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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