Optical nonlinearity in coupled exciton-phonon systems near metal surfaces

Synthetic Metals, 49-50 (1992) 167-173

167

Optical nonlinearity in coupled exciton-phonon systems near metal surfaces X. Li a n d D. L. L i n Department of Physics and Astronomy, State University of New York at Buffalo, Buffalo, N Y 14260 (USA) T h o m a s F. G e o r g e Departments of Chemistry and Physics, Washington State University, Pullman, Washington, WA 99164 (USA)

Abstract Surface effects on the optical bistability and p u m p - p r o b e process for coupled excit o n - p h o n o n systems are studied theoretically. When a laser beam is directed on an exciton-phonon coupling system inside an optical cavity, the cavity field intensity is investigated as a function of the driving field intensity for various distances between the sample and a metallic surface representing one of the mirrors of the cavity. It is found that the direction of switch of the optical bistability can be controlled by adjusting the distance from the surface. For a strong laser pump field and a weak probe field directed to this system located near a metal surface, the pump field-induced absorption spectrum is investigated for different distances between this sample and the metal surface. It is found that the position and the lineshape of the spectrum are remarkably changed due to the presence of the surface.

1. I n t r o d u c t i o n The investigation of optical properties of molecular systems near a solid surface has been of great interest from both the fundamental and practical points of view [1-4]. The study of an atom or a molecule adsorbed on a s o l i d s u r f a c e h a s a t t r a c t e d m u c h a t t e n t i o n in t h e p a s t , b o t h t h e o r e t i c a l l y a n d experimentally. Exciton-phonon coupling systems, such as organic semic o n d u c t o r s , h a v e b e e n c o n s i d e r e d [5 ] v e r y r e c e n t l y , i n s t e a d o f s i n g l e m o l e c u l a r systems located near an ideal metal surface. Surface-induced optical bistability has been discovered and its nature and origin investigated. In this paper we study the effects of the metal surface on the optical nonlinearity of an exciton-phonon system for two different processes of l a s e r - m a t t e r i n t e r a c t i o n s . I n t h e first p r o c e s s w e c o n s i d e r o p t i c a l b i s t a b i l i t y of a linear chain of polydiacetylene-toluene-sulfonate (PTS) located inside an optical cavity and oriented parallel to the metal surface of the mirrors. For the second process, we consider the pump field-induced absorption spectrum for a PTS chain located near a parallel metal surface and irradiated b y a p u m p f i e l d a n d a p r o b e ( t e s t ) field.

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168 2. Optical bistability We consider in this section a PTS chain inside an optical cavity. The chain is embedded in a non-absorbing material of dielectric constant el and is oriented parallel to the reflecting metallic mirrors that constitute the cavity. The dielectric function of the mirror is e2(~o) = 1

)p21

(1)

o~(o~+i/D

where OJpl is the plasma frequency and Fp the damping rate. Our purpose is to investigate how the optical bistability behavior is influenced by the presence of the metallic surface of one of the mirrors if the polymer is placed at a distance d smaller than the excitonic wavelength. When an incident laser b eam is directed into the cavity, excitons are excited in PTS and act as the emitting dipoles. The emitted light is reflected by the surface and the reflected light interacts in turn with the emitting excitons and changes their response to the incident light. At the same time, the excitons couple with p h o n o n mo des in PTS, which play the role of mediation in the phot o n - e x c i t o n interaction [6]. We model the excitons, p h o n o n m ode s and cavity field as damped oscillators and assume a dissipative interaction between the exciton and surface-reflected field [3]. The problem is then treated by means of Dekker's quantization procedure for dissipative systems [ 7]. In this model, the system is described by the non-Hermitian Hamiltonian: 22~ = ( ~ -- i K ) a t a + (a~x -- iTx)ctc + ~ , ( w i -- iTi)btibi + ~ , h i ctc(b~ + bi) i

i

+ i g ( a t c - c t a ) + i[p~a*Eo e x p ( - iwo t) - ~* aE~ exp(iwo t)] - p~*ctER

(2)

where a t (a), b~ (bi) and ct (c) stand for the creation (annihilation) operators for the cavity field, the ith phonon m ode and the exciton with corresponding frequencies ~, eo~ and wx, respectively. K, Tx and 3q are the damping rates for the cavity field, exciton and ith p h o n o n mode, respectively, h~ is the coupling constant for the ith phonon m ode interacting with the exciton and g is the exciton-cavity field coupling constant. The frequency and amplitude of the coherent driving field are denoted by COo and E0, respectively. The dipole m o m e n t of the emitting exciton is r e p resent ed by p = t~a with /z as its matrix element. ER stands for the reflected field at the position of the dipole; it is proportional to p and, hence, can be written as E R = E ~ c , where Er is a c number. As we shall see later, Er is in general a com pl ex quantity, so that the interaction is dissipative [8]. Since we are mainly interested in the relation between intensities of the cavity field and input field, we may take a semiclassical approach by neglecting all quantum fluctuations [8]. Thus, instead of the operators a, bi, c and c ' c , we are dealing with their m ean values a = ( a ) , fli = (bi), ~ / = ( c ) and n = (ct c).

169 With t h e s e m e a n v a l u e s defined as classical variables, the c o r r e s p o n d i n g e q u a t i o n s of m o t i o n c a n b e o b t a i n e d f r o m eqn. (2) as follows [3, 4, 9]: t~= --(iAl + K ) a + g n + E o

(3a)

/~i -- - (itoi + "Yi)[3i- l a i n

(3b)

=

-- (iA 2 + yt)~/--

(3c)

i ~ h / ( f l i + fl*)~? - g a i

(3d)

~i = - g ( a ~ * + a*~7) - 2 T t n

w h e r e w e h a v e defined the d e t u n i n g s a l : ~ ' ~ - - o ) 0 and Ae=wx-Wo+Ws, %----7x+ %. T h e s u r f a c e effect o n the e x c i t o n is reflected in the s u r f a c e i n d u c e d f r e q u e n c y shift o~s--[/~[ 2 R e ( - ~ )

(4a)

a n d the d e c a y r a t e

T h e r e f l e c t e d electric field E r in the dipole d i r e c t i o n c a n b e f o u n d by a classical a p p r o a c h . W e t a k e t h e oscillating dipole m o m e n t t o b e l o c a t e d at a d i s t a n c e d f r o m t h e s u r f a c e of a semi-infinite metal. F r o m e l e c t r o m a g n e t i c field t h e o r y , we find in a s t r a i g h t f o r w a r d m a n n e r t h a t E r ------- i - 2/~*

d u u3Rll e x p ( 2 i l ~ x ) / ( 1 - u2) ~/2

(5)

0

w h i c h h a s b e e n e x p r e s s e d in t h e unit o f %:=~e~12Ii.~[2k~. F r o m n o w o n a n d t h r o u g h o u t this p a p e r , all q u a n t i t i e s with t-~ d i m e n s i o n will b e e x p r e s s e d in yx. In the d e r i v a t i o n o f eqn. (5), w e h a v e a s s u m e d t h a t t h e dipole is p e r p e n d i c u l a r to the s u r f a c e . H e r e w e h a v e also defined ko = Wx/C, x = e~/2kod and Rll --

E l /-t2 - - E 2 ~[Z1

(6a)

E 1 ~-t2 -~- E 2 ~[Z1

/ ~ = [el/El --U2] ~

l = 1, 2

(6b)

If we i n t r o d u c e a d i m e n s i o n l e s s i n p u t field intensity Iin=g2~ol2/(ky~) 2, w e find f r o m eqns. (3) t h a t t h e m e a n e x c i t o n n u m b e r n satisfies the cubic equation Ape[1 + (As/K) e ]n s + 2Ap [ • i ( g

2 -- A 1A2)/K 2 --

A2 ]n e

+ {[ 1 + % + (g2 _ A~A2)/Kle + [A2 + (1 + %)A1/K]2}n - I i n = 0

(7)

At t h e s a m e time, t h e c a v i t y field intensity Icav= la[2/')'~ c a n b e e x p r e s s e d in t e r m s o f the i n p u t field i n t e n s i t y /in as

170 (1 + ys) 2 + (zl2 - )~n) 2

Icav= [l +,ys+g2/K_Al(A2_),pn)/K]2+[Al(l +,y~)/K+A2_Apn]2Ii~

(8)

w h e r e we have defined Ap = 2F.ih~/toi. E q u a t i o n (7) is solved n u m e r i c a l l y and the solutions are u s e d to study the tristability of the cavity field intensity in eqn. (8) u n d e r various conditions. W h e n the incident field f r e q u e n c y is on r e s o n a n c e with b o t h the cavity field and the excitonic transition, the o c c u r r e n c e o f optical bistability d e p e n d s on the location of the chain in the cavity. If the chain is far away f r o m the m i r r o r surfaces o f the cavity, eqn. (7) has only one physical solution and t h e r e is no optical bistability. If the PTS chain is n e a r a m i r r o r surface or if d is smaller t h a n the e x c i t o n i c wavelength, eqn. (7) m a y p o s s e s s t h r e e real solutions and it is possible to find optical bistability in the cavity. As d i s c u s s e d in ref. 5, this surfacei n d u c e d bistability is c a u s e d b y b o t h the f r e q u e n c y shift ~o~ and the width c h a n g e ~/~ o f the e x c i t o n i c transition due to the surface. If the distance b e c o m e s t o o small or x << 1, the surface a b s o r p t i o n b e c o m e s so s t r o n g that the lifetime o f the e x c i t o n is too s h o r t to interact strongly and nonlinearly with the p h o n o n m o d e s . Consequently, t h e r e can no l o n g e r be bistability. Off r e s o n a n c e , the b e h a v i o r d e p e n d s v e r y m u c h on the sign o f the detuning. W h e n A 1 > 0, the optical bistability is e x p r e s s e d in a n o r m a l hysteresis l o o p and the p r e s e n c e o f a surface c a n n o t c h a n g e this nature. However, a n e a r b y surface can affect the t h r e s h o l d input intensity and the c o n t r a s t o f the hysteresis loop. W h e n A I < 0, optical bistability can usually be f o u n d with the inverted h y s t e r e s i s loop, even if the sample is far away f r o m the surface. In the p r e s e n c e o f a n e a r b y surface, o u r calculations s h o w a c o m p l e t e l y new p h e n o m e n o n , namely, that the two cavity field intensities c r o s s e a c h o t h e r as the driving field intensity changes. In o u r numerical work, we have included all f o u r p h o n o n m o d e s that c o u p l e m o s t strongly to the e x c i t o n s [6, 9, 10]. T h e s e m o d e s are c h a r a c t e r i z e d by the p a r a m e t e r s to1 = 5.16, AI = 2; ~o2= 3.68, h2 = 1.66; w ~ = 2 . 9 8 , ~ 3 = 0 . 4 6 ; ~o4=2.36, A4=0.48. S o m e of the results for A1 < 0 are p l o t t e d in Fig. 1 in w h i c h we d e m o n s t r a t e the c h a n g e o f the bistable n a t u r e b y varying the distance f r o m the surface only. Figure 1 (a) shows no bistability b e c a u s e o f the small distance. As m e n t i o n e d above, the strong a b s o r p t i o n b y the surface p r e v e n t s the e x c i t o n f r o m interacting nonlinearly with p h o n o n m o d e s . T h e r e c a n n o t be optical bistability for any A~. In Fig. 1Co), w e find the interesting new p h e n o m e n o n w h e r e the two o u t p u t states c r o s s e a c h other, forming a peculiar figure e i g h t - s h a p e d loop. The cavity field shows the n o r m a l bistability as the driving field intensity increases, b u t follows the inverted loop as Im decreases. The h y s t e r e s i s l o o p b e c o m e s a triangle in Fig. l ( c ) at distance x= 2.0. After this point, the l o o p b e c o m e s i n v e r t e d and stays that w a y as the distance increases further. As d i s c u s s e d in ref. 9 in m o r e detail, in an optical cavity, the real p a r t o f the optical r e s p o n s e of PTS or the dispersion part plays a m o r e i m p o r t a n t role t h a n the a b s o r p t i o n p a r t b e c a u s e the refractive index is c h a n g e d b y the e x c i t o n s w h i c h are e x c i t e d by the input field. Our n u m e r i c a l c o m p u t a t i o n s indicate that the m a j o r c o n t r i b u t i o n to the bistability c o m e s f r o m the e x c i t o n

171 10

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

I

],~

0

2

2

4

I,~ 6

8

10

15

15

5¸

OIr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0

5

lin

I0

15

0

5

[i~

10

15

Fig. 1. Cavity field i n t e n s i t y I¢~v vs. i n c i d e n t field i n t e n s i t y Iin for v a r i o u s d i s t a n c e s x b e t w e e n t h i s s y s t e m a n d t h e m e t a l s u r f a c e w h e n A 1 = - 0 . 2 5 , w,,/Wpl=0.8, A 2 = 6 , g2/k=0.5: (a) x = 0 . 5 ; (b) x = 1.25; (c) x = 2.0; (d) x = ¢¢.

e n e r g y shift i n d u c e d b y the surface. This is in c o n t r a s t to the c a s e d i s c u s s e d in ref. 5, w h e r e the a b s o r p t i v e bistability is c a u s e d b y r e d u c e d v a c u u m fluctuations.

3. P u m p - p r o b e

process

In this s e c t i o n w e c o n s i d e r t w o l a s e r b e a m s d i r e c t e d to a c h a i n o f PTS e m b e d d e d in a m e d i u m with dielectric c o n s t a n t El a n d l o c a t e d at a d i s t a n c e d f r o m a m e t a l s u r f a c e with the dielectric f u n c t i o n c h a r a c t e r i z e d b y eqn. (1). In the s a m e w a y as t h e p r e c e d i n g section, the e x c i t o n s a n d p h o n o n s a r e m o d e l e d as d a m p e d oscillators. T h e H a m i l t o n i a n is Z = (t9,~ - i%,)c*c + ( t o - iT)bib + hctc(b t + b)

- {gc[E* exp(iwp t) + E t* e x p ( i ~ t) ] + h.c.} - g*ctER

(9)

w h e r e Ep (Et) a n d e0p (e0t) are t h e a m p l i t u d e a n d the f r e q u e n c y of the p u m p ( p r o b e ) field a n d w e h a v e o n l y i n c l u d e d o n e p h o n o n m o d e c o u p l i n g m o s t s t r o n g l y to t h e e x c i t o n s . T h e s u b s c r i p t s o f b, T a n d )t h a v e t h e r e f o r e b e e n d r o p p e d . Following a similar p r o c e d u r e as in refs. 5 a n d 1 l , we find the n o n l i n e a r s u s c e p t i b i l i t y Xt to first o r d e r in E t a n d all o r d e r s in Ep: 3

Xt = nolO[ 2 ~ p i i=1

(1 O)

172

w h e r e no is the density o f t h e e x c i t o n s and pi =

ni

with

ni

y2 _ ~ _ Ze

Z = A t + i ( 1 + Ts),

Y=Ap+eos-Axni+Y,,

~=2A2£0ni/(A2+iTAt--o)2),

superscript i is used to d e n o t e the ith r o o t o f t h e cubic e q u a t i o n with r e s p e c t to the e x c i t o n n u m b e r n: A x ---- 2A2~0/(W 2 4- T2),

A p = o) x - - (.Op,

A t = o.)t - - O)p.

The

A~n 3 - 2(zip + ~os)Axn2 + [(Zip + eos)2 + (1 + 1,~)2 ] n - Ip~7.12 = 0

(12)

T h e r e f o r e the n o n l i n e a r susceptibility i n d u c e d by t h e p u m p field AXt is t h e difference b e t w e e n the Xt with and w i t h o u t t h e p r e s e n c e o f the p u m p field, namely:

AXt= Xtbp÷o--XtlEp=O

(13)

Figure 2 s h o w s s o m e o f our n u m e r i c a l results c a l c u l a t e d f r o m eqn. ( 1 1 ) w h e r e w e h a v e u s e d zip= 6, ~o= 5.16, y = 0 . 0 4 , Ip3~pl = 0 . 2 and A = 2 . So, At----6 c o r r e s p o n d s to the c a s e w h e n the p r o b e field is o n r e s o n a n c e with the surfacefree e x c i t o n . F o r Figs. 2(a) and (b) w e h a v e c h o s e n t h e d i s t a n c e s x s u c h that the radiative lifetime is p r o l o n g e d d u e to the p r e s e n c e o f t h e m e t a l surface. In this c a s e t h e s p e c t r u m o f AXt h a s three structures, o f w h i c h the t w o side b a n d s are l o c a t e d at w t - w , = _ w. E v i d e n t l y t h e y result f r o m t h e interaction b e t w e e n the beat c o m p o n e n t o f t h e t w o incident fields and t h e 04

>-. Z o

~

0.4

(o)

O2 0.0

v~}

>,

f_

LS o

-02 -0.4

(b)

0.2 0.0

,~ - 0 . 2

~b . . . . . . .

:5

........

6 .........

5 .........

~o

A,

-o.t~o

-5

o

s

lo

A,

0.4

(c)

0.2 b

o.o .,~E --0.2 -(?.4 -10

-5

0 A,

5

10

Fig. 2. Pump field-induced absorptive part of the susceptibility vs. frequency difference between the test field and the pump field A t w h e n Ap=6.O, cox/e%1=0.8: (a) x = 0 . 5 ; (b) x = 0 . 7 5 ; (c) X~OO,

173 p h o n o n . B e c a u s e t h i s i n t e r a c t i o n is m e d i a t e d b y t h e e x c i t o n s , s u c h a n interesting phenomenon becomes remarkable only when the lifetime of the e x c i t o n s is l o n g e n o u g h . F o r F i g . 2 ( c ) , t h e v a c u u m f i e l d f l u c t u a t i o n is n o t r e d u c e d ( s o t h e l i f e t i m e o f t h e e x c i t o n s is n o t p r o l o n g e d ) a n d t h e r e a r e n o s u c h s i d e b a n d s . In a d d i t i o n , w e h a v e a l s o f o u n d t h a t t h e p o s i t i o n o f t h e d i s p e r s i o n - l i k e s t r u c t u r e in t h e s p e c t r u m o f nxt d e p e n d s o n t h e d i s t a n c e between the PTS chain and the metal surface and so does the width of the p e a k o r t h e h o l e . T h i s is d u e t o t h e f a c t t h a t t h e p r e s e n c e o f t h e s u r f a c e changes both the lifetime and the frequency of the excitons and, similar to r e f s . 1, 3 a n d 4, t h i s c h a n g e d e p e n d s o n t h e d i s t a n c e o f t h e e x c i t o n s f r o m the surface.

Acknowledgements W e a r e g r a t e f u l t o M e s s r s X. X i a a n d Z. Liu f o r t h e i r h e l p w i t h t h e n u m e r i c a l c a l c u l a t i o n s . T h i s r e s e a r c h w a s s u p p o r t e d in p a r t b y t h e Office of Naval Research.

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