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SINGULAR EFFECTS IN PHONON FOCUSING ; FORMATION OF PHONON FOCUSINGCAUSTICS
J. Hensel, R. Dynes, F. Unterwald, A. Simons
To cite this version:
J. Hensel, R. Dynes, F. Unterwald, A. Simons. SINGULAR EFFECTS IN PHONON FOCUSING ;
FORMATION OF PHONON FOCUSINGCAUSTICS. Journal de Physique Colloques, 1981, 42 (C6),
pp.C6-212-C6-214. �10.1051/jphyscol:1981661�. �jpa-00221597�
JOURNAL DE PHYSIQUE
CoZZoque C6, suppZe'ment a u n012, Tome 42, de'cernbre 1981 page C6-212
SINGULAR EFFECTS IN PHONON FOCUSING ; FORMATION OF PHONON FOCUSING CAUSTICS
J . C . Hensel, R.C. Dynes, F.C. Unterwald and A.L. Simons Be22 L a b o r a t o r i e s , Murray H i Z Z , New J e r s e y 07974, U.S.A.
A b s t r a c t - We have conducted a comprehensive i n v e s t i g a t i o n i n t o s i n g u l a r e f f e c t s i n phonon focusing. Measured focusing angular d i s t r i b u t i o n s and networks of c a u s t i c s a r e i n e x c e l l e n t agreement w i t h theory.
Phonon focusing is t h e tendency of t h e f l u x of phonons emanating from a l o c a l - i z e d s o u r c e t o c o n c e n t r a t e , owing t o e l a s t i c a n i s o t r o p y , along c e r t a i n d i r e c t i o n s i n a c r y s t a l . This phenomenon h a s been recognized f o r some time.' Only r e c e n t l y , how- e v e r , we found2 t h a t t h e r e a r e c e r t a i n c a s e s where t h e phonon i n t e n s i t y becomes extremely i n t e n s e , almost s i n g u l a r i n n a t u r e . This paper g i v e s a b r i e f account of t h e s e i n v e s t i g a t i o n s .
The o r i g i n of focusing can be v i s u a l i z e d by r e f e r r i n g t o t h e slowness s u r f a c e
+ +
-fw(q) = w i ~ ( ~ ) and q being, r e s p e c t i v e l y , t h e frequency and wavevector of t h e phonon) shown i n Fig. 1. The h i g h l y convoluted shape i s r e s p o n s i b l e f o r a p l e t h o r a of focusing s t r u c t u r e . The d i r e c t i o n a l i t y of t h e phonon f l u x i s d e f i n e d by t h e
+ +
group v e l o c i t y v = V w(q), a v e c t o r normal t o t h e s u r f a c e . This f a c t i m p l i e s t h a t
g q
f o c u s i n g can be i d e n t i f i e d w i t h a c e r t a i n geo-- metry p r o p e r t y of t h e s u r f a c e , namely t h e Gaussian c u r v a t u r e K, t h e product
K ~0 2
Kt h e
~two p r i n c i p a l c u r v a t u r e s of t h e s u r f a c e . The r e l a t i o n s h i p i s A-' = q K/cos6, where 2 A i s t h e
f o c u s i n g enhancement f a c t o r and 6 is t h e a n g l e
+ -+
between q and v A becomes l a r g e where curva- g'
t u r e i s small. Indeed, when K
=0, a s i n d i c a - t e d i n Fig. 1 by heavy s o l i d curves c a l l e d c r i t i c a l l i n e s , t h e r e occur s i n g u l a r i t i e s i n f o c u s i n g .
The experiments c o n s i s t e d of angular F i g . 1. T2 slowness s u r f a c e . s c a n s w i t h t h e geometries shown i n F i g . 2. A
s m a l l b a l l i s t i c phonon s o u r c e
(%50 u m diam. l a s e r s p o t ) and s m a l l d e t e c t o r (80 x 80 vm2 bolometer) y i e l d e d an angular r e s o l u t i o n of
%25 minutes of a r c .
An angular d i s t r i b u t i o n of phonon i n t e n s i t y i n t h e (1i0) p l a n e of Ge, obtained w i t h t h e s e t u p i n F i g . 2a, i s shown i n Fig. 3 . I t s most prominent f e a t u r e s a r e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981661
s t r o n g , s h a r p l y d e f i n e d beams of focused phonons cen- t e r e d on t h e [1007, [1111, and [1101 a x e s . T h i s drama-
c4BOLO.
t i c s t r u c t u r e i s t y p i c a l of t r a n s v e r s e phonons; f o r
-METAL FILM
l o n g i t u d i n a l phonons whose slowness s u r f a c e i s more n e a r l y s p h e r i c a l t h e behavior i s r e l a t i v e l y g e n t l e ( s e e , e.g., Fig. 4 ) . These beans m a n i f e s t a f i n e s t r u c t u r e
( b l
s e e n a s s p i k e s , which d e r i v e from c r i t i c a l p o i n t s
METAL FILM
, (where K
=9) on t h e slowness s u r f a c e .
A c a l c u l a t i o n of t h e f o c u s i n g i n t h e (170) p l a n e of Ge is g i v e n i n F i g . 4. (The u n i t s s c a l e d i r e c t l y w i t h t h e d a t a i n F i g . 3.) E f f e c t s of d e t e c t o r geometry
L 2 c r n - J
a r e included i n t h e c a l c u l a t i o n ; t h e s o l i d c u r v e s a r e f o r t h e p r e s e n t geometry, whereas t h e d o t t e d c u r v e s Fig. 2. Experimental r e p r e s e n t a l a r g e r bolometer used i n e a r l i e r work. 2
geometry.
One s e e s e x c e l l e n t agreement between c a l c u l a t i o n and d a t a .
F u r t h e r i n s i g h t i n t o t h e n a t u r e of f o c u s i n g i s provided by t h e c a u s t i c s , 2- dimensional p l o t s of t h e l o c i of t h e f o c u s i n g peaks. A remarkable example i s t h e (100) p a t t e r n i n F i g . 5 o b t a i n e d w i t h t h e scanning scheme3 i n F i g . 2b. One s e e s a n i n t r i c a t e p a t t e r n e x h i b i t i n g o v e r a l l t h e &-fold symmetry of t h e [loo] a x i s . Both p o s s i b l e t o p o l o g i c a l c l a s s e s of phonon c a u s t i c s a r e exemplified i n F i g . 5. The
*WW 8 IN DEGREES
F i g . 3. Angular d i s t r i b u t i o n s : experimental
Coo11 c1;a c1401
ANQE 8 IN DEGREES
F i g . 4. Angular d i s t r i b u t i o n s :
theory.
JOURNAL DE PHYSIQUE
PNGLE + I IN ODREE3
