Download 11. Stimulated Brillouin Scattering

Stimulated Brillouin Scattering
Interaction of light wave with acoustic waves
A light wave interacting with an acoustic wave through the refractive index (dielectric
constant) moving grating set up by the acoustics.
k1  k2
k1
ks
k2
Moving strain
grating
k1
ks
k2
ks
k1
k2
The acoustic waves set up a displacement u in the media. The strain is the time derivative of
the displacement with position.
The Electromagnetic wave equation
 2 Ei (r , t )  
2
2
E
(
r
,
t
)


( PNL ) i
i
t 2
t 2
For Brillouin scattering, the nonlinear polarization source is cause by the changes in dielectric
constant by the traveling acoustic waves. The acoustic waves are in turn driven by the beat
waves of the electric field of light.
Acoutic waves driven by electromagnetic waves
The change in the dielectric constant caused by
u
the strain
is
x
u
(1)
x
where  is a constant quantifying the changes in
dielectric constant..
Strain 
u
x
  
U(x1)
The changes in electrostatic energy density is
1 u 2
given by  
E . When the field is varying,
2 x
the enegy density changes. The electrostrictive
pressure associated with the energy change is the work divided by the strain.
1
p   E 2
2
Thus the force applied to a unit volume is the gradient of pressure or
 E 2
F
2 x
The equation of motion for acoustic waves driven by a force is given by
 2u
u
 2u   2
 2   T 2 
E
t
2 x
t
x
Where T and  are the elastic constant and mass density.
The speed of acoustic waves is vs =
T

How to understand equation (4)
Two electric fields and acoustic field are in the form of plane waves:
U(x2)
x
(2)
(3)
(4)
1
E1 (r1 )e j (1t  k1 r )  cc
2
1
E2 (r , t )  E1 (r2 )e j ( 2 t  k 2 r )  cc
2
1
u (r , t )  us (rs )e j ( s t  k s r )  cc
2
E1 (r , t ) 
(5)
(6)
(7)
The r’s are the distance measured along the direction of propagation. The Laplacians are
simply the second order derivatives along the r’s.
Substituting the waveforms into (4), and neglecting the second order derivatives of the
acoustic filed for slow-varying with position.

 2
du s  j (st ks r )
2

 e
 cc
( j s  s )u s  T  k s u s  2 jk s
dr
s




  
 

E2 (r2 ) E1* (r1 )e j (w 1 )t ( k2 k1 )r   cc
8 rs

(8)
The phase factors on two sides must be the same in order to have any meaningful effect
 s  2  1

 
k s  k 2  k1
(9)

( E 2 E1* )  k s E 2 E1* ( slow varying
rs
Eq (8) can be simplified as
With these constraints, and the assumption that
within one wavelength of acoustic waves,
du s (rs )  2 2
j s 
jk s
u s (rs )  
  k s v s   s2 
E 2 (r2 ) E1* (r1 )
(10)
drs

8



This relation governs the development of the acoutics waves with a source that is driven by
electro-magnetic waves.
2ik s s2
Electromagnetic wave equation driven by acoustic waves
 2 Ei (r , t )  
2
2
E
(
r
,
t
)


( PNL ) i
i
t 2
t 2
The nonlinear polarization caused by the acoustic wave is given by , from (1)
(11)
PNL i   E  E (r, t ) u(r, t )
(12)
ri
`
From (11) and using slow-varying envelope for the electric field along the propagation
direction of a plane wave
 dE1 (r1 )  j (1t k1 r )
2
 cc  j 2 PNL i
k1
e
dr
t
1


(13)
Combining (12) and (13) and by choosing terms that satisfies conservation of momentum and
energy:
 dE1 (r1 )  j (1t k1 r )
 2 
 *  j (  s t  k s r
  E 2 e j (2t k2 r )
 j
us e
k1
e
2 
dr
4

r

t
1
s





i
(13)
This equation govens the generation of E1 by the interation of E2 with acoustic wave u.
For  s  2  1 , Eq (13) can be simplified as
du s*
dE1
j12
*
k1

E 2 ( jk s u 
)
(14)
dr1
4
drs
For acoustic wave whose amplitude do not change quickly, the derivative with respect to rs
can be neglected,
dE1  12

E2 k s u *
dr1
4k1
(15)