions (Eisenmenger and
Dayem [7]) or superconducting bolometers (v. Gutfeld
and Nethercot [8]). Secondly, the required electromagnetic pulse power can best be produced by means
of a far infrared gas laser, and thirdly, the concentration
of the electric energy at the crystal surface must be
done in a different way.
Recent experiments by Grill and Weis [9, 10] have
shown that coherent phonons can be generated by
surface excitation at 0.891 THz and even at 2.53 THz
using the chopped radiation of far infrared lasers. The
laser radiation was focussed onto the surface of a
quartz crystal by means of a spherical mirror (Fig. 1).
In this technique no electromagnetic resonator is used
at all. The disadvantage of this arrangement consists
in a lower efficiency in transforming electromagnetic
power into hypersound in comparison with a resonance structure. On the other hand a true broadband
transformation is achieved, not limited by an electromagnetic resonance. An advantage too is the possibility
of moving the excitation spot over the crystal surface,
which allows new experiments exploiting the strong
collimation of the generated phonon beam. Moreover,
by changing the angle of incidence and direction of
polarization of the exciting laser beam, coherent phonons with prescribed polarization can be produced.
In order to get an insight in these new possibilities,
the surface excitation ofhypersound due to an incident
plane electromagnetic wave is considered. After discussing the boundary-value problem, numerical results
will be presented.
2. Boundary-Value Problem
Given is a semi-infinite piezoelectric crystal. The surface may be oriented arbitrarily relative to the Cartesian crystal coordinates xl =X, x 2 = Y and x3 = Z in
which usually the components of material tensors are
specified. We choose a new Cartesian vector basis ~,
~ , J~ and furthermore spherical coordinates 0 and q5
according to Fig. 2. The plane of incidence of an incoming electromagnetic wave is fixed by the angle ~b,
the angle of incidence in this plane by 0. An arbitrary
polarization can be splitted in a linear polarization ~ll
parallel to the plane of incidence and a linear polarization ~± perpendicular. We define as polarization
vector the direction of the electric displacement
vector/}.
O. Weis: Surface Excitation of Hypersound
3
INCIDI
Po' div/4 = 0
(2 c)
div/) = 0,
(2 d)
ELECT
WAVE
1, enT2
GOUNB
have to be solved simultaneously together with the
stress equation of motion
Uiv ~ ' - p - ~ = 0 ,
(3)
where
F(f, t)= ~(f, t)
PLANE OF
INCIDENCE
VACUUM
(MEDIUM 0 )
PIEZOELECTRIC
CRYSTAL (MEDIUM I )
Fig. 2. Schematic representation of waves generated by an incident
plane electromagnetic wave at the surface of a piezoelectric crystal
illustrating the definition of several quantities used in the text
(4)
denotes the particle velocity and g the particle displacement vector• The strain tensor can be expressed
by
t) =
+
(5)
within the linear elastic theory.
If a linear polarized plane wave falls onto the surface
of a pure dielectric crystal, one reflected and two
transmitted linear polarized electromagnetic waves
are generated (an inhomogeneous wave is not possible
in the crystal under these circumstances, since the
electromagnetic wavelength in the crystal is always
smaller than in vacuum).
If the half space is a piezoelectric crystal, a coupling
exists between electrical and mechanical fields. Assuming adiabatic conditions the material relations
can be written in linear approximation [11]
We are interested in solutions which represent plane
monochromatic waves, for which the electric displacement has the form
b,(f, t)=Q~- b~. e- i(~"~-'°'° + c.c.
(6)
and the particle velocity
#,(f, t)=#4,. ~,. e -i(G'~-~'t) +c.c..
(7)
~_= 4 ~ : ~ _ _ b . 3f/
(la)
Eqs. (6) and (7) show that the electromagnetic polarization is given by #~,, the mechanical polarization
by #~. Since the fields are coupled, both wave vectors
are equal in the infinite piezoelectric medium:
/~= 3 ~ : ~ + ~ / " / ) / ( o
(lb)
Q=
where the electric displacement b (f, t) and the strain
tensor ~(f, t) are taken as independent field variables
and the stress tensor J(f, t) and the electric field/~(f, t)
as dependent variables. 4~--45D is the tensor of elastic
constants evaluated at constant electric displacement,
3f/ the appropriate piezoelectric tensor and ~/-=~ the
dielectric impermeability tensor evaluated at constant
strain. In order to set up the boundary equations we
need plane wave solutions for a piezoelectric crystal.
The propagation of plane waves in a lossless piezoelectric crystal was first investigated by Kyame [12].
In order to get the basic formula for these waves we
have to discuss this problem in short• This will be done
in a slightly different notation and as far as possible
an invariant formulation will be preferred:
Due to the coupling of electrical and mechanical
fields in a piezoelectric crystal, Maxwell's equations
(assuming no magnetic effects, no electric currents
and charges)
rot/~ = - Po"/~
(2 a)
rot H = D
(2 b)
-
o4c
G),
(8)
where we have introduced the wave normal Q and
corresponding phase velocity G(Q) for the polarization "a".
Introducing the expression (7) in Eqs. (4) and (5) leads
to
Q (f, t) = Q.. (Q/i co)- e- ~ ; -
°'') + c.c.
(9)
and
t)= - 1/(2. c.G)). { G .
• bo. e- i(~"'7-°° + c.c..
(10)
On the other hand expression (6) together with (10)
introduced into the material Eqs. (1 a) and (lb) give
the whole stress field as well as the electric field of the
plane wave:
~-.(~,t)= --W:~nQ.. C / c . G ) + ~ . .
(11)
• e-i(lG'F-a~t)-t-C.C.
• e-i(~.~-~.t)+
~h.b.}
c.c.
(12)
4
Z. Physik B 21 (1975)
Eq. (2d) states that electromagnetic polarization is
pure transverse:
~.
G=o.
(13)
Eq. (2 c) shows that the magnetic field/t¢ if, t) is pure
transverse, too. With this knowledge the Maxwell
equation (2b) yields for the magnetic field/~¢(F, t) of
the plane wave the expression
t:t,(f,t)=c¢(Y,).(GxT~,).D~.e-"~°'~-°~'°+c.c.
(14)
c,2 (G) and hence five piezoelectric plane waves exist
for each special wave front normal ~,.
~
If the piezoelectric tensor 3f~ is negligible, V and W
vanish and we get a reduction of (17) into two separate
eigenvalue equations of order three resp. two, which
are wellknown from wave propagation in a pure
dielectric crystal.
The eigenvalue equation of third order
{ F ( G ) - p" c2} • ~7, = 0
determines the three elastic waves ~ = L, T1, T2 with
polarization vectors OL,1,~rlLOrs and phase velocities % >cr~ >Cr2. The phase velocities can only be
computed numerically as roots of the secular determinant and in the next step the polarization vectors
d~ can be determined from (20).
The eigenvalue equation of second order
Finally, the remaining two Eqs. (3) and (2a) can be
written by using the expression (9), (11), (12) and (13):
{G ~e:~,~-p. c~(~,). ~ } . ,~°
+~.
aft. ~,./5~. G(Y,)= 0,
(15)
(.,13.olco
{16)
where we have introduced the velocity of light in
vacuum co. From Eqs. (15) and (16) the polarization
vectors Y~,~and ~ , , the phase velocities G(e,) as well
as the corresponding ratio of Fourier amplitudes
/)~/~ can be calculated• For this purpose a Cartesian
vector basis ea,
. . . . .e2,
. . . .e3 is defined (Fig. 3) with e3 = d,.
If we take the components of ~¢ and ~ in this vector
basis, we get the following homogeneous system of
five equations (Y~¢. f, = ~ , . ~ = 0 is always valid !):
//qi - p . c~ //12
. . . . G
F~.2. . . ~ 2 - P .c~2
~3"
3
. .
dc.
. V~dc¢
. .
\v;
.1/C¢ . . V22/Co;
. .
~t
2
2
vldco
~1,-c°/~o
V;3/Ca
~h2
....
,~
= [irk'/]
+"
= [ e i .e...:.
\
(Christoffel tensor),
(18)
and
W ( ~ , ) - ~ o . V(d,).
(19)
The determinant of the coefficients of the five independent homogeneous Eqs. (17) is of order five in
e-;~
--
~
I e2
nIle3
(
2
~22--C¢/C0
2
]
~,,)
,
\e~
.~
=0
(21)
2
tt
#!
tt
tt
2
,2
(G/Co) = (qll + tl22)/2 +_{(,H - tl22)/4 +,~ 2 }
1/2
(22)
where q~k
" = e~
~" • ~?
~ • ek
~" are the tensor components in the
Cartesian vector basis of Fig. 3. The angle 7~ and y. of
]
G" (eg,. el)
.....
G" (%~" e2)
\
l =°.
(17)
~ Do/~o.(O~,.O;) l
,/~o " (e~, " e2
.... 7
the corresponding polarization vectors are now determined from (21) leading to
tg G = (/'~22- - hi'l)/(2 • rU2)
_~_
V(d,)=-3h.e~.=[hik3]
,,
,1,2
,,
describes two pure electromagnetic waves a = I , II
with polarization YI'1"ei~ ± e , = e~. The secular determinant yields the two phase velocities G - - % qi:
~122- c,/c
4C. .:.e.n. e k ]
22
,
#712
w - ;col
Here we have introduced for convenience the tensors:
/~(G)=G • ,g " G = [ F ~ ] ="[ ~ , , . / ~ . e k ]
,,
Wll. Ca
W~I. c,
. . . .
X /
12" c,
W~2 "G
G3--P "C~
G3
\v;
(20)
tt
,
2
.
{(,11
-r/22)/(2
t/i'2)z + 1} 1/2.
(23)
If piezoelectric coupling exists, i.e. aft+0, the sound
wave is accompanied by a small electromagnetic
energy flow and vize versa the electromagnetic wave
possesses a mechanical part too. This gives rise to a
change in phase velocities and from (17) the relative
change can easily be estimated giving
ZICq/Cq~--(:0" h2/( 2 . cq2 - rl " p)
(mechanical wave type),
eI
A cdca~ ~o" hZ/( 2" Ck2 • rl" p)
Fig. 3. Double-primed vector base for defining the polarization
vector ~
(electromagnetic wave type),
~Jp
(24)
O. Weis: Surface Excitation of Hypersound
5
where h is a typical piezoelectric constant, tl the mean
impermeability and Cq, Ck the phase velocities computed from (20), resp. (22). Taking values from Table 1
the relative change in quartz is less than one percent
for the sound waves and absolutely negligible for the
electromagnetic waves.
Table 1. Material constants in quartz used in calculating the power
conversion for surface excitation of hypersound. These components
are given in the Cartesian crystal coordinates x~ = X, x 2 = Y, x 3 = Z
and are chosen in accordance with the IRE 1949 S t a n d a r d s [ l l ]
for right-handed quartz
Elastic constants
(Ref. 15)
c~ = 87.49 • 109 N / m 2
q2 =
6.244.109 N / m 2
q3=
11.91 . 1 0 9 N / m 2
q 4 ~ - 18.09 • 109 N / m 2
c3~= 107.2 , 1 0 9 N / m 2
c¢4= 57.98 • 109 N / m 2
Piezoelectric constants
(Ref. 15)
d~ = -2,31 • 10 -12 C/N
d ~ = - 0 , 7 2 7 . 1 0 -t2 C/N
respectively:
h H = - 4 . 4 4 .109N/C
hi4=
1.055.109 N / C
Dielectric constants
(Ref. 16)
e~~ = 4.436 (o) = 0)
e3a =4.639 (~o=0)
respectively:
~h~ =0.2254
Mass density
q33 =0.2156
p = 2 650 kg/rn 3
The ratio of Fourier amplitudes ~//5~ can be estimated
in the same way and yields
f) ~/v~ ~ ~o" h/Ol" ca)
(mechanical wave type),
~/b~ ~ h/(p . Ck)
(electromagnetic wave type).
(25t
electromagnetic, pure mechanical and mixed term
in (27):
Sel-mag : Smech : Smixed
[
1
: eo" h2" cg/(P "rl" c~): e0 "h2/(P "~l"c~)
I
[(electromagnetic wave types)
-[Eo.h2/(p.q.c2):~
:e 0 h2/(p.tl.c 2)
1
I
((mechanical wave types).
(28)
For sound waves in quartz the electromagnetic and
mixed part is in the order of 1 ~o of the whole flux,
whereas for the electromagnetic wave types the mechanical part as well as the mixed part are absolutely
negligible.
Knowing all types of plane waves and their field in a
piezoelectric crystal we next have to consider the
boundary problem outlined in Fig. 2. At the boundary
an incoming plane electromagnetic wave may excite
reflected electromagnetic waves and transmitted piezoelectric waves. We will denote the field quantities of
waves in vacuum by a superscript (0) and of piezoelectric waves by (1). All these generated waves are
determined by the electromagnetic and mechanical
boundary conditions:
For a free surface the mechanical boundary conditions
demand the vanishing of all mechanical forces for all
points at the surface and all times t, i.e.
0~ 2
~(1) /
t
~/
as
(xl, x2,
0, t).e3
or in components
0 = Y~ ~'~. a~ ~(x;, xl, 0, t). ~;
i = 1, 2, 3,
If we cast the exact ratio following from solving (17)
in the form
respectively:
~ = a~./5~
0 = ~ ~s~'~~~,..
, ~a.~• o,~
~"~ ~~'"~ ~ '1 , x l , o, t)/c~ (~,~)
resp. /5~ = b e - b~,
(26)
ff
the real coefficients as are small for a plane wave of
electromagnetic type and the b~ are small for sound
waves in comparison to the other modes as follows
from (25).
From the time independent Poynting vector
+ De (Xx, x2, 0, t). h: ei e3 }
i = 1, 2, 3,
(29 a)
where we have used the relation (11).
As electromagnetic boundary condition we require
the continuity of the two tangential components of/~
and/~ at the boundary x; = 0:
Co. F, {P2 × b~(~i, xl, 0, t)}- ~',
= ( ~ , x ~,) x ~. ~ . c~(~,). 2.1~5,1~/~o
ff
i=1,2,
+ ( ( ~ x ~,) x 3~:~,~ + ~o. ~ . ~ }
•
{/5"- ~+/5~-~*}
(29 b)
(27)
one can estimate by means of (25) the relative contributions to the whole energy flux due to the pure
O-
O"
+~l'b~)(x't,x'z,O,t)} "el,
i=1,2.
(29 c)
6
Z. Physik B 21 (1975)
The seven scalar boundary equations (29) require that
all waves involved have the same frequency and the
same spatial dependence at the boundary.
If the dielectric displacement of the incident plane
electromagnetic wave (rr = " 0 " ) is given by
b~o°) (f, t) = e~o"/30' e- i. (~,o.;. ko-,O.0 + C.C.
(30 a)
the phase matching at the surface is satisfied, if each
of the generated waves fulfills the following law of
reflection resp. refraction:
Y.~. gJc~(~,~)= ~,o" F,/Co
i= 1, 2.
(31)
This condition sorts out from all imaginable plane
waves only seven permissible outgoing waves having
all the wave normal in the plane of incidence:
Two mirror-like reflected waves may be generated
with polarization parallel or perpendicular to the
plane of incidence
/)~o) (r, t) = ~1./5~. e- i"~a"'"~"ko-,O-,)+ C.C.
(30 b)
O(2O)(f, t) = eg2" Oz' e-i.(~,~.;. ~o- o,.,) + c.c.
(30 c)
together with two transmitted waves of electromagnetic
type
/~1)(~, t ) = ~ 1 '/)I'
e-i'(g"l'f~°/cI(~"')-"t)+C,C.
D~I)( f, t)=egll" Oil" e-i'(enn'7"~°/cn(enI1)--°~'t)q-C'C
(30d)
(30e)
and three piezoelectric waves. Since sound velocities
differ by a factor 10 -5 from light velocity, Eq. (31) tells
us that the wave normals of the excited sound waves
are always perpendicular to the surface independent
of the direction of the incident electromagnetic wave.
Hence the particle velocities of the piezoelectric sound
waves G = L , T1, T 2 have the form
~(Ll) (F, t) = e~L" ~L" e- i. ~(~ ' ~ / ~ 1 - 0 + C.C.
(30 f)
~(t!(~,t)
T1
= e~
~Tl
(30g)
^
" DTI
.e-~'°~'(~'~/~r~(~)-~)+c.c.
~)2 (r, t)= egr 2 • VT2" e-i'~°'(F34/cr2(~'~)-t)+c.c.
(30h)
Introducing the plane wave expressions (30) in the
boundary conditions (29) and taking into account the
relations (26) as well as G~=g.ra = e , r2 = -g~, we end
with the following system of seven inhomogeneous
equations (32)
F, {a~. ~'~G:~e:o,o O~dc~(O,~)+~. 3h:e, G}" bdbo
¢=I, II
+
X
¢=L, T1.T2
- F, (~,,ox~).~',.bo/bo+
{-~',G:~e:~;GIc~(~;)+b~.~.%~;~;I.~dbo=o,
i=1, 2,3,
(32a)
i= 1, 2,
(32b)
Y, (Gx~a~).~',.c~(~,,~)/Co.G/bo
~=1,2
~=l, II
-
y"
b~. (F3 x ~ ) . g~. G(F3)/c 0 • ~dbo =(~,0 x e~o)" Fi,
a=L, T1,T2
- E g~o" ~',"b~//)o+ E {~" ~¢+ao- ~o" 3~:y.o Yq~/G(~.~)}' F~. Do/D 0
G=1,2
G=I,II
+
E
{_ q. 3~:~; e~o/co(~;)+ b~. q. G } O',.~o/bo =~o. ~'~,
i= 1, 2,
(32 c)
a=L, T1,T2
which determine, if the amplitude/) o of the incident
electromagnetic wave is known, the amplitudes /5~,
/5 z of the reflected pure electromagnetic waves, the
amplitudes f)~, /5~i of both piezoelectric waves of
electromagnetic type, and finally, the three amplitudes
OL, br~ and ~r2 of the generated sound waves.
In the next step the other field amplitudes may be
determined for each piezoelectric sound wave.
Subsequently the ratio of energy fluxes, i.e. the power
conversion from the incident electromagnetic wave
--(en2 X eg2)' el
-(G~ x ~a)' ~i
- ~ 1 " Oi
- ~ . ~i
--~2' gl
-- e~2" e2
into a sound wave of special polarization, can be
calculated from relation (27).
This procedure is simplified, if all terms of (32) which
have a coefficient a~ or b~ can be regarded negligible
small and if the piezoelectric contribution in the last
term of (32 c) can be neglected (in quartz this term is
about 1 ~/o compared to the others). In this special
case, the system (32) splits up in a system of four and
another system of three coupled equations. The system
of four linear equations
el • (enl x el~l)" cl(enl)/co
~'2 "(~,~x ~0 "c, GO/co
ei" ~" M
el (enIIXegll)'Ctl(enii)/Co
I I/~)2//T)°~
/
(33)
O. We\s: Surface Excitation of Hypersound
7
corresponds to a pure electromagnetic determination
of the resulting electric displacement /}i +/}H at the
boundary x; =0, a procedure wellknown from crystal
optics. The remaining three equations in (32) are due
to the mechanical boundary conditions (29 a) which
can be written with the electric displacement/}i+/9i~
in the form:
~' ~ • 6; =
0=ei"
{~:4d--(/~iq-])ii
) • 3~h } : e i e~'3 , ~'
i=1,2, 3.
(34)
Here the strain ~ is fully attributed to the generated
sound waves, i.e. we have the following system of three
coupled linear equations
were determined at room temperature by ultrasonic
methods. Since it is to be expected that material constants of continuum acoustics have to be modified
in quartz at frequencies beyond 10~ Hz, where phonon
dispersion effects may become noticeable [14], the
use of these material constants will give only a first
insight in the surface excitation of hypersound at these
very high frequencies• The special sets of material
constants used in numerical calculations of this chapter
are compiled in Table 1.
In the special case of quartz, the piezoelectric coupling
is very small and therefore it seems reasonable to
e~ e3 c:e3 Q/cL(Q) el e3: c:e3 erl/crl(e3)
e2 e3 : C:ea en/crl(e3)
\e3 e3. e3 eL/cL (e3) 6; 6; :¢e: Y; 6ra/Crl (6'3) ~; ~;.'e:6; ~2/c~(6;)/
\~dDo/
/ (Q~. b~/bo + Q n bn/bo) • 3~:6i Q~
= |(Q~ b~/bo + Qxi b./bo) • 3h:6 i e ; } .
\(&i b,/bo -~6gI1 b./bo) ~r,: ~; ~;j
•
Solving (35) yields the relative amplitudes ~,/b0.
These can be introduced in the following expression
for the power conversion
PdP0
=
I(N,(f, t)}'[/I <So(f, t)Yl
=E0" 16~'~"4& ~ Q ~ I " I~./b0 I2/(c. • Co)
=% . p . IQIbol z . ~V~dco,
(36)
where we have introduced the absolute value of the
group velocity ~,, of a sound wave with polarization
a and wave vector 0 according to
~
= 6g~. 4e: ~g~6,/(p. c~ (6,)).
(37)
The expression (36) follows directly from (27) under
the assumption already made that the piezoelectric
constants are small.
The formula derived in this chapter gives the general
solution of the boundary problem allowing only a
numerical evaluation. Of course, for special crystals,
special crystal orientations and special directions of
the exciting electromagnetic wave a further reduction
can be achieved.
3. Numerical Results for Hypersound Excitation
in Quartz
Material constants for quartz are available from literature 1-13]. The most accurate sets of elastic constants
(35)
calculate the propaganon properties for all five piezoelectric waves for any direction 6, in quartz not from
the exact equations (17) but from the simpler equations (20) and (21). Instead of equations (32) also the
simpler equations (33), (35) and (36) can be used without making a substantial error.
Using this approximation the power conversion from
the incident electromagnetic plane wave into hypersound waves was calculated numerically for the five
different crystal cuts appearing as faces at the two
quartz cubes shown in Fig. 4. The results of calculations are plotted in Fig. 5 to Fig. 9 for several planes
of incidence being specified by the angle ¢. The ratio
of the energy fluxes of generated hypersound to the
incoming electromagnetic wave is given as radius
vector for each direction of the incoming plane wave,
i.e. the radius vector gives the power conversion of
incident waves having wave normals in this direction
and a polarization either parallel (II) or perpendicular
(.1_) to the plane of incidence. The conversion is small,
about 10 .6 or less, and depends strongly on crystal
cut and angle of incidence.
Let us now discuss ha more detail the hypersound
excitation at an X-face [17] (Fig. 5). At normal incidence the fast transverse wave T1 and the slow
transverse wave T2 can only be excited. The generation
8
Z. Physik B 21 (1975)
"•Z•
Y-SiO
BC~32.2o"~
IZ
¢
=
0
T1
~
~
D
~
'
_
(257°1
~
i z 'gT21a'ooI
,iY
¢= 90 °
;¥
,,Z
Tlll
4
Fig. 6. Power conversion at the Y-face of a quartz crystal
_
~,IX
uX
Z-SiO
I 0:, 0o
T 2 ± ~
,,~'
L
~.Z
.0;0oo,oo
~=90 °
'
~
~
~
~
~
j,T2
1
~
* y qb=30 °
__
yi
gL I23'5°)
2
XL II
~=60 °
o
32.°2
4 ~~. _ ~ - - . -L ~ . - -
¢m : wL = 5.75 krn/s D
CT1 = WTl = 5.12 km/s
CT2= WT2=3.32 krn/s
,Y" X--7 I
wL = 6.57 km/s
WT1 = 4.79 krn/s
WT2 = 4.29 km/s
Fig. 4. Definition of some crystal faces in quartz and the polarization
of sound waves having a wave normal perpendicular to these faces
X- Si02
2
(':(I0~271~uX
l
w L = 6.36 km/s
w T = 4.90 km/s
~TIH
T
2
~
gT (17"3°)I t '
2
TM
TII
Fig. 5. Power conversion at the X-face of a quartz crystal. Phase
velocity g.(el) and group velocity w.(el) coincide for each of these
three generated sound waves
of the longitudinal wave L demands a /5-field component parallel to the X-axis which only exists at
oblique incidence and with a parallel electromagnetic
polarization. Whereas the excitation of the weak
longitudinal wave is independent of the chosen plane
of incidence, generation of transverse hypersound
waves occurs only if the polarization vector of the
incident electromagnetic wave possesses a c o m p o n e n t
in the XY-plane. The Poynting vectors SL, St1 and
St2 are parallel to the wave normal. As a consequence,
if we consider the excitation of hypersound over a
limited region of the crystal surface (for example, over
the area of the spot of a focussed laser beam), the
7,z
Y
X
S
Fig.?. Power conversion at the Z-face of a quartz crystal. The
direction of the Poynting vector of the generated transverse sound
waves is marked at the refraction cone as function of the direction
of the/}-component of the incoming electromagnetic wave in the
X Y-plane
excited sound propagates independent of polarization
exactly normal to the X-face in form of a collimated
beam.
The three sound waves excited at a Y-face (Fig. 6)
have all different directions of energy flux deviating
O. Weis: Surface Excitation of Hypersound
9
AC-SiO 2
wL = 7.02km/s
WT1 = 3.81 km/s
= 3.32kmls
WT2
(~ : 0 °
O-Oz- I
AC
IIZX
¢ :-*30°
Lx
,#c
1
~
i
[
{
/
X/~'~.T1"
--'2107 ~ - T 2 ~
* =-'t60°
IIAC
l
L~/~~,,
II AC
2
z2,,
qb: 90°
, u~-/~I~TI"L
l ''X
,['fT2*
IIAC L
~-T2,,
Fig. 8. Power conversion at the AC-face of a quartz crystal
X
BC-Si 02
w L = 6.36kmls
WT1 = 5A2 km/s
WT2 = 3.80 km/s
¢=0 °
Z
~
o-,~-'-
. -YY
T1
~
/ ~e~o~ST2[7"5°)
s7.8o"--.~1 SL~6~°I ,-
,L
u BC
2.10-7
~ = ±30°
/IBC
,
T,$
2.10-7
(
',.22/J
HZ
qb=z60 °
¢ = 90°
T1.L~
IIBC
2'10-7
k~T2 ''
I
{ T2.~"~T2~
~
.X
T
I
~
IIBC
2'10-7
Fig. 9. Power conversion at the BC-face of a quartz crystal
more than 23 ° from the Y-axis. The irradiation of a
Y-face leads to relative strong longitudinal sound
waves becoming strongest at normal incidence and
with an electromagnetic polarization parallel to the
X-axis.
With a Z-face (Fig. 7) no longitudinal waves can be
generated at all. The power of the pure transverse
sound waves is independent of an arbitrary rotation
around the threefold Z-axis. Transverse waves with
wave normals along the Z-axis are degenerate in the
long-wavelength limit ~ =0, they have the same phase
velocity c r independent of their special direction of
polarization in the XY-plane. But if one turns the
polarization continuously over an angle of 180° in
this plane, the corresponding Poynting vectors form
a whole circular cone with a half-angle of 17.3° around
the Z-axis. This "conical refraction" was studied in
quartz by means of ultrasound by McSkimin and
Bond [18] in detail. In Fig. 7 the direction of the
Poynting vector Sr is marked at the cone as function
of the sound polarization, respectively direction of
particle velocity ~r, which on the other hand is uniquely
determined by the c o m p o n e n t Dproj. of the whole
electric displacement vector at the Z-face. From this
figure the direction of the Poynting vector can be
taken if the polarization vector of the incident electromagnetic wave is given. The described degeneracy of
the two transverse sound waves is lifted at finite frequencies and one left- and one right-circularely polarized wave remain as was shown experimentally by
Pine [19]. This splitting becomes more and more
pronounced the higher the hypersound frequencies
are. It should be observable in pulse experiments at
very high frequencies.
Surface excitation of hypersound at a AC-face is
treated in Fig. 8, at a BC-face in Fig. 9. Both faces
are perpendicular to each other, their face normals
are parallel to the polarization vectors of the pure
transverse waves propagating along the X-axis. At
the A C and BC-face a pure transverse wave can be
excited with polarisation parallel to the X-direction.
These waves have the same phase velocity (and group
velocity) as have the two pure transverse waves with
wave normal along X-axis. These properties follow
directly from the invariant expression for the phase
velocities:
~. c2(~.)=~ J.:4a:~. ~
(38)
by interchanging the directions of wave normal ~,
and polarization 6~ which, as we already know, are
perpendicular to each other for e , = e l . Only these
two pure transverse waves excited at a A C and BCface have an energy flux perpendicular to the surface,
whereas for the other waves the Poynting vector
deviates some degree from the face normal.
4. Conclusion
This paper has demonstrated how power and polarization of hypersound, excited at the surface of a piezoelectric crystal by an incident electromagnetic wave,
10
is related to the crystal properties within continuum
acoustics. The detailed analysis revealed that, under
the assumption of undamped waves, the excitation
of hypersound by means of plane electromagnetic
waves is only possible at the surface and not in the
bulk of a piezoelectric monocrystal. As a consequence,
the shape of detected phonon pulses is not influenced
by the excitation process but is only a function of
the applied electromagnetic pulse, of the response of
the phonon detector, of dispersion and of interaction
processes occuring during propagation between the
spot of excitation and the position of the phonon
detector.
Numerical results for power conversion at five faces
of quartz show that power conversion is typically
- 6 5 dB or less.
We may conclude that this new technique of hypersound excitation is not very effective. But in practice
this disadvantage is not a serious limitation, if the
exciting electromagnetic wave is focussed onto the
crystal, since in that case a narrow, highly collimated
beam is emitted which takes the whole vibrational
energy over large distances to small detecting areas.
Moreover, detectors can be used which absorb the
incoming phonons completely. It should be noted that
the excitation is broadband mechanically as well as
electrically and that the spot of phonon emission can
be moved easily over the crystal face.
Z. Physik B 21 (1975)
2. Baranskii, K.N.: Kristallograpbiya 2, 299 (1957) and Doklady
Akad. Nauk S.S.S.R. 114, 517 (1957) (English Translation:
Soviet Physics-Crystallography 2, 296 (1957) and Soviet PhysicsDoklady 2, 237 (1957)
• 3. B~mmel, H.E., Dransfeld, K.: Phys. Rev. Letters 1, 234 (1958)
4. Jacobsen, E.H.: J. Acoust. Soc. Am. 32, 949 (1960)
5. Ilukor, J.O., Jacobsen, E.H.: Science 153, 1113 (1966)
6. Ilukor, J.O., Jacobsen, E.H.: In: Physical Acoustics (W.P.
Mason, ed.), Vol. V, p. 221. New York: Academic Press 1968
7. Eisenmenger, W., Dayem, A.H.: Phys. Rev. Letters 18, 125
(1967)
8. v. Gutfeld, R.J., Nethercot, A.H.: Phys. Rev. Letters 12, 641
(1964)
9. Grill, W., Weis, O.: Satellite Symposium of the 8th International
Congress on Acoustics on "Microwave Acoustics", Lancaster
1974, Editor: E.R. Dobbs and J.W. Wigmore, p. 179. Published
by Institute of Physics, London
10. Grill, W., Weis, O.: Phys. Rev. Letters (to be published)
11. IRE Standards on Piezoelectric Crystals. Proc. IRE 37, 1378
(1949)
12. Kyame, J.J.: J. Acoust. Soc. Am. 21, 159 (1949)
13. Hellwege, K.H.: Landolt-B~Srnstein, Group III, Vol. L BerlinHeidelberg-New York: Springer 1966
14. Elcombe, M.M.: Proc. Phys. Soc. 91, 947 (1967)
15. Bechmann, R.: Proc. Phys. Soc., London B64, 323 (1951)
16. Russell, E.E., Bell, E.E.: J. Opt. Soc. Am. 57, 341 (1967)
17. The power-conversion diagram given in Ref. 9 was calculated
with a different set of elastic, dielectric and piezoelectric constants, especially the piezoelectric constants h ~ was assumed to
be larger
18. McSkimin, H.J., Bond, W.L.: J. Acoust. Soc. Am. 39, 499 (1966)
19. Pine, A. S.: J. Acoust. Soc. Am, 49, 1026 (1969)
O. Weis
References
1. Langevin, P.: Proc~d6 et appareil d'emission et de r6ception des
ondes elastiques sous-marines ~t l'aide des propri6t6s pi6zo61ectriques du quartz. French Patent Nr. 505 703 (1918)
Institut fiir Angewandte Physik der Universit/it Heidelberg
D-6900 Heidelberg 1
Albert- ~berle-Stral3e 3-5
Federal Republic of Germany