Документ взят из кэша поисковой машины. Адрес оригинального документа : http://qilab.phys.msu.ru/papers/Appl%20Phys%20B34%20167-170(1984).pdf Дата изменения: Mon Feb 4 18:39:09 2008 Дата индексирования: Mon Oct 1 20:05:11 2012 Кодировка:

Appl. Phys. B 34, 167-170 (1984)

Applied

Physics B

,..o,,,physics C ~hemisby

9 Springer-Verlag 1984

Saturation S p e c t r o s c o p y o f Coherent R a m a n Scattering in M o l e c u l a r G a s e s
V. N . Z a d k o v , N . I. K o r o t e e v , M. V. R y c h o v , a n d A. B. F e o d o r o v Physics D e p a r t m e n t , M o s c o w State University, SU-117234 M o s c o w , U S S R Received 8 November 1983/Accepted 12 February 1984 Abstract. W e present the t h e o r y o f t w o - p h o t o n R a m a n s a t u r a t i o n o f a two-level R a m a n t r a n s i t i o n s t u d i e d b y a n i n d e p e n d e n t C A R S p r o c e s s . T h e m a i n g o a l h e r e is t o p r o b e t h e s a t u r a t e d h o m o g e n e o u s R a m a n line shape. I t is s h o w n t h a t t h e r e a p p e a r s a s a t u r a t i o n d i p w i t h a w i d t h d e t e r m i n e d b y t h e r e l a x a t i o n t i m e T~. I n t h e c a s e o f D o p p l e r - b r o a d e n e d line t h e c o h e r e n t R a m a n s a t u r a t i o n s p e c t r o s c o p y m a y b e u s e d t o d e t e r m i n e b o t h t h e T~ a n d T2 r e l a x a t i o n times. P A C S : 42.65 C q , 33.80 S a t u r a t i o n a b s o r p t i o n s p e c t r o s c o p y has b e c o m e a powerful t o o l for studying Doppler-free electron a b s o r p t i o n s p e c t r a [ 1 , 2 ] . T h i s w o r k is d e v o t e d t o t h e possible e x t r a p o l a t i o n o f the s a t u r a t i o n technique t o t h e field o f R a m a n - a c t i v e t r a n s i t i o n s in t h e v i b r a t i o n a l s p e c t r a o f molecules. T w o intense p u m p waves are n e e d e d to s a t u r a t e the t r a n s i t i o n u n d e r s t u d y , a n d t h e p r o b i n g is t o b e c a r r i e d o u t b y coherent anti-Stokes R a m a n spectroscopy ( C A R S ) [3, 4 ] . T h i s p r o b l e m h a s e a r l i e r b e e n d i s c u s s e d in [ 5 ] a n d it h a s b e e n s o l v e d t h e r e within t h e incoherent a p p r o x i m a t i o n taking into a c c o u n t only t h e p o p u l a t i o n c h a n g e s u n d e r t h e influence o f bih a r m o n i c p u m p waves. It h a s b e e n s h o w n t h a t a specific p e c u l i a r i t y a p p e a r s i n t h e c e n t e r o f t h e line w i t h a w i d t h p r o p o r t i o n a l t o T2 ~. H o w e v e r , t h e c o n c l u s i o n s m a d e in [ 5 ] a r e n o t a c c u r a t e e n o u g h . I t is n e c e s s a r y t o t a k e i n t o a c c o u n t c o h e r e n t effects, a n d m o s t o f all, t h e p o p u l a t i o n o s c i l l a t i o n s in t h e t w o - l e v e l system [6]. This p a p e r p r e s e n t s t h e c a l c u l a t i o n o f t h e C A R S signal l i n e - s h a p e c a r r i e d o u t f o r t h e c a s e o f c o h e r e n t effects. An additional n a r r o w dip with the width p r o p o r t i o n a l t o T~- ~ is s h o w n t o a p p e a r in t h e C A R S s p e c t r u m b o t h in the cases o f D o p p l e r - b r o a d e n e d a n d h o m o g e n e o u s l y b r o a d e n e d lines.
1. M a i n Equations

E q u a t i o n s for t h e p o p u l a t i o n s difference n a n d t h e coherent amplitude Q o f molecular vibrations are the following ones [ 3 ] :

dn n - 1 d r 4 T~ -

1 { O c t ' ~ 2dQ 2 t i f 2 \ ~ J E dr'
1 (3~_~ E2 n
= \oQ)

(1) (2)

d2Q
dt +

2 dQ

H e r e f2 is t h e f r e q u e n c y o f t h e t r a n s i t i o n u n d e r s t u d y , T1 a n d T2 a r e t h e v i b r a t i o n a l e n e r g y a n d p h a s e r e l a x a t i o n t i m e s , r e s p e c t i v e l y , e(Q) is a p o l a r i z a b i l i t y . T h e field E in (1 a n d 2) is a s u p e r p o s i t i o n o f t w o p a i r s o f plane waves, namely, a p a i r o f p u m p w a v e s with t h e f r e q u e n c i e s o)1 a n d o)2 a n d a p a i r o f p r o b e w a v e s w i t h t h e f r e q u e n c i e s co~ a n d co~. F o r the collinear interaction:

E = ~ ( A l e - i , ~ l t + Aze-iO~2t+c.c.) -2w.l,~'!~A' - i o m . A,.~,2 imt+c.c.) 9 ~ e
(3)

T h e s t e a d y - s t a t e s o l u t i o n o f s y s t e m o f (1, 2) is a s s u m e d to have the following form:
Q _ - -~ l2f l 0 ~ - - i ( ( o 1 _~" o ) 2 ) t-[rl ~' o/ /)

e - i ( o ~ i - o~)t + c . c . ,

(4)

n = no + 89 e - iE(,~- o,2)-( m - o,~)i,+ c.c., T h e p r o b l e m is c o n s i d e r e d w i t h i n t h e a p p r o x i m a t i o n o f a t w o - l e v e l s y s t e m a n d fixed p u m p a n d p r o b e fields. w h e r e co1
a= - 0 2 ' ~ ~'~ ~e~o ] - - ( 1 ) 2 .

(5)

Let

[((D1 -- ( D 2 ) - - ~ ] T2, A ' = [(e)i - - c o l ) - - O] 7"2


168

V . N . Z a d k o v e t al.

and

w h i c h a r e a n a l o g o u s t o (6-7):

r t T2IA~A2I 2 ( 0 ~ ' ] 2

G=

16MhQ

\OQJ '

r t T21A'IA'2[2 ( 0 o ~ z g= 16Mhf2 \OQJ '

Q~

-i-A

'

(ll)
(12)

( s a t u r a t i o n p a r a m e t e r s f o r p u m p a n d p r o b e waves, respectively). F r o m (2-5) t h e e x p r e s s i o n s f o r t h e c o h e r e n t a m p l i t u d e s result in
A A* .3- n O A t A t *

' A ' * - - r~~ Qo' = ~ ~) _ i ~ A-7

as well a s t h e e x p r e s s i o n s f o r no a n d rio:

Q o - 4MO \ ~ J

~

'

1
(6) noG I+-I+A 2 '

(13)

Q; = ~

- i - A'

(7)

[I +i(A-A')~z

" G/2 7-,~J~*

I t is e a s y t o o b t a i n t h r e e c o m p l e x linear e q u a t i o n s f o r no, no, n* w h i c h m a y b e r e d u c e d t o t h e r e a l e q u a t i o n s b y the following substitution: r~o= rT; + i~7;,

=notzi ,
-1 G 1+-I+A 2

(14)

~*= ~ ; - i ~ ; .

(8)

S u b s t i t u t i n g (13, 14) i n t o (12) we o b t a i n f o r t h e c o h e r e n t a m p l i t u d e Q~:

O~ 4 ~

W e m a k e t h e s a m e s u b s t i t u t i o n in (7) a n d c a l c u l a t e QoT h e t h i r d - o r d e r n o n l i n e a r s u s c e p t i b i l i t y Z(3)R is p r o p o r t i o n a l t o Q~, l e a d i n g t o
/CAllS ~" IZ(3)RI 2

1 + i ( A - A ' ) T~

iG/2 "A'* (15)

G 1 + i ( A - - A ' ) ~2 4 ii~ /2
fib + n~
, A ' 2 ..]_ 1

2no +
= (~(3)R)2

(9)

T h e r e s u l t i n g e x p r e s s i o n for/CARS is

where

Iz<3)RI _ 2
ICARS~ ~

NT2 ( 0 ~ 2

(10)

(1+
\

4G]2
I+A2J AG/2
(16)

N being t h e n u m b e r o f m o l e c u l e s in t h e u n i t volume. E x p r e s s i o n (9) s h o w s t h a t t h e C A R S signal line p r o f i l e d e p e n d s b o t h o n t h e s t a t i c p o p u l a t i o n difference c h a n g e no a n d o p t i c a l n u t a t i o n s c a u s e d b y t h e t e r m s p r o p o r t i o n a l t o ti~ a n d r~. W e d o n o t w r i t e d o w n t h e c l o s e d - f o r m a n a l y t i c a l s o l u t i o n o f (9) b e c a u s e o f it is cumbersome. H o w e v e r , the general analytical s o l u t i o n of(9) c a n easily b e a n a l y s e d n u m e r i c a l l y a n d t h e C A R S signal l i n e s h a p e c a n b e c a l c u l a t e d e x a c t l y f o r a r b i t r a r y v a l u e s o f g/G a n d T1/T r a t i o s . 2 W h e n gIG ~ 1 r a t h e r simple a n a l y t i c a l e x p r e s s i o n f o r ICARs(A3 c a n b e o b t a i n e d . I n this c a s e t h e s o l u t i o n o f t h e p r o b l e m is t o b e s e a r c h e d i n t h e f o r m o f a p e r t u r b a t i o n - t h e o r y e x p a n s i o n series b a s e d o n a s m a l l p a r a m e t e r g/G. A s s u m i n g t h a t [Q~[~ IQol, [~ol~ Ino]a n d l e a v i n g o n l y t h e t e r m s o f t h e first o r d e r o f smallness, i.e. p r o p o r t i o n a l t o 9/G, w e o b t a i n t h e f o l l o w i n g e q u a t i o n s

(1 lu

G/2 ~2

~2
9

2. The Homogeneous CARS Signal Lineshape Analysis
The calculations have s h o w n t h a t t h e general s o l u t i o n o f (9) a n d t h e p e r t u r b a t i o n t h e o r y s o l u t i o n (16) a r e t h e s a m e w h e n 9/G <~ . 1, g ~ 1. O F i g u r e 1 illustrates t h e influence o f t h e s a t u r a t i o n p a r a m e t e r G o n t h e l i n e s h a p e o f t h e C A R S signal in t h e case o f a h o m o g e n e o u s l y b r o a d e n e d R a m a n transition. T h e s p e c t r a l line is c l e a r l y s e e n t o h a v e a p u r e l y L o r e n t z i a n profile in the a b s e n s e o f p u m p w a v e s (G = 0). T h e i n c r e a s e o f G l e a d s t o t h e f o r m a t i o n o f a


Saturation Spectroscopy o f Coherent R a m a n Scattering
I

169

=

>.t.-

~z 0 . 5
bA t-Z 09 rr

~ 7 0.5

0 -5 -I.5

I

I

0 A'

1.5
h e h o m o g e n e o u s C A R S signal i o n effect b y i n d e p e n d e n t b i various values o f the saturation co~- f2)T2

o -5

- 2.5

0 A'

2.5

5

F i g . 1. T h e d e f o lineshape due to harmonic Raman p a r a m e t e r G: 0; 1

rmation of t the saturat p u m p i n g for ; 20 A ' = (co] -

F i g . 3. T h e l i n e s h a p e o f s a t u r a t e d h o m o g e n e o u s C A R S - s i g n a l spectral profile for v a r i o u s v a l u e s o f A: 1 A = 0 ; 2 A = l . T~/T2= IO, G = 2

0
< A=

A=0
o.~ ~=0

0.5
7

~
O'3

).-

13s

C)

0
-5 - 1.5 0 1.5 5

c) 0
A'

-10

-5

0

5

10

/t
F i g . 4. T h e l i n e s h a p e o f D o p p l e r - b r o a d e n e d s a t u r a t e d C A R S signal b a n d profile for v a r i o u s gas t e m p e r a t u r e s : 1 T ~ - 0 ( u = 0 ) ;

F i g . 2. T h e i l l u s t r a t i o n o f t h e d e p e n d e n c e o f t h e d i p w i d t h i n t h e s a t u r a t e d C A R S s i g n a l l i n e s h a p e u p o n t i m e T1 ( h o m o g e n e o u s l y b r o a d e n e d R a m a n t r a n s i t i o n ) : ?,= T1/T2= 1 a n d 10; G = 2

2 T---300K(v,0). T1/T2=10; G = 2

dip, the d e p t h o f w h i c h grows w i t h the increase o f G a n d reaches the m a x i m u m value at G = 2 . W i t h G g r o w s f u r t h e r , t h e d i p d e p t h lessens a n d t h e f r e q u e n c y i n t e r v a l b e t w e e n the t w o m a x i m a in the profile o f the /CARs(A') l i n e s h a p e i n c r e a s e s . A s o n e c a n s e e i n F i g . 2, t h e w i d t h o f t h e d i p is p r o p o r t i o n a l t o T1- a. I t m e a n s t h a t t h e d i p a p p e a r s d u e t o t h e p r o c e s s e s o f p o p u l a t i o n c h a n g e s , s i n c e T1 is n o t h i n g b u t the c h a r a c t e r i s t i c t i m e o f these changes ( c o m p a r e w i t h t h e r e s u l t s o f [6]). I n o u r case t h e oscillations o f t h e p o p u l a t i o n difference w i t h t h e f r e q u e n c y A - A ' [-terms w i t h no a n d ~7] i n (5)] m o d u l a t e t h e n e t p o p u l a t i o n difference w i t h t h e s a m e frequency. W h e n A' t e n d s t o A t h e f r e q u e n c y o f p o p u l a t i o n oscillations decreases a n d the index o f the m o d u l a t i o n i n c r e a s e s . A t A"= A w e o b t a i n t h e d e e p e s t h o l e in t h e h o m o g e n e o u s C A R S signal line profile, t h e w i d t h o f w h i c h is ~ T1-1. F i g u r e 3 illustrates the influence o f p u m p - w a v e d e t u n i n g .A o n t h e l i n e p r o f i l e o f t h e " s a t u r a t e d " C A R S signal. W i t h A d e t u n i n g o f f r e s o n a n c e t h e d i p shifts f r o m t h e c e n t e r o f t h e line.

3. The Doppler-Broadened Saturated CARS Lineshape Analysis
The case parti purp subs results o b t a i n e d a b o v e c a n be g e n e r a l i z e d t o t h e o f a n i n h o m o g e n e o u s l y b r o a d e n e d line a n d , in c u l a r , t o a D o p p l e r - b r o a d e n e d one. F o r this o s e it's necessary t o m a k e the following t i t u t i o n i n (9), o r (16):

A - - , A - ( k l - k 2 ) v T2, d'-(kl T2 ,

w h e r e ~ is a v e c t o r o f t h e t h e r m a l - m o t i o n v e l o c i t y o f a molecule u n d e r study, a n d t h e n to integrate the expression o b t a i n e d over o using the Maxwell d i s t r i b u t i o n o f m o l e c u l a r velocities. We d o n o t write o u t the general a n a l y t i c a l s o l u t i o n o b t a i n e d f o l l o w i n g t h i s s c h e m e b e c a u s e i t is c u m b e r s o m e , too. T h e results o f c o m p u t e r calculations for different values o f t h e r m a l velocities o f molecules a r e r e p r e s e n t e d i n F i g . 4. T h e o b v i o u s c o n s e q u e n c e o f t h e t h e r m a l - v e l o c i t y d i s t r i b u t i o n is t h e b r o a d e n i n g o f t h e s p e c t r a l line. T h e p r o f i l e o f t h e dip, h o w e v e r ,


170 c h a n g e s in a m o r e c o m p l i c a t e d m a n n e r : n o w it c o n s i s t s of the two holes (one inside a n o t h e r ) with characteristic w i d t h s p r o p o r t i o n a l t o T2-1 a n d T~-1, r e s p e c t i v e l y . T h u s the c o h e r e n t s a t u r a t i o n s p e c t r o s c o p y of R a m a n transitions provides a possibility for simultaneous m e a s u r e m e n t s o f b o t h T1 a n d T2 t i m e s . B o t h the existence a n d d e p t h o f the s a t u r a t i o n dip p r o p o r t i o n a l t o T:- : s t r o n g l y d e p e n d o n t h e g e o m e t r y o f the i n t e r a c t i o n o f t h e light waves. T h e p i c t u r e j u s t d r a w n ( F i g . 4) a r e r e l e v a n t f o r t h e c a s e o f c o - d i r e c t i o n a l p r o p a g a t i o n o f t h e p u m p i n g a n d p r o b i n g waves. I n t h e o p p o s i t e case o f c o u n t e r - p r o p a g a t i n g p u m p i n g a n d p r o b i n g waves the s a t u r a t i o n dip with the w i d t h -,~TI-* v a n i s h e s a n d t h a t w i t h t h e w i d t h ~ Tz- 1 s u r v i v e s . I t is t h e l a s t d i p t h a t h a s b e e n e x p e r i m e n t a l l y d e t e c t e d b y O w y o u n g a n d E s h e r i c k [7] in a D o p p l e r b r o a d e n e d Qo1(2) R a m a n l i n e i n d e u t e r i u m gas. H o w e v e r , they were u n a b l e to detect the m u c h n a r r o w e r s a t u r a t i o n d i p w i t h t h e w i d t h ,-~ T~- 1 i n a c o directional g e o m e t r y because of the lack of the needed very high spectral resolution. 4. D i s c u s s i o n T h e results o b t a i n e d a b o v e allow o n e to consider the c o h e r e n t s a t u r a t i o n s p e c t r o s c o p y as a m e t h o d f o r d e t e r m i n i n g t h e r e l a x a t i o n t i m e s T1 a n d T2. I n o r d e r t o e v a l u a t e the real p a r a m e t e r s o f lasers n e e d e d t o p e r f o r m t h e s a t u r a t i o n C A R S e x p e r i m e n t , let us c o n s i d e r t h e t y p i c a l R a m a n t r a n s i t i o n , t h e single r o t a t i o n a l t r a n s i t i o n So(l) in the h y d r o g e n gas (T1 = 0 , 1 ~ts, T 2 = 2 n s a t p = l a t m ) . A c c o r d i n g t o [83 t h e s a t u r a t i o n p a r a m e t e r v a l u e G ~ 10 c a n e a s i l y b e a c h i e v e d w h e n e x p e r i m e n t i n g w i t h t h e R a m a n l i n e . T h i s v a l u e G is m o r e t h a n e n o u g h t o o b s e r v e t h e d i s c u s s e d effects since t h e m a x i m u m d e p t h o f t h e h o l e o c c u r s a t G = 2 (in t h i s c a s e t h e l a s e r p o w e r s n e e d e d a r e a b o u t P 1 = 1 M W , P 2 = 100 k W ) . T h e w i d t h o f t h e d i p ~ T1-1 is Am ( F W H M ) ~ 0 . 0 0 0 2 c m - 1 a n d t h e h o m o g e n e o u s l i n e w i d t h is AC%o ( F W H M ) m 0 . 0 0 0 9 c m - 1 , w h i c h is p r o p o r t i o n a l t o T2-1. A t t h e s a m e t i m e t h e D o p p l e r w i d t h o f t h e l i n e is A~od = 0 . 0 0 5 c m - ~. T h e s p e c t r a l r e s o l u t i o n (less t h e n o r e q u a l t o 0 . 0 0 0 2 c m - 1) n e e d e d t o d e t e c t b o t h T1 a n d T2 f r o m t h e p r o f i l e o f t h e D o p p l e r - b r o a d e n e d S o ( l ) line in h y d r o g e n gas c a n b e o b t a i n e d w i t h t h e use o f a p u l s e d a m p l i f i e r c h a i n o f a s t a b i l i z e d cw d y e laser o u t p u t , developed b y Drell and C h u [9] and by O w y o u n g [ 7 , 103.

v . N . Zadkov et al. I t is c l e a r , h o w e v e r , t h a t t h e r e e x i s t s a n o p p o r t u n i t y o f using the t i m e - d o m a i n - s p e c t r o s c o p y technique instead o f t h e f r e q u e n c y - d o m a i n e o n e , as i t d o e s in e v e r y o t h e r n o n l i n e a r s p e c t r o s c o p y s c h e m e [33. I n this p a r t i c u l a r case o n e s h o u l d detect the d e p e n d e n c e o f t h e i n t e n s i t y o f t h e C A R S signal o n the d e l a y t i m e b e t w e e n s a t u r a t i n g a n d p r o b i n g pulses, t h e p e r i o d o f fast o s c i l l a t i o n s in this d e p e n d e n c e giving t h e r e l a x a t i o n t i m e T2 a n d t h e d e c a y t i m e g i v i n g t h e r e l a x a t i o n t i m e T1. F o r o u r e x a m p l e o f t h e S o ( l ) l i n e i n H 2 g a s t h e d e l a y t i m e n e e d e d s h o u l d b e less t h a n o r e q u a l t o ~ 1 gs, a n d t h e p u l s e d u r a t i o n s s h o u l d b e s h o r t e r t h a n o r e q u a l t o 100 ps. I n conclusion, we s h o u l d like to p o i n t o u t s o m e possible applications of the predicted s a t u r a t i o n dip w i t h t h e w i d t h ~ T1-1. T h e g a s cell w i t h t h e s a t u r a t e d R a m a n l i n e c a n b e u s e d i n s i d e t h e l a s e r c a v i t y as a f r e q u e n c y n a r r o w i n g a n d stabilizing e l e m e n t in the s a m e m a n n e r as a n o r d i n a r y L a m b - d i p s a t u r a t i o n cell is u s e d f o r t h e s a m e p u r p o s e s . T h e d i f f e r e n c e i s t h a t t h e w i d t h o f t h e R a m a n s a t u r a t i o n d i p ~ T1 1 c a n b e m a d e m u c h n a r r o w e r than the L a m b a b s o r p t i o n dip can ( ~ T f 1) i f t h e w i d t h s o f s a t u r a t i n g f i e l d s i n t h e R a m a n c e l l a r e s u f f i c i e n t l y n a r r o w ( A V l , 2 <~r 1-1).
Acknowledgements. The authors would like to gratefully acknowledge helpful discussions with S.A. Akhmanov, A.I. Burshtein, S. M. Gladkov, and S. Yu. Nikitin.

References 1. V . S . Letokhov, V . P . Chebotaev: Nonlinear Laser Spectroscopy, Springer S e r . Opt. Sci. 4 (Springer, Berlin, Heidelberg, New York 1975) 2. S . G . Rautian, G.I. Smirnov, A . M . Shalagin: Nonlinear Resonanses in the Spectra o f Atoms and Molecules (Nauka, Novosybirsk 1979) 3. S.A.Akhmanov, N.I. Koroteev: M e t h o d s o f Nonlinear Optics in Light Scattering Spectroscopy (Nauka, Moscow 1981) 4. A. Weber (ed.): Raman Spectroscopy o f Gases and Liquids, Topics Current Phys. 11 (Springer, Berlin, Heidelberg, New York 1979) 5. D.N. Kozlov, V.V. Smirnov, V.I. Fabelinskii: Soy. Phys. Dokl. 24, 369 (1979) 6. E.V. Baklanov, V.P. Chebotaev: S o v . Phys. JETP 60, 552 (1971) 7. A. Owyoung, P. Esherick: Opt. Lett. 5, 421 (1980) 8. A.M. Brodnikovsky, V . N . Zadkov, M.G. Karimov, N.I. Koroteev: Opt. Spektrosk. 54, 385 (1983) 9. P. Drell, S. Chu: Opt. Commun. 28, 343 (1979) 10. A. Owyoung: In Laser Spectroscopy IV, ed. by H. Walther, K.W. Rothe, Springer Ser. Opt. Sci. 21 (Springer, Berlin, Heidelberg, New York 1979) p. 175