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Transient threshold power of stimulated Brillouin Raman scattering
Volume 34A, number 6
PHYSICS LETTERS
TRANSIENT
THRESHOLD
POWER
RAMAN
OF
5April1971
STIMULATED
BRILLOUIN
SCATTERING
M. MAIER and G. RENNEH Physik-Department der Technischen Universit~t Miinchen, Germany Received 3March 1971 Measured threshold power values are in good agreement with a transient theory, even for laser pulses long compared to the relaxation time of the material excitation.
In r e c e n t y e a r s the steady state behavior of s t i m u l a t e d BriUouin (SBS) and R a m a n s c a t t e r i n g (SRS) in the s a t u r a t i o n r a n g e has been shown to be in excellent a g r e e m e n t with theory [1, 2]. The t r a n s i e n t onset of these p r o c e s s e s , however, has not yet r e c e i v e d detailed quantitative i n v e s t i g a tions. In this l e t t e r we r e p o r t on t i m e r e s o l v e d m e a s u r e m e n t s (resolution 0.Snsec) of the t h r e s hold power of SBS and SRS which were c a r r i e d out u s i n g a giant pulse ruby l a s e r [1] (pulse d u r a tion 17nsec) and a mode locked ruby l a s e r [2] (1.Tnsec). SRS was i n v e s t i g a t e d in H2-gas ( p r e s s u r e p ~ 50 arm) and SBS in N 2 - g a s (i0 = 70 to 100 arm), liquid CC14, acetone, ethyl alcohol, methyl alcohol, ethyl ether and H20. The e x p e r i m e n t a l s e t up has been d e s c r i b e d e l s e w h e r e [1, 3]. T y p i c a l o s c i l l o s c o p e t r a c e s of the i n c i d e n t (PL) and t r a n s m i t t e d (PT) l a s e r power and the B r i l louin power (PB) a r e shown in the i n s e r t of fig. 1 for ethyl alcohol. A s h a r p b r e a k i s o b s e r v e d in the t r a n s m i t t e d l a s e r power P T at the t i m e of the onset of SBS. When P L exceeds this power level, the t h r e s h o l d power Pth t [4], a steady state is r e a c h e d rapidly [1]. B r e a k s in the t r a n s m i t t e d power were found in a l a r g e v a r i e t y of s u b s t a n ces; the t i m e s of the break, tth, r a n g e d f r o m the beginning of the p u l s e to its m a x i m u m . An effective gain factor gt~ was d e t e r m i n e d f r o m the t h r e s h o l d power P t h in the following way. The b r e a k in the t r a n s m i t t e d l a s e r power o c c u r s when the s c a t t e r e d light power is amplified to a p p r o x i m a t e l y 1% of the l a s e r • power. F o r ex* ponential a m p l i f i c a t i o n the gain g t h P t h l / F n e c e s In previous time-integrated measurements of the Stokes light the threshold power was defined as the maximum power necessary to generate a definite amount of scattered light [5].
0
-10 0 -',10 TIME t (nsec)
i
o/
o//~
/~/ o ~>!O~oo+ o Oo ~r,o4~
U < i,
a v~
_z 2 0
~ e
o ~
•
_
_ 1
.
.
.
. 10
~ 100
GoC II-t o
Fig. 1. Normalized effective gain g/g*th as a function of GoC/I~to . GO is the maximum steady §~ate gain, I~to the normalized laser pulse duration and C ~ 1.2 (for -t o < tth < 0). SRS in H2-gas: • (giant pulse laser), O (mode locked laser); SBS in N2-gas: +70 arm, x80 atm, * 90 arm, ~ 100 arm;[] liquid CC14, Ilacetone, A ethyl alcohol, A ethyl ether, VH20 , (} methyl alcohol (all giant pulse laser). Insert: Oscilloscope pictures of the incident (PL) and transmitted (PT) laser power and the Brillouin power (PB) in ethyl alcohol. s a r y to obtain this s c a t t e r e d power is Gth. The value of Gth is e s t i m a t e d f r o m spontaneous s c a t t e r i n g data to be ~ 24 for B r i l l o u i n s c a t t e r i n g in the liquids and N2-gas , and ~ 33 for R a m a n s c a t t e r i n g in H2-gas. I is the length of the m e d i u m , F the a r e a of the l a s e r beam [3] (I/F ~ 3 × 104 cm-1). A s s u m i n g a G a u s s i a n t i m e dependence for the l a s e r p u l s e / L i t ) = Io exp {-(t/to)2 } we calculated an a p p r o x i m a t e , a n a l y t i c a l e x p r e s s i o n for the effective gain factor u s i n g a t r a n s i e n t theory [4, 6]. If r is the damping constant of the density f l u c t u 299
Volume 34A, number 6
PHYSICS L E T T E R S
ations or m o l e c u l a r v i b r a t i o n s , /So and I S the i n i t i a l and final value of the s c a t t e r e d light, r e spectively, then the t r a n s i e n t gain G is a p p r o x i mately given by (for -t o < t < 0)
G(t)
= In [ I s ( t , l )/Iso
X
] = (E(t)r /IL (t))
(1)
{(l+4gll2(t)~ 1/2 rE(t)
!
FE (t)
/
{
] - 1 +ln2 I"
Here g is the steady state gain factor and E the energy per unit a r e a of the l a s e r pulse at t i m e t, i.e. t
E(t)=
f IL(t')dt' . -oo
It is c l e a r l y seen f r o m eq. (1) that the r e l e v a n t p a r a m e t e r s for the a m p l i f i c a t i o n a r e the steady state gain gI L and the quantity FE/IL, the r a t i o of the l a s e r energy n o r m a l i z e d to the damping t i m e l/F to the i n s t a n t a n e o u s l a s e r i n t e n s i t y . It is i n t e r e s t i n g to c o n s i d e r the l i m i t i n g c a s e s of eq. (1): a) for glIL2/FE << 1, the steady state gain gill is obtained; b ) f o r glIL2/FE >> 1, we have the u s u a l t r a n s i e n t gain 2(glFE) 1/2 [q]. F r o m eq. (1) the effective gain factor g~h at the threshold (t = tth, IL(tth) =Ith), is d e r i v e d [41. F o r a G a u s s i a n l a s e r pulse we obtain g/g*th as a function of the n o r m a l i z e d pulse d u r a t i o n Fto and the m a x i m u m steady state gain Go = glol.
(2)
g = .G°C
rto
Here C is given by
Io Eth
300
l I,+er,
II,
\ to-J}
5 April 1971
where erf stands for the e r r o r function. If the threshold intensity I t h is found before the pulse m a x i m u m (in the r a n g e - t o < t < 0), the p a r a m e t e r C is approximately equal to 1.2. In this region a c o m p a r i s o n of the approximate solution (2) with n u m e r i c a l r e s u l t s obtained from an exact theory [7] shows a g r e e m e n t within a few p e r cent. This is seen from fig. 1 where g/g*th is plotted as a function of GoC/Ft o (solid line f r o m eq. (2), broken line f r o m n u m e r i c a l c a l c u l a t i o n s a c c o r d i n g to ref. [7]). The e x p e r i m e n t a l points c o r r e s p o n d to SRS in H2-gas and to SBS in N 2gas and a l a r g e n u m b e r of liquids. T h e r e is good a g r e e m e n t between theory and e x p e r i m e n t s over the e n t i r e r a n g e of GoC/Ft o values (three o r d e r s of magnitude). Fig. 1 shows c l e a r l y that g/g*th r i s e s with GoC/Fto, i.e. with i n c r e a s i n g steady state gain Go and d e c r e a s i n g n o r m a l i z e d pulse duration F t o the p r o c e s s b e c o m e s m o r e t r a n s i e n t . It should be e m p h a s i z e d that even for light pulses long c o m p a r e d to the damping time of the m a t e r i al excitation the t h r e s h o l d power for s t i m u l a t e d s c a t t e r i n g is d e t e r m i n e d by t r a n s i e n t behavior, e.g. for SBS in liquids we have F t o = 10 to 25 (CVo/rt o = 5 to 20) and g/g*th ~ 2.5. The authors would like to thank P r o f e s s o r W. K a i s e r and K. Darde for valuable d i s c u s s i o n s .
References [1] M. Maier, Phys. Rev. 166 (1968) 113. [2] D. yon der Linde, M. Maier and W. Kaiser, Phys. Rev. 178 (1969) 11. [3] M. Maier and G. Renner, to be published. [4] K. Dar6e and W. Kaiser, Phys. Rev. Letters to be published. [5] E. E. Hagenlocker, R.W. Minck and W. G. Rado, Phys. Rev. 154 (1967) 226; W. Heinicke and G. Winterling, App!. Phys. Letters 11 (1967) 231. [6] W. Rother, Z.Naturforsehung 25a (1970) 1120. [7] R. L. Carman, F. Shimizu, C.S. Wang and N. Bloembergen, Phys. Rev. A2 (1970) 60.
PHYSICS LETTERS
TRANSIENT
THRESHOLD
POWER
RAMAN
OF
5April1971
STIMULATED
BRILLOUIN
SCATTERING
M. MAIER and G. RENNEH Physik-Department der Technischen Universit~t Miinchen, Germany Received 3March 1971 Measured threshold power values are in good agreement with a transient theory, even for laser pulses long compared to the relaxation time of the material excitation.
In r e c e n t y e a r s the steady state behavior of s t i m u l a t e d BriUouin (SBS) and R a m a n s c a t t e r i n g (SRS) in the s a t u r a t i o n r a n g e has been shown to be in excellent a g r e e m e n t with theory [1, 2]. The t r a n s i e n t onset of these p r o c e s s e s , however, has not yet r e c e i v e d detailed quantitative i n v e s t i g a tions. In this l e t t e r we r e p o r t on t i m e r e s o l v e d m e a s u r e m e n t s (resolution 0.Snsec) of the t h r e s hold power of SBS and SRS which were c a r r i e d out u s i n g a giant pulse ruby l a s e r [1] (pulse d u r a tion 17nsec) and a mode locked ruby l a s e r [2] (1.Tnsec). SRS was i n v e s t i g a t e d in H2-gas ( p r e s s u r e p ~ 50 arm) and SBS in N 2 - g a s (i0 = 70 to 100 arm), liquid CC14, acetone, ethyl alcohol, methyl alcohol, ethyl ether and H20. The e x p e r i m e n t a l s e t up has been d e s c r i b e d e l s e w h e r e [1, 3]. T y p i c a l o s c i l l o s c o p e t r a c e s of the i n c i d e n t (PL) and t r a n s m i t t e d (PT) l a s e r power and the B r i l louin power (PB) a r e shown in the i n s e r t of fig. 1 for ethyl alcohol. A s h a r p b r e a k i s o b s e r v e d in the t r a n s m i t t e d l a s e r power P T at the t i m e of the onset of SBS. When P L exceeds this power level, the t h r e s h o l d power Pth t [4], a steady state is r e a c h e d rapidly [1]. B r e a k s in the t r a n s m i t t e d power were found in a l a r g e v a r i e t y of s u b s t a n ces; the t i m e s of the break, tth, r a n g e d f r o m the beginning of the p u l s e to its m a x i m u m . An effective gain factor gt~ was d e t e r m i n e d f r o m the t h r e s h o l d power P t h in the following way. The b r e a k in the t r a n s m i t t e d l a s e r power o c c u r s when the s c a t t e r e d light power is amplified to a p p r o x i m a t e l y 1% of the l a s e r • power. F o r ex* ponential a m p l i f i c a t i o n the gain g t h P t h l / F n e c e s In previous time-integrated measurements of the Stokes light the threshold power was defined as the maximum power necessary to generate a definite amount of scattered light [5].
0
-10 0 -',10 TIME t (nsec)
i
o/
o//~
/~/ o ~>!O~oo+ o Oo ~r,o4~
U < i,
a v~
_z 2 0
~ e
o ~
•
_
_ 1
.
.
.
. 10
~ 100
GoC II-t o
Fig. 1. Normalized effective gain g/g*th as a function of GoC/I~to . GO is the maximum steady §~ate gain, I~to the normalized laser pulse duration and C ~ 1.2 (for -t o < tth < 0). SRS in H2-gas: • (giant pulse laser), O (mode locked laser); SBS in N2-gas: +70 arm, x80 atm, * 90 arm, ~ 100 arm;[] liquid CC14, Ilacetone, A ethyl alcohol, A ethyl ether, VH20 , (} methyl alcohol (all giant pulse laser). Insert: Oscilloscope pictures of the incident (PL) and transmitted (PT) laser power and the Brillouin power (PB) in ethyl alcohol. s a r y to obtain this s c a t t e r e d power is Gth. The value of Gth is e s t i m a t e d f r o m spontaneous s c a t t e r i n g data to be ~ 24 for B r i l l o u i n s c a t t e r i n g in the liquids and N2-gas , and ~ 33 for R a m a n s c a t t e r i n g in H2-gas. I is the length of the m e d i u m , F the a r e a of the l a s e r beam [3] (I/F ~ 3 × 104 cm-1). A s s u m i n g a G a u s s i a n t i m e dependence for the l a s e r p u l s e / L i t ) = Io exp {-(t/to)2 } we calculated an a p p r o x i m a t e , a n a l y t i c a l e x p r e s s i o n for the effective gain factor u s i n g a t r a n s i e n t theory [4, 6]. If r is the damping constant of the density f l u c t u 299
Volume 34A, number 6
PHYSICS L E T T E R S
ations or m o l e c u l a r v i b r a t i o n s , /So and I S the i n i t i a l and final value of the s c a t t e r e d light, r e spectively, then the t r a n s i e n t gain G is a p p r o x i mately given by (for -t o < t < 0)
G(t)
= In [ I s ( t , l )/Iso
X
] = (E(t)r /IL (t))
(1)
{(l+4gll2(t)~ 1/2 rE(t)
!
FE (t)
/
{
] - 1 +ln2 I"
Here g is the steady state gain factor and E the energy per unit a r e a of the l a s e r pulse at t i m e t, i.e. t
E(t)=
f IL(t')dt' . -oo
It is c l e a r l y seen f r o m eq. (1) that the r e l e v a n t p a r a m e t e r s for the a m p l i f i c a t i o n a r e the steady state gain gI L and the quantity FE/IL, the r a t i o of the l a s e r energy n o r m a l i z e d to the damping t i m e l/F to the i n s t a n t a n e o u s l a s e r i n t e n s i t y . It is i n t e r e s t i n g to c o n s i d e r the l i m i t i n g c a s e s of eq. (1): a) for glIL2/FE << 1, the steady state gain gill is obtained; b ) f o r glIL2/FE >> 1, we have the u s u a l t r a n s i e n t gain 2(glFE) 1/2 [q]. F r o m eq. (1) the effective gain factor g~h at the threshold (t = tth, IL(tth) =Ith), is d e r i v e d [41. F o r a G a u s s i a n l a s e r pulse we obtain g/g*th as a function of the n o r m a l i z e d pulse d u r a t i o n Fto and the m a x i m u m steady state gain Go = glol.
(2)
g = .G°C
rto
Here C is given by
Io Eth
300
l I,+er,
II,
\ to-J}
5 April 1971
where erf stands for the e r r o r function. If the threshold intensity I t h is found before the pulse m a x i m u m (in the r a n g e - t o < t < 0), the p a r a m e t e r C is approximately equal to 1.2. In this region a c o m p a r i s o n of the approximate solution (2) with n u m e r i c a l r e s u l t s obtained from an exact theory [7] shows a g r e e m e n t within a few p e r cent. This is seen from fig. 1 where g/g*th is plotted as a function of GoC/Ft o (solid line f r o m eq. (2), broken line f r o m n u m e r i c a l c a l c u l a t i o n s a c c o r d i n g to ref. [7]). The e x p e r i m e n t a l points c o r r e s p o n d to SRS in H2-gas and to SBS in N 2gas and a l a r g e n u m b e r of liquids. T h e r e is good a g r e e m e n t between theory and e x p e r i m e n t s over the e n t i r e r a n g e of GoC/Ft o values (three o r d e r s of magnitude). Fig. 1 shows c l e a r l y that g/g*th r i s e s with GoC/Fto, i.e. with i n c r e a s i n g steady state gain Go and d e c r e a s i n g n o r m a l i z e d pulse duration F t o the p r o c e s s b e c o m e s m o r e t r a n s i e n t . It should be e m p h a s i z e d that even for light pulses long c o m p a r e d to the damping time of the m a t e r i al excitation the t h r e s h o l d power for s t i m u l a t e d s c a t t e r i n g is d e t e r m i n e d by t r a n s i e n t behavior, e.g. for SBS in liquids we have F t o = 10 to 25 (CVo/rt o = 5 to 20) and g/g*th ~ 2.5. The authors would like to thank P r o f e s s o r W. K a i s e r and K. Darde for valuable d i s c u s s i o n s .
References [1] M. Maier, Phys. Rev. 166 (1968) 113. [2] D. yon der Linde, M. Maier and W. Kaiser, Phys. Rev. 178 (1969) 11. [3] M. Maier and G. Renner, to be published. [4] K. Dar6e and W. Kaiser, Phys. Rev. Letters to be published. [5] E. E. Hagenlocker, R.W. Minck and W. G. Rado, Phys. Rev. 154 (1967) 226; W. Heinicke and G. Winterling, App!. Phys. Letters 11 (1967) 231. [6] W. Rother, Z.Naturforsehung 25a (1970) 1120. [7] R. L. Carman, F. Shimizu, C.S. Wang and N. Bloembergen, Phys. Rev. A2 (1970) 60.