- Email: [email protected]
Surface plasmon enhanced Brillouin scattering from metal films
S o l i d S t a t e Communications, Vol.36, p p . 9 9 5 - I 0 0 0 . Pergamon P r e s s L t d . 1980. P r i n t e d i n G r e a t B r i t a i n .
Sb~FACE PLASHDN ENHANCED 8RILLOUIH SCATTERING FROH HETAL FILHS H. Fukui and O. Toda Department o f E l e c t r o n i c E n g i n e e r i n g U n i v e r s i t y o f Tokushima Tokusblma, J a p a n 770 V . C . ¥ . So and Department University Toronto, Canada
G . I . Stegeman* of Physics of Toronto Ontario HSS 1AT
( R e c e i v e d 26 June 1980 by A. A. Haradudln)
The theory o f B r t l l o u i n s c a t t e r i n g i n a p r i s m - m e t a l f i l e , - a i r (Kretschmenn) geometry i s a n a l y z e d and n u m e r i c a l e s t i m a t e s a r e g i v e n f o r a LaSF9 p r l s m - A g f i l m - e l f c o n f i g u r a t i o n . S i g n a l enhancement o f s e v e r a l o r d e r s o f magnitude i s p r e d i c t e d under c o n d i t i o n s o f c o u p l l n g to s u r f a c e p l a s m e n - l l k e f i e l d s .
1.
a c o u s t i c boundary c o n d i t i o n s a t b o t h f i l m interfaces. The s c a t t e r i n g f o r m a l i s m i n c l u d e s the e l A a t o o p t l c e f f e c t i n b o t h media a s w e l l as the c o r r u g a t i o n e f f e c t at both boundaries. C o n s i d e r t h e geometry i n F i g . I . The "p"-polarlzed incident fields (appropriate for coupling to s u r f a c e plasmons) are w r i t t e n as
Introduction
S e v e r a l i n v e s t i g a t o r s have r e c e n t l y d e v e l o p e d new t h e o r i e s f o r g r i l l o u l n s c a t t e r l n g a t t h e s u r f a c e s of opaque L-5 and t r a n s p a r e n t 6-7 m e d i a . A c c o r d i n g to t h e s e , t h e r e a r e two s c a t t e r i n g mechanisms, the vell-knoun e l a s t o o p t l c e f f e c t , as w e l l a s t h e c o r r u g a t i o n e f f e c t due t o acoustlcally-created surface ripple. Although the surface corrugation effect is a relatively weak s c a t t e r i n g mechanism, i t can p l a y a domin a n t r o l e I ' 1 2 i n opaque medla c h a r a c t e r i z e d by l a r g e v a l u e s o f d i e l e c t r i c c o n s t a n t , f o r example, in metals. I t can be e n h a n c e d by i n c r e a s i n g t h e amplltude of the incident or scattered light f l e l d s , t h e a c o u s t i c d i s p l a c e m e n t normal t o t h e s u r f a c e , or a combination of b o t h . I t i s y e l l - k n o w n 13 t h a t s u r f a c e plasmen p o l a r l t o n s can be e x c i t e d a t t h e a l r - m e t a l i n t e r f a c e o f a p r l s m - m e t a l f i l m - a l r sample u s i n g attenuated-total-reflectlon (ATR) m e t h o d s . The electric fields associated wlth these p o l a r l t o n s I~ can be much l a r g e r t h a n t h e I n c l d e n t f i e l d s used t o e x c i t e them. These p r o p e r t i e s have a l r e a d y b e e n u t i l i z e d i n e n h a n c i n g n o n l i n15_16P e a r hen°mane i n v o l v i n g s u r f a c e pl~mmons. In t h i s p a p e r we show t h a t c o u p l i n g t o s u r f a c e - p l a s m o n - l l k e modes c a n a l s o l e a d to l a r g e enhancements of B r i l l o u i n s c a t t e r i n g i n m e t a l l i c media.
2.
k,
i [~t-klX+kz(Z-h) ]
.&
E i - E0(~ + ~ - ~)e
(I)
Z
f o r the l i g h t
Er "
Er(i
i n c i d e n t through the prism,
k. " ~-- i ) e
i[wt-k, x-k (z-h) ] z
(2)
Z
for light
reflected
_ , = Eml(~ ÷ i - -
back i n t o the p r i s m , end
i[~t-k.x]
+ az
i(~t-~xl
-
i)e (3)
i_~_ + v-=2(~ for the l i g h t 0, 2
=
oz
i)e i n t h e m e t a l f i l m where
k,,2-cm(~0)~2/c2
and
kz2 = rp(~j)~2/c2-kll 2.
( H e n c e f o r t h t h e s u b s c r i p t s and s u p e r s c r i p t s p and m r e f e r to t h e p r i s m end metal r e s p e c tively.) The e l e c t r i c f i e l d a m p l i t u d e s Er, E 1 and ¥m2' n o r m a l l z e d to E0 a r e d l s p l a y e d as a
Theoretical Analysis
f u n c t i o n o f t h e i n c i d e n t a n g l e ¢0 i n F i g . 2 f o r The a p p r o a c h used i n t h e p r e s e n t work i s identical to that reported previously in r e f e r e n c e s 2 and 7. Normal phonou modes a r e f o u n d which s a t i s f y ( I ) t h e a c o u s t i c wave e q u a t i o n i n b o t h t h e film end t h e p r i s m , end (2) t h e "Present Address:
t h e m a t e r i a l p a r a m e t e r s l i s t e d i n Table I . The l a r g e enhancement i n Em2 , the f i e l d which is a
maximum at t h e m e t a l - a l r i n t e r f a c e ,
corresponds t o t h e c r e a t i o n o f s u r f a c e p l u m o n - l l k e modes a t t h a t boundary f o r i n c i d e n t a n g l e s n e a r 8 . Hence P B r i l l o u l n s c a c t e r l n ~ from t h e r m a l phonons n e a r t h a t i n t e r f a c e (0c E ~ ) s h o u l d be e n h a n c e d by up
Optical Sciences Center University of Arizona Tucson, A r i z o n a 85721
U.S.A. 995
Vol. 36, No. 12
BRILLOUIN SCATTERING FROM METAL FILMS
996
to f o u r o r d e r s o f magnitude. The phonon modes which p a r t i c i p a t e i n the scattering process are those of a film deposited on a s e e L t - l n f i n i t e s u b s t r a t e , and have been t r e a t e d p r e v i o u s l y by Rowell. 7 T h e i r s t r u c t u r e depends upon the f r e q u e n c y regimo of i n t e r e s t and
-Z
AIR
C
r
L
the relative shear and longitudinal wave velocities in the prism and film. Here we shall restrict our attention to the "x-z" polarized bulk modes (and hence to "p"-polarlzed s c a t t e r e d l i g h t )
characterized
METAL
by t h e f r e q u e n c y (~) range
~LP>fl>~Lm w i t h flLp = qlvL p and f l j p r i s m and m e t a l l o n g i t u d i n a l d e f i n e d by VLP and v j
h
= qmvLm.
The
wave v e l o c i t i e s
are
respectively,
and qm i s
the (x-) component o f the phonon w a v e v e c t o r p a r a l l e l to the s u r f a c e s . (Other frequency regions will be considered in a subsequent, mere detailed publication.) In general, the phonon displacement field density in O* - R space, i.e. in a volume element do. dR centred at qs, R is of the form
+2
-
i
m i [ R t - q u x ] ~ ~ ( V e ) e i q ~ m ( v ' ) z + cc
u = N0
Fig. i
Schematic diagram for the a n a l y s i s .
in
e
~
v'=l
(4)
m
t h e m e t a l and
i =~u
P i[~t-%x] ~ 0 e
iq p ( v ' ) z
v'
~ ~(
)e
~
+co
(5)
v'=l p
16
in the prism.
Here u0 is the mode amplitude
d e n s i t y , the p a r a m e t e r s ~ ( v ' ) d e s c r i b e the mode p o l a r l z a t l o n and the summation v ' i s taken o v e r a l l s o l u t i o n s to the a c o u s t i c wave e q u a t l o n c h a r a c t e r i z e d by a g i v e n v a l u e of ql and ft. S i n c e R
ld
the p r i s m (fl>flTP=q v ? )
XsT
.
b
4PP G(2a)3vTp(fl2-Gpz ~ T"
(6)
whereKB, T andpp are Boltzmann's constant,
tu 16
Fig. 2
and
The i n c i d e n t
electric field amplitudes
the
t e m p e r a t u r e and the p r i s m d e n s i t y r e s p e c t i v e l y . The c o m p l e t e e x p r e s s i o n s f o r t h e p a r a m e t e r s i n e q u a t i o n s 4-10 a r e c o m p l i c a t e d , can be d e r i v e d from r e f e r e n c e 7, and w i l l be g i v e n i n d e t a i l in a subsequent paper. The s c a t t e r e d f i e l d s were e v a l u a t e d by c a l c u l a t i n g the p o l a r i z a t i o n s o u r c e s c r e a t e d by the e l a s t o o p t l c e f f e c t and s o l v i n g the p o l a r l z a t i o n - d r i v e n wave e q u a t i o n s u b j e c t to the e l e c t r o m a g n e t i c boundary c o n d i t i o n s a c r o s s the acouettcally-corrugated interfaces. As o u t l i n e d i n r e f e r e n c e s 2 and 7, a p a r t i c u l a r s o l u t i o n to the Inhomogeneous wave e q u a t i o n was found i n each medium and t h e t a n g e n t i a l f i e l d s
Er, Eml and Bs2, n o r m a l i z e d to E0 v e r s u s
Em'P(z=0,h) and Hm'P(z=0,h) were c a l c u l a t e d on
the i n c i d e n t
each s i d e o f t h e f i l m
a n g l e ~0"
x
y
boundaries
( a t z=O and
Vol.
36,
scattering (Fig. 1). For l i g h t t h e p r i s m a t t h e a n g l e ~s
Table I Ag
LaSF9
1.76 x 103 m/s
3.O4 x 103 m/s
vL
3.79 x 103 m/s
5.54 x 103 m/s
o
1.O64 x 104 Kgm/m 3
4.53 x 103 Kgm/m 3
~S
E
-16.34 - i0.144
3.46
Cll
1.527 x 1011 N/m 2
1.39 x 1011 N/m 2
c44
3.287 x iO IO N/m 2
4.18 x IO IO N/m 2
Pll
(Cm-l)/¢m 2 = Pll
O.24
P44
O
O.O15
T
300 °K
m
3.66 x 1015 r a d / s
(~=O.5145 ~m)
List of material param:ters the numerical calculations.
used i n
z-h). The B r i l l o u i n s c a t t e r e d f i e l d s c o r r e s p o n d to s o l u t i o n s o f t h e homogeneous wave e q u a t i o n with a m p l l t u d e s a d j u s t e d t o e n s u r e t h a t t h e t o t a l fields satisfy the usual electromagnetic boundary conditions across the rippled surfaces. ~ l e r e s u l t a n t f i e l d a m p l l t u d e due t o t h e e l a s t o o p t l c e f f e c t can be w r i t t e n as Ee " I & ~ A I [ E / ( h ) - E x P ( h ) ] + A2Exm(O) + A3[Hym(h)-HyP(h) ] + A4Hym(O) } where the p a r a m e t e r s A i d e s c r i b e
(7)
the coupling
b e t w e e n t h e homogeneous and inhomogeneous solutions at the boundaries, l.e. the scattering g e o m e t r y , and & i s a f i l m r e s o n a n c e t e r m . The s c a t t e r e d f i e l d due t o t h e s p a t i a l and t e m p o r a l m o d u l a t i o n o f t h e i n c i d e n t l i g h t f i e l d s by t h e acoustically-created surface ripple is Ec = 4-~AB(h,Cp)Sm(h) - B(0,1)~m(O) } Uom
(8)
where 5m(h) and ~m(0) a r e t h e normal s u r f a c e displacements at the film boundaries. The t e r m s B(z,c) deter~ne the strength of the corrugation mechanism, and a r e p r o p o r t i o n a l 2 , 6 - 7 t o t h e optical field amplitudes at the interfaces, to £-¢ where c - 1 a t z=O and ¢ - ¢ a t z - h , and t o m p v a r i o u s o t h e r o p t i c a l , a c o u s t i c a l and g e o m e t r i c factors. Due t o t h e l a r g e i n c i d e n t f i e l d e n h a n c e m : n t ( F i g . 2) a t t h e s u r f a c e p l a s : o n coupling angle, large enhancements in the Exm(O), Hym(O) and B(O,I) t e r ~
(9)
2)e
where c~ - kQS+ksin+0 , Es - Ee+Ec, u s = w±R,
incident light intensity
are also
expected. The d i r e c t i o n s c h o s e n f o r o b s e r v i n g t h e Brillouin spectrum correspond closely to back-
(I 0) as
l(ms)
ms2¢ I~sl
3.
Numerical Calculations
0.27
PI2
+k-~ z
out of
Introducing the kl a - ksin~ s and k zs , kcos~s concept of the frequency spectrum, 2,17 we finally obtain the Brillouln spectrum per unit s o l i d a n g l e (ARs)-Unit f r e q u e n c y (~s)-unit
4o0
Table I.
" ES(~
radiated
i[mst+ktSx-kzS(z-h)]
kms vT
h
997
BRILLOUIN SCATTERING FROM METAL FILMS
No. 12
(10)
and D i s c u s s i o n
N u m e r i c a l c a l c u l a t i o n s were - : d e f o r a s i l v e r f i l m d e p o s i t e d on a LaSF9 g l a s s p r i s m w i t h i n c i d e n t O.5145 ~m r a d i a t i o n . The v a l u e s of the material constants used are presented in Table I. Since the elastooptic constants of LaSF9 g l a s s a r e n o t known, t h o s e o f dense f l i n t
g l a s s t8 were used i n s t e a d : this is reasonable since the elastooptic constants of different glasses are of comparable value. Two s c a t t e r i n g g e o m e t r i e s were e x a m i n e d , b o t h of which l e d t o t h e same v a l u e o f ql and h e n c e i n v o l v e d s c a t t e r i n g from t h e sam: phonon fields. In the first, the directions of the i n c i d e n t and s c a t t e r e d l i g h t were c h o s e n to c o r r e s p o n d to c o u p l i n g to s u r f a c e plasmon modes In t h e s e c o n d , n e i t h e r o p t i c a l f i e l d c o u p l e s s t r o n g l y t o s u r f a c e plasmon : o d e s . The c a l c u l a t e d B r t l l o u t n s p e c t r u m f o r b o t h g e o m : t r l e s i s shown i n F i g . 3. When t h e i n c i d e n t and s c a t t e r e d f i e l d s a r e c o u p l e d to s u r f a c e p i s s : o n modes, an e n h a n c e m e n t o f two t o three orders of magnitude Is predicted. The calculated signal strength is several orders of m a g n i t u d e l a r g e r t h a n t h a t m e a s u r e d by S a n d e r c o c k 1! i n h i s e x p e r i m : n t s on B r i l l o u i n s c a t t e r i n g on r e f l e c t i o n from m : t a l s u r f a c e s and h e n c e s h o u l d b e r e a d i l y o b s e r v a b l e . The spectrum in the frequency region considered is b r o a d and r e l a t i v e l y f e a t u r e l e s s , a characteri s t i c t y p i c a l of B r i l l o u i n s c a t t e r i n g from m : t a l l t c media. 1-10 S t r u c t u r e a p p e a r s n e a r = quvLP(RL p) a t w h i c h f r e q u e n c y t h e Bragg condition for scattering of the incident r e f l e c t e d l i g h t by l o n g i t u d i n a l waves i n the prism is satisfied. The s p e c t r u m i s f u r t h e r c o m p l i c a t e d t h e r e by a change i n the d e n s i t y o f s t a t e s 2 , ? ( = l / q P) when the l o n g i t u d i n a l waves i n t h e p r i s m change from an e v a n e s c e n t t o a
propagating character, grillo~Ln scattering from R a y l e i g h and Sezava 19 waves s h o u l d a l s o be e n h a n c e d i n t h i s s u r f a c e plasmon g e o m : t r y , and t h i s a s p e c t w i l l be r e p o r t e d l a t e r . The r e g i o n o f t h e sample which d o m i n a t e s t h e s c a t t e r i n g was i n v e s t i g a t e d i n o r d e r t o v e r i f y t h e r o l e o f t h e s u r f a c e p l a s m : n s . The f i e l d s a s s o c i a t e d w i t h e a c h i n t e r f a c e were calculated,
i.e.
998
BRILLOUIN SCATTEEING FROM METAL FILMS
(p.--- (Po: Oo[:~z'] lo'
i
=3724"
~T ( 10" ~o/s~c) FIB. 3
The s c a t t e r i n g
cross-sectlon
l(Ws)/
lO&~sA~e as a [ u n c t l o n of f r e q u e n c y shift
~s-~O - ~ f o r t~ao s c a t t e r i n g
geometrles.
V o l . 36, No. 12
Vol. 36, No. 12
BKI.LLOUI.N SCATTERING FRON F~TAL F ILI4S
999
1o+
E~o~ lci
J
9u.t t.&-
O ILl n.-
la
J o~
tt5
ld (D+ =3724"
([)o=3O53"
9
1of
o~
1.35
.Q [ld' ~,=c] Pt.g. 4
The s c a t t e r e d
[ZT(h)l and shift
~T.
field
IzZ(o)l
contributions ver,,-
(a) $O,~m - Op,
(b) 4~0,* s It 0p.
frequency
BRILLOUIN SCATTERING FROM METAL FILMS
I000
ET(0) -~A2Exm(0)
~=RLP and
+ A4Hym(0)
_ !4B(O'i)~m(°)uoml
(ii)
which is dominated by scattering at and near the metal-air interface and ET(h) = ~Al[Exm(h)-ExP(h)] + A3[Hym(h)-HyP(h)] + 41B(h,~p)~m(h)u0 m}
(12)
which arises from scattering in the prism and metal, at and near the metal-prism boundary. As indicated in Fig. 4, the alr-metal interfaces dominate the scattering when surface plasmons are excited; otherwise the prism and the prism-metal interface region is principally responsible for the Brillouln spectrum. This result verifies that the excitation of surface plasmons is the cause for the large enhancements in the Brillouln signal. The dominant scattering mechanism, elastooptlc versus corrugation was also examined. With the exception of the frequency regions
f~Lm, t h e s u r f a c e r i p p l e
Vol. 36, No. 12 mechanism
dominates the scattering cross-section. This is in a g r e e m e n t w i t h p r e v i o u s c a l c u l a t i o n s I-5 f o r s e m l - l n f i n i t e m e t a l s c h a r a c t e r i z e d by o p t i c a l s k i n d e p t h s much l e s s t h a n a w a v e l e n g t h . In s,,m~=ry, we have shown t h a t B r i l l o u i n s c a t t e r i n g from m e t a l f i l m s i n a n ATR geometry i s e n h a n c e d when t h e o p t i c a l f i e l d s c o u p l e to surface plasmon-like excitations at the metal-alr boundary. The p r e d i c t e d s i g n a l l e v e l s a r e comparable to that of Brillouin scattering from bulk media and experiments to verify these calculations a r e in progress. Acknowledgements - We would l l k e t o t h a n k Dr. N. Rowell f o r u s e f u l d i s c u s s i o n s and a l s o acknowledge Mr. K. Muguruma f o r h i s h e l p i n t h e numerical calculations. One o f us ( G . I . S . ) thanks his colleagues at Stanford for their hospitality d u r i n g p a r t o f t h i s work. The financial support of the Canadian Natural Sciences a n d E n g i n e e r i n g R e s e a r c h Counc11 i s g r a t e f u l l y acknowledged.
References
IR. Loudon, P h y s . Rev. L e t t . 40, 581 ( 1 9 7 8 ) . 2N.L. Rowell and G . l . Stegemen, S o l i d S t a t e Comm. 2 6 , 809 ( 1 9 7 8 ) ; P h y s . Rev. BI8, 2598
3(1978).
K.R. Subbaswamy and A.A. N a r a d u d l n , Phys. Rev. B18, 4181 ( 1 9 7 8 ) . ~V. B o r t o l a n i , F. N i z z o l i and G. S a n c o r o , Phys. Rev. Lett. 4 1 , 39 ( 1 9 7 8 ) . 5A. Derv£sch a n d R. Loudon, J . Phys. C: S o l i d S t a t e Phys. 11, L291 ( 1 9 7 8 ) . 6G.I. Stegeman, J . Appl. P h y s . ~ , 5624 ( 1 9 7 8 ) . 7N.L. Rowell, 1978, Ph.D. T h e s i s , U n i v e r s i t y o f Toronto (unpublished). 8J.G. D i l and E.M. Brody, P h y s . Rev. B14, 5218 (1976). 9S. ~ttshra and R. B r a y , P h y s . Rev. L e t t . 3__99, 222 ( 1 9 7 7 ) ; S o l i d S t a t e Comm. 32, 621 ( 1 9 7 9 ) ; I b i d 33, 281 ( 1 9 8 0 ) . IOv. B o r t o l a n i , F. N i z z o l i , G. SanCoro, A . M a r v i n
and J . R . S a n d e r c o c k , Phys. Rev. L e t t . 4 3 , 224 (1979). l l J . R . S a n d e r c o c k , Solid State Comm. 26, 547
(1978).
12R. T. H a r l e y and P.A. F l e u r y , J . Phys. C: S o l i d S t a t e P h y s . 12, L863 ( 1 9 7 9 ) . 13For e x a m p l e , s e e E. B u r s t e i n , W.P. Chen, Y . J . Chen and A. H a r t s t e i n , J . Vac. S c i . T e c h n o l . 11, 1OO4 ( 1 9 7 4 ) . I ~ H . J . Simon, D.E. ~Lttche11 and J . G . Watson, Amer. J . Phys. 43, 630 ( 1 9 7 5 ) . 15H.j. Simon, R.E. Benner and J . G . Rako, Opt. Comm. 23, 245 ( 1 9 7 7 ) . t 6 y . j . Chen and E. B u r s t e i n , Nuovo Cimento 39B, 807 ( 1 9 7 7 ) . 17G.B. Benedek and K. F r l t s e h , Phys. Rev. 149, 647 ( 1 9 6 6 ) . ISR.J. P r e s s l e y , 1971 CRC Handbook o f L a s e r s (The Chemical Rubber Co.) p . 481.
Sb~FACE PLASHDN ENHANCED 8RILLOUIH SCATTERING FROH HETAL FILHS H. Fukui and O. Toda Department o f E l e c t r o n i c E n g i n e e r i n g U n i v e r s i t y o f Tokushima Tokusblma, J a p a n 770 V . C . ¥ . So and Department University Toronto, Canada
G . I . Stegeman* of Physics of Toronto Ontario HSS 1AT
( R e c e i v e d 26 June 1980 by A. A. Haradudln)
The theory o f B r t l l o u i n s c a t t e r i n g i n a p r i s m - m e t a l f i l e , - a i r (Kretschmenn) geometry i s a n a l y z e d and n u m e r i c a l e s t i m a t e s a r e g i v e n f o r a LaSF9 p r l s m - A g f i l m - e l f c o n f i g u r a t i o n . S i g n a l enhancement o f s e v e r a l o r d e r s o f magnitude i s p r e d i c t e d under c o n d i t i o n s o f c o u p l l n g to s u r f a c e p l a s m e n - l l k e f i e l d s .
1.
a c o u s t i c boundary c o n d i t i o n s a t b o t h f i l m interfaces. The s c a t t e r i n g f o r m a l i s m i n c l u d e s the e l A a t o o p t l c e f f e c t i n b o t h media a s w e l l as the c o r r u g a t i o n e f f e c t at both boundaries. C o n s i d e r t h e geometry i n F i g . I . The "p"-polarlzed incident fields (appropriate for coupling to s u r f a c e plasmons) are w r i t t e n as
Introduction
S e v e r a l i n v e s t i g a t o r s have r e c e n t l y d e v e l o p e d new t h e o r i e s f o r g r i l l o u l n s c a t t e r l n g a t t h e s u r f a c e s of opaque L-5 and t r a n s p a r e n t 6-7 m e d i a . A c c o r d i n g to t h e s e , t h e r e a r e two s c a t t e r i n g mechanisms, the vell-knoun e l a s t o o p t l c e f f e c t , as w e l l a s t h e c o r r u g a t i o n e f f e c t due t o acoustlcally-created surface ripple. Although the surface corrugation effect is a relatively weak s c a t t e r i n g mechanism, i t can p l a y a domin a n t r o l e I ' 1 2 i n opaque medla c h a r a c t e r i z e d by l a r g e v a l u e s o f d i e l e c t r i c c o n s t a n t , f o r example, in metals. I t can be e n h a n c e d by i n c r e a s i n g t h e amplltude of the incident or scattered light f l e l d s , t h e a c o u s t i c d i s p l a c e m e n t normal t o t h e s u r f a c e , or a combination of b o t h . I t i s y e l l - k n o w n 13 t h a t s u r f a c e plasmen p o l a r l t o n s can be e x c i t e d a t t h e a l r - m e t a l i n t e r f a c e o f a p r l s m - m e t a l f i l m - a l r sample u s i n g attenuated-total-reflectlon (ATR) m e t h o d s . The electric fields associated wlth these p o l a r l t o n s I~ can be much l a r g e r t h a n t h e I n c l d e n t f i e l d s used t o e x c i t e them. These p r o p e r t i e s have a l r e a d y b e e n u t i l i z e d i n e n h a n c i n g n o n l i n15_16P e a r hen°mane i n v o l v i n g s u r f a c e pl~mmons. In t h i s p a p e r we show t h a t c o u p l i n g t o s u r f a c e - p l a s m o n - l l k e modes c a n a l s o l e a d to l a r g e enhancements of B r i l l o u i n s c a t t e r i n g i n m e t a l l i c media.
2.
k,
i [~t-klX+kz(Z-h) ]
.&
E i - E0(~ + ~ - ~)e
(I)
Z
f o r the l i g h t
Er "
Er(i
i n c i d e n t through the prism,
k. " ~-- i ) e
i[wt-k, x-k (z-h) ] z
(2)
Z
for light
reflected
_ , = Eml(~ ÷ i - -
back i n t o the p r i s m , end
i[~t-k.x]
+ az
i(~t-~xl
-
i)e (3)
i_~_ + v-=2(~ for the l i g h t 0, 2
=
oz
i)e i n t h e m e t a l f i l m where
k,,2-cm(~0)~2/c2
and
kz2 = rp(~j)~2/c2-kll 2.
( H e n c e f o r t h t h e s u b s c r i p t s and s u p e r s c r i p t s p and m r e f e r to t h e p r i s m end metal r e s p e c tively.) The e l e c t r i c f i e l d a m p l i t u d e s Er, E 1 and ¥m2' n o r m a l l z e d to E0 a r e d l s p l a y e d as a
Theoretical Analysis
f u n c t i o n o f t h e i n c i d e n t a n g l e ¢0 i n F i g . 2 f o r The a p p r o a c h used i n t h e p r e s e n t work i s identical to that reported previously in r e f e r e n c e s 2 and 7. Normal phonou modes a r e f o u n d which s a t i s f y ( I ) t h e a c o u s t i c wave e q u a t i o n i n b o t h t h e film end t h e p r i s m , end (2) t h e "Present Address:
t h e m a t e r i a l p a r a m e t e r s l i s t e d i n Table I . The l a r g e enhancement i n Em2 , the f i e l d which is a
maximum at t h e m e t a l - a l r i n t e r f a c e ,
corresponds t o t h e c r e a t i o n o f s u r f a c e p l u m o n - l l k e modes a t t h a t boundary f o r i n c i d e n t a n g l e s n e a r 8 . Hence P B r i l l o u l n s c a c t e r l n ~ from t h e r m a l phonons n e a r t h a t i n t e r f a c e (0c E ~ ) s h o u l d be e n h a n c e d by up
Optical Sciences Center University of Arizona Tucson, A r i z o n a 85721
U.S.A. 995
Vol. 36, No. 12
BRILLOUIN SCATTERING FROM METAL FILMS
996
to f o u r o r d e r s o f magnitude. The phonon modes which p a r t i c i p a t e i n the scattering process are those of a film deposited on a s e e L t - l n f i n i t e s u b s t r a t e , and have been t r e a t e d p r e v i o u s l y by Rowell. 7 T h e i r s t r u c t u r e depends upon the f r e q u e n c y regimo of i n t e r e s t and
-Z
AIR
C
r
L
the relative shear and longitudinal wave velocities in the prism and film. Here we shall restrict our attention to the "x-z" polarized bulk modes (and hence to "p"-polarlzed s c a t t e r e d l i g h t )
characterized
METAL
by t h e f r e q u e n c y (~) range
~LP>fl>~Lm w i t h flLp = qlvL p and f l j p r i s m and m e t a l l o n g i t u d i n a l d e f i n e d by VLP and v j
h
= qmvLm.
The
wave v e l o c i t i e s
are
respectively,
and qm i s
the (x-) component o f the phonon w a v e v e c t o r p a r a l l e l to the s u r f a c e s . (Other frequency regions will be considered in a subsequent, mere detailed publication.) In general, the phonon displacement field density in O* - R space, i.e. in a volume element do. dR centred at qs, R is of the form
+2
-
i
m i [ R t - q u x ] ~ ~ ( V e ) e i q ~ m ( v ' ) z + cc
u = N0
Fig. i
Schematic diagram for the a n a l y s i s .
in
e
~
v'=l
(4)
m
t h e m e t a l and
i =~u
P i[~t-%x] ~ 0 e
iq p ( v ' ) z
v'
~ ~(
)e
~
+co
(5)
v'=l p
16
in the prism.
Here u0 is the mode amplitude
d e n s i t y , the p a r a m e t e r s ~ ( v ' ) d e s c r i b e the mode p o l a r l z a t l o n and the summation v ' i s taken o v e r a l l s o l u t i o n s to the a c o u s t i c wave e q u a t l o n c h a r a c t e r i z e d by a g i v e n v a l u e of ql and ft. S i n c e R
ld
the p r i s m (fl>flTP=q v ? )
XsT
.
4PP G(2a)3vTp(fl2-Gpz ~ T"
(6)
whereKB, T andpp are Boltzmann's constant,
tu 16
Fig. 2
and
The i n c i d e n t
electric field amplitudes
the
t e m p e r a t u r e and the p r i s m d e n s i t y r e s p e c t i v e l y . The c o m p l e t e e x p r e s s i o n s f o r t h e p a r a m e t e r s i n e q u a t i o n s 4-10 a r e c o m p l i c a t e d , can be d e r i v e d from r e f e r e n c e 7, and w i l l be g i v e n i n d e t a i l in a subsequent paper. The s c a t t e r e d f i e l d s were e v a l u a t e d by c a l c u l a t i n g the p o l a r i z a t i o n s o u r c e s c r e a t e d by the e l a s t o o p t l c e f f e c t and s o l v i n g the p o l a r l z a t i o n - d r i v e n wave e q u a t i o n s u b j e c t to the e l e c t r o m a g n e t i c boundary c o n d i t i o n s a c r o s s the acouettcally-corrugated interfaces. As o u t l i n e d i n r e f e r e n c e s 2 and 7, a p a r t i c u l a r s o l u t i o n to the Inhomogeneous wave e q u a t i o n was found i n each medium and t h e t a n g e n t i a l f i e l d s
Er, Eml and Bs2, n o r m a l i z e d to E0 v e r s u s
Em'P(z=0,h) and Hm'P(z=0,h) were c a l c u l a t e d on
the i n c i d e n t
each s i d e o f t h e f i l m
a n g l e ~0"
x
y
boundaries
( a t z=O and
Vol.
36,
scattering (Fig. 1). For l i g h t t h e p r i s m a t t h e a n g l e ~s
Table I Ag
LaSF9
1.76 x 103 m/s
3.O4 x 103 m/s
vL
3.79 x 103 m/s
5.54 x 103 m/s
o
1.O64 x 104 Kgm/m 3
4.53 x 103 Kgm/m 3
~S
E
-16.34 - i0.144
3.46
Cll
1.527 x 1011 N/m 2
1.39 x 1011 N/m 2
c44
3.287 x iO IO N/m 2
4.18 x IO IO N/m 2
Pll
(Cm-l)/¢m 2 = Pll
O.24
P44
O
O.O15
T
300 °K
m
3.66 x 1015 r a d / s
(~=O.5145 ~m)
List of material param:ters the numerical calculations.
used i n
z-h). The B r i l l o u i n s c a t t e r e d f i e l d s c o r r e s p o n d to s o l u t i o n s o f t h e homogeneous wave e q u a t i o n with a m p l l t u d e s a d j u s t e d t o e n s u r e t h a t t h e t o t a l fields satisfy the usual electromagnetic boundary conditions across the rippled surfaces. ~ l e r e s u l t a n t f i e l d a m p l l t u d e due t o t h e e l a s t o o p t l c e f f e c t can be w r i t t e n as Ee " I & ~ A I [ E / ( h ) - E x P ( h ) ] + A2Exm(O) + A3[Hym(h)-HyP(h) ] + A4Hym(O) } where the p a r a m e t e r s A i d e s c r i b e
(7)
the coupling
b e t w e e n t h e homogeneous and inhomogeneous solutions at the boundaries, l.e. the scattering g e o m e t r y , and & i s a f i l m r e s o n a n c e t e r m . The s c a t t e r e d f i e l d due t o t h e s p a t i a l and t e m p o r a l m o d u l a t i o n o f t h e i n c i d e n t l i g h t f i e l d s by t h e acoustically-created surface ripple is Ec = 4-~AB(h,Cp)Sm(h) - B(0,1)~m(O) } Uom
(8)
where 5m(h) and ~m(0) a r e t h e normal s u r f a c e displacements at the film boundaries. The t e r m s B(z,c) deter~ne the strength of the corrugation mechanism, and a r e p r o p o r t i o n a l 2 , 6 - 7 t o t h e optical field amplitudes at the interfaces, to £-¢ where c - 1 a t z=O and ¢ - ¢ a t z - h , and t o m p v a r i o u s o t h e r o p t i c a l , a c o u s t i c a l and g e o m e t r i c factors. Due t o t h e l a r g e i n c i d e n t f i e l d e n h a n c e m : n t ( F i g . 2) a t t h e s u r f a c e p l a s : o n coupling angle, large enhancements in the Exm(O), Hym(O) and B(O,I) t e r ~
(9)
2)e
where c~ - kQS+ksin+0 , Es - Ee+Ec, u s = w±R,
incident light intensity
are also
expected. The d i r e c t i o n s c h o s e n f o r o b s e r v i n g t h e Brillouin spectrum correspond closely to back-
(I 0) as
l(ms)
ms2¢ I~sl
3.
Numerical Calculations
0.27
PI2
+k-~ z
out of
Introducing the kl a - ksin~ s and k zs , kcos~s concept of the frequency spectrum, 2,17 we finally obtain the Brillouln spectrum per unit s o l i d a n g l e (ARs)-Unit f r e q u e n c y (~s)-unit
4o0
Table I.
" ES(~
radiated
i[mst+ktSx-kzS(z-h)]
kms vT
h
997
BRILLOUIN SCATTERING FROM METAL FILMS
No. 12
(10)
and D i s c u s s i o n
N u m e r i c a l c a l c u l a t i o n s were - : d e f o r a s i l v e r f i l m d e p o s i t e d on a LaSF9 g l a s s p r i s m w i t h i n c i d e n t O.5145 ~m r a d i a t i o n . The v a l u e s of the material constants used are presented in Table I. Since the elastooptic constants of LaSF9 g l a s s a r e n o t known, t h o s e o f dense f l i n t
g l a s s t8 were used i n s t e a d : this is reasonable since the elastooptic constants of different glasses are of comparable value. Two s c a t t e r i n g g e o m e t r i e s were e x a m i n e d , b o t h of which l e d t o t h e same v a l u e o f ql and h e n c e i n v o l v e d s c a t t e r i n g from t h e sam: phonon fields. In the first, the directions of the i n c i d e n t and s c a t t e r e d l i g h t were c h o s e n to c o r r e s p o n d to c o u p l i n g to s u r f a c e plasmon modes In t h e s e c o n d , n e i t h e r o p t i c a l f i e l d c o u p l e s s t r o n g l y t o s u r f a c e plasmon : o d e s . The c a l c u l a t e d B r t l l o u t n s p e c t r u m f o r b o t h g e o m : t r l e s i s shown i n F i g . 3. When t h e i n c i d e n t and s c a t t e r e d f i e l d s a r e c o u p l e d to s u r f a c e p i s s : o n modes, an e n h a n c e m e n t o f two t o three orders of magnitude Is predicted. The calculated signal strength is several orders of m a g n i t u d e l a r g e r t h a n t h a t m e a s u r e d by S a n d e r c o c k 1! i n h i s e x p e r i m : n t s on B r i l l o u i n s c a t t e r i n g on r e f l e c t i o n from m : t a l s u r f a c e s and h e n c e s h o u l d b e r e a d i l y o b s e r v a b l e . The spectrum in the frequency region considered is b r o a d and r e l a t i v e l y f e a t u r e l e s s , a characteri s t i c t y p i c a l of B r i l l o u i n s c a t t e r i n g from m : t a l l t c media. 1-10 S t r u c t u r e a p p e a r s n e a r = quvLP(RL p) a t w h i c h f r e q u e n c y t h e Bragg condition for scattering of the incident r e f l e c t e d l i g h t by l o n g i t u d i n a l waves i n the prism is satisfied. The s p e c t r u m i s f u r t h e r c o m p l i c a t e d t h e r e by a change i n the d e n s i t y o f s t a t e s 2 , ? ( = l / q P) when the l o n g i t u d i n a l waves i n t h e p r i s m change from an e v a n e s c e n t t o a
propagating character, grillo~Ln scattering from R a y l e i g h and Sezava 19 waves s h o u l d a l s o be e n h a n c e d i n t h i s s u r f a c e plasmon g e o m : t r y , and t h i s a s p e c t w i l l be r e p o r t e d l a t e r . The r e g i o n o f t h e sample which d o m i n a t e s t h e s c a t t e r i n g was i n v e s t i g a t e d i n o r d e r t o v e r i f y t h e r o l e o f t h e s u r f a c e p l a s m : n s . The f i e l d s a s s o c i a t e d w i t h e a c h i n t e r f a c e were calculated,
i.e.
998
BRILLOUIN SCATTEEING FROM METAL FILMS
(p.--- (Po: Oo[:~z'] lo'
i
=3724"
~T ( 10" ~o/s~c) FIB. 3
The s c a t t e r i n g
cross-sectlon
l(Ws)/
lO&~sA~e as a [ u n c t l o n of f r e q u e n c y shift
~s-~O - ~ f o r t~ao s c a t t e r i n g
geometrles.
V o l . 36, No. 12
Vol. 36, No. 12
BKI.LLOUI.N SCATTERING FRON F~TAL F ILI4S
999
1o+
E~o~ lci
J
9u.t t.&-
O ILl n.-
la
J o~
tt5
ld (D+ =3724"
([)o=3O53"
9
1of
o~
1.35
.Q [ld' ~,=c] Pt.g. 4
The s c a t t e r e d
[ZT(h)l and shift
~T.
field
IzZ(o)l
contributions ver,,-
(a) $O,~m - Op,
(b) 4~0,* s It 0p.
frequency
BRILLOUIN SCATTERING FROM METAL FILMS
I000
ET(0) -~A2Exm(0)
~=RLP and
+ A4Hym(0)
_ !4B(O'i)~m(°)uoml
(ii)
which is dominated by scattering at and near the metal-air interface and ET(h) = ~Al[Exm(h)-ExP(h)] + A3[Hym(h)-HyP(h)] + 41B(h,~p)~m(h)u0 m}
(12)
which arises from scattering in the prism and metal, at and near the metal-prism boundary. As indicated in Fig. 4, the alr-metal interfaces dominate the scattering when surface plasmons are excited; otherwise the prism and the prism-metal interface region is principally responsible for the Brillouln spectrum. This result verifies that the excitation of surface plasmons is the cause for the large enhancements in the Brillouln signal. The dominant scattering mechanism, elastooptlc versus corrugation was also examined. With the exception of the frequency regions
f~Lm, t h e s u r f a c e r i p p l e
Vol. 36, No. 12 mechanism
dominates the scattering cross-section. This is in a g r e e m e n t w i t h p r e v i o u s c a l c u l a t i o n s I-5 f o r s e m l - l n f i n i t e m e t a l s c h a r a c t e r i z e d by o p t i c a l s k i n d e p t h s much l e s s t h a n a w a v e l e n g t h . In s,,m~=ry, we have shown t h a t B r i l l o u i n s c a t t e r i n g from m e t a l f i l m s i n a n ATR geometry i s e n h a n c e d when t h e o p t i c a l f i e l d s c o u p l e to surface plasmon-like excitations at the metal-alr boundary. The p r e d i c t e d s i g n a l l e v e l s a r e comparable to that of Brillouin scattering from bulk media and experiments to verify these calculations a r e in progress. Acknowledgements - We would l l k e t o t h a n k Dr. N. Rowell f o r u s e f u l d i s c u s s i o n s and a l s o acknowledge Mr. K. Muguruma f o r h i s h e l p i n t h e numerical calculations. One o f us ( G . I . S . ) thanks his colleagues at Stanford for their hospitality d u r i n g p a r t o f t h i s work. The financial support of the Canadian Natural Sciences a n d E n g i n e e r i n g R e s e a r c h Counc11 i s g r a t e f u l l y acknowledged.
References
IR. Loudon, P h y s . Rev. L e t t . 40, 581 ( 1 9 7 8 ) . 2N.L. Rowell and G . l . Stegemen, S o l i d S t a t e Comm. 2 6 , 809 ( 1 9 7 8 ) ; P h y s . Rev. BI8, 2598
3(1978).
K.R. Subbaswamy and A.A. N a r a d u d l n , Phys. Rev. B18, 4181 ( 1 9 7 8 ) . ~V. B o r t o l a n i , F. N i z z o l i and G. S a n c o r o , Phys. Rev. Lett. 4 1 , 39 ( 1 9 7 8 ) . 5A. Derv£sch a n d R. Loudon, J . Phys. C: S o l i d S t a t e Phys. 11, L291 ( 1 9 7 8 ) . 6G.I. Stegeman, J . Appl. P h y s . ~ , 5624 ( 1 9 7 8 ) . 7N.L. Rowell, 1978, Ph.D. T h e s i s , U n i v e r s i t y o f Toronto (unpublished). 8J.G. D i l and E.M. Brody, P h y s . Rev. B14, 5218 (1976). 9S. ~ttshra and R. B r a y , P h y s . Rev. L e t t . 3__99, 222 ( 1 9 7 7 ) ; S o l i d S t a t e Comm. 32, 621 ( 1 9 7 9 ) ; I b i d 33, 281 ( 1 9 8 0 ) . IOv. B o r t o l a n i , F. N i z z o l i , G. SanCoro, A . M a r v i n
and J . R . S a n d e r c o c k , Phys. Rev. L e t t . 4 3 , 224 (1979). l l J . R . S a n d e r c o c k , Solid State Comm. 26, 547
(1978).
12R. T. H a r l e y and P.A. F l e u r y , J . Phys. C: S o l i d S t a t e P h y s . 12, L863 ( 1 9 7 9 ) . 13For e x a m p l e , s e e E. B u r s t e i n , W.P. Chen, Y . J . Chen and A. H a r t s t e i n , J . Vac. S c i . T e c h n o l . 11, 1OO4 ( 1 9 7 4 ) . I ~ H . J . Simon, D.E. ~Lttche11 and J . G . Watson, Amer. J . Phys. 43, 630 ( 1 9 7 5 ) . 15H.j. Simon, R.E. Benner and J . G . Rako, Opt. Comm. 23, 245 ( 1 9 7 7 ) . t 6 y . j . Chen and E. B u r s t e i n , Nuovo Cimento 39B, 807 ( 1 9 7 7 ) . 17G.B. Benedek and K. F r l t s e h , Phys. Rev. 149, 647 ( 1 9 6 6 ) . ISR.J. P r e s s l e y , 1971 CRC Handbook o f L a s e r s (The Chemical Rubber Co.) p . 481.