Т К А 5 ^ .
KFKI-1985-76
G . P . D J O T Y A N / J 1S ■ B A K O S , Т . J U H Á S Z
GENERATION OF FREQUENCY TUNED PICOSECOND PULSES BY FOUR-WAVE MIXING
H u n g a ria n A cadem y o f S c ie n c e s C E N T R A L
R E S E A R C H
I N S T I T U T E F O R P H Y S I C S
B U D A P E S T
2017
KFKI-1985-76 PREPRINT
GENERATION OF FREQUENCY TUNED PICOSECOND PULSES BY FOUR-WAVE MIXING
G.P. DJOTYAN*, J.S. BAKOS, T. JUHÁSZ**
Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary
*Institute for Physics of Condensed Matter University of Erevan of Armenian S.S.R.
*‘Technical University of Budapest,
Institute of Physics, Department of Experimental Physics H-1521 Budapest, XI. Budafoki ut. 8.
Submitted to Appl. Phys. B.
HU ISSN 0368 5330
ABSTRACT
The parametric nonstationary four wave phase conjugation process is investigated in nonlinear media without absorption and inertia. The reflected phase conjugated pulse is shortened by shortening the duration of one of the reference pulse propagating in the same direction. In noncollinear scheme of four wave phase conjugation the frequency of the reflected short pulse can be tuned by tuning the frequency of the signal wave of much longer duration than the one of the reflected and the reference short pulse beam.
АННОТАЦИЯ
Исследован нестационарный режим четырехволнового параметрического обра
щения волнового фронта /ОВФ/ в безинерционной непоглощающей нелинейной среде.
Показана возможность укорочения импульса обращенной волны за счет укорочения импульса попутной опорной волны. При этом в неколлинеарной схеме невырожденно
го режима ОВФ предложен способ перестройки частоты генерируемого короткого им
пульса обращенной волны за счет перестройки частоты сигнальной волны, имеющей существенно ббльшую длительность импульса.
KIVONAT
Abszorpció és tehetetlenség nélküli nemlineáris közegben megvizsgáljuk a parametrikus nemstacionarius négyhullámu keverés folyamatát. A reflektált fáziskonjugált hullám követi a legrövidebb impulzus alakját, a frekvenciája pedig a beeső "jel" hullám frekvenciáját. így hangolható pikoszekundumos impulzusok állíthatók elő.
1. I N T R O D U C T I O N
T h e s c h e m e of p h a s e c o n j u g a t i o n b y f o u r w a v e m i x i n g is w e l l k n o w n [1]. T h e s i g n a l w a v e is r e f l e c t e d w i t h p h a s e c o n j u g a t i o n if it is m i x e d w i t h two r e f e r e n c e w a v e s p r o p a g a t i n g in o p p o s i t e to direction to each other in t h e n o n l i n e a r m a t e r i a l . T h i s p r o c e s s c a n
b e t h e o r e t i c a l l y e a s i l y d e s c r i b e d if the i n t e r a c t i n g w a v e s are s t a t i o n a r y s i n u s o i d a l w a v e s .
T h e p r o c e s s of f o u r - w a v e m i x i n g is i n v e s t i g a t e d f o r the c a s e of t h e i n t e r a c t i n g b e a m s a r e p u l s e s of d i f f e r e n t d u r a t i o n and d i f f e r e n t f r e q u e n c i e s (see a l s o [2,3]). It c a n b e fo u n d , t h a t t h e r e f l e c t e d p u l s e c a n be s h o r t e n e d b y s h o r t e n i n g o n e p u l s e of the r e f e r e n c e b e a m a n d t h a t t h e f r e q u e n c y of t h e r e f l e c t e d b e a m c a n b e t u n e d b y t u n i n g the f r e q u e n c y o f t h e s i g n a l b e a m .
2. COLLI NEAR B E A M S
T h e s c h e m e of the i n t e r a c t i n g b e a m c a n b e s e e n i n Fig. 1.
It is s u p p o s e d t h a t t h e m e d i u m is t r a n s p a r e n t at f r e q u e n c i e s of the b e a m s a n d t h e r e l a x a t i o n t i m e o f the m e d i u m is s h o r t e r t h a n t h e d u r a t i o n o f the s h o r t e s t p u l s e in the i n t e r a c t i o n . T h e e l e c t r i c a l f i e l d s t r e n g t h of t h e f o u r i n t e r a c t i n g w a v e t a k e s the f o l l o w i n g f o r m
1 •» , . ч i(k.r-cot)
= 2- j A j (£ ' t)e 3 - + c . c w h e r e
E . -3
2
T h e a m p l i t u d e o f t h e r e f e r e n c e waves, t h e s i g n a l a n d the r e f l e c t ed w a v e s a r e A ^ ( w ^ ) , a n d А^(о)^) r e s p e c t i v e l y , e j , к ^ are t h e p o l a r i s a t i o n v e c t o r , t h e w a v e v e c t o r and the f r e q u e n c i e s . T h e p o l a r i s a t i o n of the w a v e s are t h e same a n d A ^ , A2, A ^ a r e g i v e n at the b e g i n n i n g . T h e n t h e r e f l e c t e d w a v e a m p l i t u d e f u l l f i l s t h e e q u a t i o n
i y ei A k
z z
) -A*(t
(1 )
in t h e a p p r o x i m a t i o n of t h e s l o w l y v a r y i n g a m p l i t u d e .
V j (j = l , 2 , 3,4) a r e t h e g r o u p v e l o c i t i e s of the w a v e s w h e r e v ^ = V2=v.
у is t h e f a c t o r c o n t a i n i n g t h e s u s c e p t i b i l i t y of t h i r d o r d e r of t h e n o n l i n e a r m a t e r i a l . Лк is t h e p r o j e c t i o n o f t h e p h a s e
z
m i s m a t c h Ak = (^3+ ^4) o n the z axi s , w h i c h is the d i r e c t i o n of p r o p a g a t i o n o f t h e b e a m s . If ш1= ш2= ш ' 0)3=01+6 a n d o>^=oi-6, Лк=
= - ( к з + к ^ ) . T h e m i s m a t c h is m i n i m u m if t h e d i r e c t i o n of p r o p a g a t i o n o f t h e s i g n a l w a v e is o p p o s i t e t o t h e p r o p a g a t i o n d i r e c t i o n o f t h e r e f l e c t e d w a v e i.e.
Дк = Ak = k.-k-, = —— —n (u>- 6) - n(o>+6)
z 4 3 c c
w h e r e n(o>) is t h e i n d e x o f r e f r a c t i o n o f the m a t e r i a l . T h e s o l u t i o n o f t h e e q u a t i o n (1) a t z = 0 (see Fig. 1) h a s t h e form
A A
2
Fig. 1,
Collinear scheme of nondegenerated parametric four wave phase conjugation
3
- , - .. . _ , iAk z'-A, ( t - z7 (^+^— ) ) -A* (t-z' (— +— ) ) A . ( z = 0 , t ) = i y A - (t )Jdz r z 1' v v. 3 V v 4 v.
(2) w h e r e
(- - — )L << T V v 4
w a s s u p p o s e d . L is t h e l e n g t h o f the n o n l i n e a r m a t e r i a l a n d т is the w i d t h of the p u l s e A^. In t h e f o l l o w i n g , t a k e t h e a m p l i t u d e o f the b e a m s A ^ = A^. If т is m u c h les s t h a n the d u r a t i o n o f the s i g n a l (t< <t^) and t h e p r o p a g a t i o n t i m e b a c k and f o r t h t h r o u g h the m e d i u m , t h e n the d u r a t i o n o f the r e f l e c t e d p u l s e x^ is r o u g h l y
e q u a l to . Let us investigate this effect in more detail* Suppose
- as an e x a m p l e - t h a t the i n t e r a c t i n g w a v e s are o f G a u s s i a n t y p e
2 2
A ^ 2 = A ^ 2e x p ( — 2~) ; A3(t) = e x p ( 2---- ) (3)
T x3
At is t h e t i m e lag b e t w e e n t h e s i g n a l (A3 ) a n d t h e r e f e r e n c e (A. 9 ) beams. T h e i n t e n s i t y o f t h e r e f l e c t e d w a v e I .(o,t) =
1. r z 2 ^
= I A ^ (o ,t ) I is g i v e n a c c o r d i n g to the e x p r e s s i o n (2) as
I 4 (o,t) =
jl L
32 10*X20 30 ■ v (1-5)
■ exp<
2 2
4t 4At
2 2
T T 3
2 2 2
■г -Дк -v*
16 (1-5) f>
erf </ 2 T
2L ^ i v A k -т , 5,2L
— t - --- + xr l— - t +
v n гк 2 v
8/2
v Ak • x + 2 • At + i - ^ --- )
4/2"
“
> + erf< /2
— i
X _
. v- Ak* x ‘
■l--- 8 / 2
, 5 ,. о,,. .vAk* x , + -^(t-2At - x--- ) 2
2 4/2"
(41
4
w h e r e
о x -+-2
erf(X) = — / e dt;
/тт о
It c a n b e s e e n f r o m e x p r e s s i o n (4) t h a t t h e i n t e n s i t y of the r e f l e c t e d w a v e d e c r e a s e s w i t h i n c r e a s i n g m i s m a t c h Дк in the c a s e o f the f o u r w a v e m i x i n g d i s c u s s e d . T h e b a n d w i d t h o f t h e p o s s i b l e f r e q u e n c y t u n i n g c a n b e g i v e n a s
5 = << 1
Aw c
n (w +6) + n (w — <$) 1
T • V (5)
vdiich m e a n s t h a t t h e b a n d w i d t h o f f r e q u e n c y t u n i n g is a b o u t t h e s a m e as t h e b a n d w i d t h o f the G a u s s i a n r e f e r e n c e b e a m o f w i d t h т .
In t h e c a s e of " s h o r t " n o n l i n e a r m e d i u m i.e. if t>^— t h e2L e x p r e s s i o n (4) is t h e s a m e as t h a t w e l l k n o w n e x p r e s s i o n g i v e n f o r the q u a s i t a t i o n a r y s c a t t e r i n g .
In t h e o p p o s i t e c a s e of " l o n g " n o n l i n e a r m e d i u m i.e. if 2L
t < < — w e c a n g e t f r o m e x p r e s s i o n . (4)
T4 ( ° »fc) = ~у2 I Y I ' I i o ’ I2 0* I 3 0 T *V
exp< 4t 4 At'
T 3
2 2 2 T •Дк *v
16 (1-5) (6)
e r f </2
T
2L v
1Хд к ^ _ + S (2L+2.at+iy: tk^T2
8/2 2 v 4/2
It c a n b e s e e n f r o m e x p r e s s i o n (6) t h a t the p u l s e w i d t h o f t h e r e f l e c t e d w a v e is t h e s a m e as t h a t o f t h e r e f e r e n c e w a v e t.
12 -12
T h e b a n d w i d t h o f f r e q u e n c y t u n i n g A w - l O H z if t= 1 0 sec.
C o n s e q u e n t l y - in c a s e of t h e c o l l i n e a r f o u r w a v e m i x i n g d i s c u s s e d - it is p o s s i b l e to g e n e r a t e r e f l e c t e d p u l s e o f p i c o s e c o n d d u r a t i o n a n d w i t h c o n j u g a t e d w a v e front. B u t the r a n g e of f r e q u e n c y t u n i n g is v e r y r e s t r i c t e d . T o w i d e n the t u n i n g r a n g e l e t u s i n v e s t i g a t e t h e c a s e of n o n c o l l i n e a r f o u r w a v e m i x i n g .
5 3. N O N C O L L I N E A R F O U R W A V E M I X I N G
T h e g e o m e t r y of n o n c o l l i n e a r f o u r - w a v e m i x i n g is g i v e n in Fig. 2. H e r e w e h a v e s u p p o s e d t h a t w a v e v e c t o r s o f t h e i n t e r a c t i n g b e a m s a r e in t h e XZ plane. T h e s o l u t i o n of the e q u a t i o n (1) is g i v e n in tha t c a s e as
Fig. 2.
Noncottinear scheme of nondegenerated para
metric four wave phase conjugation process.
h.=h.2+h-2=-3+--4
A4( z = o , x , t ) = i y / A 1 <x. x s i n ( ß + a /2) „ ”1 Lcos ( ß+a/2 )~| .
v C
л] <J
■A2^
t >x s i n ( ß - g /2) ; V
1 _ c o s (ß-ot/2)
>" A 3 ^
t x s i n g 1 + cosß
-v 4 V 3 J
In case o f i n t e r a c t i n g b e a m s o f G a u s s i a n t y p e g i v e n b y e x p r e s s i o n (3) a n d in the m o s t i n t e r e s t i n g c a s e o f " l o n g " n o n -
2L
l i n e a r m e d i u m (t<<— ) t h e i n t e n s i t y o f the r e f l e c t e d w a v e is g i v e n as
6
I4(o,X,t)=|Y , i •i «I • — .T2-v2 10 20 30 32 T
exp<
8(t-x/v4•(ß+a/2•v 4/v))
(7)
e x p í- m , « 4 r, 2 /2 * L4
e r f ( - --- — )
T • v
a and ß are the a n g l e s b e t w e e n t h e v e c t o r s k^, k^ a n d k 3 , k4 r e s p e c t i v e l y (see Fig. 2) a n d
f(t,x) = <-
(v-v3 )
~2vZ t - x
2vv. v(ß- •2 ) “v 3(6+ 2) + A t r
F u r t h e r m o r e v 4= v w a s a l s o s u p p o s e d . T h e r e l a t i o n b e t w e e n the a n g l e a a n d ß is g i v e n b y the p h a s e s y n c h r o n i s m Лк=0 f r o m w h e r e
. a /. 2 ß Штг = + /Sin тг s i n
2 -
2,6.
c o s (-^) (8)
F o r s i m p l i c i t y t h e d i s p e r s i o n o f t h e m e d i u m is n e g l e c t e d in the r a n g e ы + б .
If the f r e q u e n c y o f f s e t 6 is c h a n g e d i.e. the f r e q u e n c y of the s i g n a l b e a m u s t u n e d (ш+б) a n d t h e a n g l e b e t w e e n t h e s i g n a l a n d the r e f l e c t e d b e a m is n o t c h a n g e d (3= c o n s t . ) the a n g l e b e t w e e n the r e f e r e n c e b e a m s has to b e c h a n g e d a c c o r d i n g to the e x p r e s s i o n (8). C o n s e q u e n t l y the f r e q u e n c y of the p i c o s e c o n d r e f l e c t e d b e a m c a n be t u n e d in t h e w h o l e r a n g e (— <1) o f the
0) f r e q u e n c y t u n i n g o f t h e s i g n a l beam.
If the s p e c t r u m of the s i g n a l b e a m is w i d e the f r e q u e n c y of the p i c o s e c o n d r e f l e c t e d b e a m c a n b e t u n e d b y c h a n g i n g t h e a n g l e
6(6) a c c o r d i n g to t h e e x p r e s s i o n
s m 1 = . 2 a
- v s m % - A2 / 2
б /Ш
(9)
7
g o t t e n f rom e x p r e s s i o n (8) s u p p o s i n g a = c o n s t . H e r e . a . . a
- a i s i n ^ < о -íüJSin-^
is a l s o s u p p o s e d .
4. C O N C L U S I O N
T h e s t u d y o f t h e n o n s t a t i o n a r y n o n d e g e n e r a t e f o u r w a v e m i x i n g p r o c e s s sho w s t h a t it is p o s s i b l e to g e t p i c o s e c o n d p u l s e s w i t h t u n a b l e f r e q u e n c y b y t u n i n g the f r e q u e n c y o f c o m p a r a t i v e l y long
(nanosecond) p u l s e of t h e s i g n a l beam. M o r e o v e r t h e w a v e f r o n t o f t h i s p i c o s e c o n d " r e f l e c t e d " p u l s e s in t h e fou r w a v e m i x i n g p r o c e s s is conj u g a t e d in r e s p e c t to the s i g n a l w a v e of t h e "long"
p u l s e .
R E F E R E N C E S
[1] A. Y a r i v , D.M. P e p p e r , Opt. L e t t . , 1, 16 (1977) [2] W.W. R i g r o d , R.A. F i s h e r , B.I. F e l d m a n , Opt. L e t t . ,
5, 105 (1980)
[3] B.Ya. Z e l ' d o v i c h , M.A. O r l o v a , V.V. S h k u n o v , Doki. Akad.
N a u k S S S R 252, 592 (1980)
I* г
4
CS. 7 7 t
Ц i
•f
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Budapest, 1985. augusztus hó
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