MODE CONVERSION COEFFICIENTS OF ACOUSTIC WAVES AT A CRYSTAL-VACUUM INTERFACE

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MODE CONVERSION COEFFICIENTS OF

ACOUSTIC WAVES AT A CRYSTAL-VACUUM

INTERFACE

J. Throwe, W. Bron

To cite this version:

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JOURNAL D E P H Y S I Q U E

CoZZoque C 6 , supple'ment au n o 12, Tome 4 2 , ddcernbre 1981 page c6-849

MODE CONVERSION COEFFICIENTS OF ACOUSTIC WAVES

AT

A CRYSTAL-VACUUM

INTERFACE

J. Throwe and W.E. Bron

Indiana University, Bloomington, IN 47405, U.S.A.

Abstract.-Recent work by ~aborek' @ has shown that the reflection of

acoustic phonons from a sapphire-vacuum interface is divided between a specular component which is predicted by elastic theory and a diffuse

component. Knowledge of the magnitude and direction of the specular

component is useful in the design of experiments to study the frequency

dependence of the reflection process. We have generated mode conversion

coefficients for infinite plane waves traveling initially in the x direction and incident at various angles on perfect surfaces in quartz,

sapphire, and ZnO. For waves propagating in the xy-plane, ZnO behaves as

an isotropic solid whereas quartz, because of its anisotropy, exhibits new transverse reflections beyond the critical angle.

Coherent collimated phonons pulses can be generated via the piezoelectric

interaction of an FIR laser beam with a crystal surface. As a first approximation

the reflection of such a pulse from the opposite surface after traversing the crystal can be treated as the reflection of an infinite plane elastic wave from a perfectly flat crystal-vacuum interface. The conditions governing such a reflec-

tion have been treated extensively in the literature2 r 3 r4 and can be summarized as

follows :

1. The component of the incident wave vector parallel to the surface is

conserved.

2. The group velocities of the reflected waves must point into the

crystal, although their wave vectors need not do so.

3. The vacuum exerts no stresses on the crystal.

In general, the energy of the incident wave is divided among reflections

from each of the three acoustic branches. Beyond certain critical angles of

incidence, however, it is no longer possible to satisfy the first two conditions

for all three branches. It is then necessary to introduce evanescent modes in

order to maintain the stress free boundary condition. These modes carry no energy away from the surface, leaving the incident energy to be divided between the remaining bulk wave reflections.

Figures la, 2a and 3a show the energy reflection coefficients in ZnO.

sapphire and quartz for T2 waves traveling in the x direction incident on planes

containing the z-axis. Because it has hexagonal symmetry ZnO is elastically

isotropic with respect to rotation about the z-axis4 (Figure!b). In the xy-plane

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JOURNAL DE PHYSIQUE

Figure

1.

a) Energy reflection caefficients for T2 phonons traveling in the

x direction in ZnO. b) Slowness surface for ZnO in the xy-plane showing

possible reflected wave vectors (solid arrows) for a T2 wave (dashed arrow)

incident at ZOO to the surface normal. The

TI

reflection has been omitted.

surface

/

\ A

Figure 2. a) Energy reflection coefficients for T2 phonons traveling in the

x direction in sapphire. b) Slowness surface for sapphire in the xy-plane

showing possible reflected wave vectors (solid arrows) for a T2 wave (dashed arrow) incident at 40° to the surface normal. There is no possible L reflection.

the Tl mode is polarized perpendicular to the plane of incidence and thus is not

coupled to the other modes by reflection. Accordingly, the energy of the incident

T2 wave is therefore divided between T2 and L reflections. Beyond the T S L

critical angle of 28O the only bulk wave available for reflection is the T2 mode,

i.e., with an energy reflection coefficient of 1.

In sapphire the transverse modes are polarized at oblique angles to the

plane of incidence. The incident energy is therefore shared by all three

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