Download Raman-Nath diffraction

Chapter 12. Interaction of Light and Sound
12.0 Introduction
Acousto-Optic(AO) effect :
# Effect of change in the index of refraction of medium (crystal)
by an Acoustic wave
# Acoustic wave  Photoelastic effect  Change in refractive index
Reference :
A. Ghatak, K. Thyagarajam, “Optical Electronics”, Cambridge Univ. Press
A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons
Nonlinear Optics Lab.
Hanyang Univ.
Photoelastic effect
: Mechanical strain  Index of refraction
x2 y2 z 2
Index ellipsoid for Principal axes : 2  2  2  1
n1 n2 n3
Strain tensor elements, S :
S xx  u
x
, S yy  v

y

, S zz  w
S xy   u   v
 S yx

y

x


S yz  v   w   S zy
z  y 
S xz  u  w  S zx
z
x





z
S1  S xx , S 2  S yy , S3  S zz : Normal strain
S4  S yz , S5  S zx , S6  S xy : Shear strain
where, u, v, w : displacements along the x, y, z axes
Nonlinear Optics Lab.
Hanyang Univ.
Change in index of refraction due to the mechanical strain :
6
1
( 2 )i  pij S j (i1,,6)
n
j 1
where, Pij : Elasto-Optic (Strain Optic) Coefficient (6x6 matrix)  Table 9.1 / 9.2
The equation of the index ellipsoid in the presence of a strain field :
 1
 2 1
 2 1






x  2  pij S j  y  2  p2 j S j  z  2  p3 j S j 
 n1 j

 n2 j
  n3 j

 2 yz  p4 j S j  2 xz p5 j S j  2 xy p6 j S j 1
2
j
j
j
Nonlinear Optics Lab.
Hanyang Univ.
Example) Sound wave propagating along the z direction in water
Sound wave : w( z , t ) Azˆcos(t  Kz )
S1 S xx u/x0, S 2 S yy v/y0, S3 S zz w/z KAsin( t Kz)Ssin( t Kz)
S 4 S yz (v/z )(w/y)0, S5 S zx (u/z )(w/x)0, S6 S xy (u/y)(v/x)0
Elasto-Optic Coefficient for the water (isotropic, Table 9.1) :
 p11
p
 12
 p12

pij  0

0


0

p12
p11
p12
p12
p12
p11
0
0
0
0
0
0
0
0
1
( p11  p12 )
2
0
0
0
0
1
( p11  p12 )
2
0
0
0
0





0



0


1
( p11  p12 )
2

0
0
0
1
1
)


(
) 2  p12 Ssin( t  Kz),
1
n2
n2
1
( 2 )3  p11Ssin( t  Kz),
n
1
( 2 ) 4,5, 6 0
n
 (
The new index ellipsoid :
1

1

1

x 2  2  p12 Ssin( t  Kz)  y 2  2  p12 Ssin( t  Kz)  z 2  2  p11Ssin( t  Kz) 1
n

n

n

Nonlinear Optics Lab.
Hanyang Univ.
Example) y-polarized Shear wave propagating along the z direction in Ge
Sound wave : v( z , t ) Ayˆ cos(t  Kz )
S1 S xx u /x0, S 2 S yy v/y 0, S3 S zz w/z 0,
S 4 S yz (v/z )(w/y ) KAsin( t  Kz )S sin( t  Kz ),
S5 S zx (u /z )(w/x)0, S 6 S xy (u /y )(v/x)0
Elasto-Optic Coefficient for the Ge (cubic, Table 9.1) :
 p11
p
 12
p
pij  12
0
0

 0
p12
p11
p12
0
0
0
p12
p12
p11
0
0
0
0
0
0
p44
0
0
0
0
0
0
p44
0
0 
0 
0 

0 
0 

p44 
1
1
1
)


(
)


(
)3 0,
1
2
n2
n2
n2
1
( 2 ) 4  p44 Ssin( t  Kz),
n
1
( 2 )5, 6 0
n
 (
The new index ellipsoid :
1 2 2 2
( x  y  z )2 yzp44 Ssin( t  Kz)1
2
n
Nonlinear Optics Lab.
Hanyang Univ.
Acousto-Optic effect
Bragg diffraction & Raman-Nath diffraction
L
Vector Representation

light wave
acoustic wave
# Spread angle of Acoustic wave : ~
# Diffraction angle of Light :  B ~
# Dimensionless parameter : Q 


L
2 n
4B 2L


n2
Raman-Nath diffraction
: acoustic wave vector
has an angular distribution
Bragg diffraction
: acoustic wave vector
is well defined
 1 : Bragg diffractio n
 1 : Raman-Nath diffractio n
Nonlinear Optics Lab.
Hanyang Univ.
Example) Water, n=1.33, =6MHz (vs=1,500 m/s), =632.8 nm
vs /250m
L  2 n/( 2)2 cm : Raman -Nath Regime
 2 cm
: Bragg Regime
Bragg diffraction :
Single order diffraction
Raman-Nath diffraction :
Multiple order diffraction
Nonlinear Optics Lab.
Hanyang Univ.
Raman-Nath diffraction
Moving periodic refractive index grating : nz, t   n0  n sin t  Kz 
Consider L is small enough so that the medium behave as a thin phase grating,
2
 
nz, t L  0  1 sin( t  Kz)

where, 12/n0L, 1=(2/)n0L
The transmitted field on the plane x=L :
Et  E0 ei[t 0 1 sin(t  Kz )]
e
i sin


im
J
(

)
e
 m
m  
Et  E0ei (t 0 ) [ J 0 (1 )  J1 (1 ){ei  e i }  J 2 (1 ){e 2i  e 2i }        ]
where,
  (t  Kz)
Nonlinear Optics Lab.
Hanyang Univ.
Et0  E0 J 0 (1 )ei[t k ( x  L ) 0 ]
Et  E0 J 0 (1 )ei (t 0 )
 E0 J1 (1 )[e
i (   ) t  kz 0 ]
 E0 J 2 (1 )[e
e
i (  2  ) t  2 kz 0 ]
i (   ) t  kz 0 ]
e
amplitude reduction
]
i (  2  ) t  2 kz 0 ]
 
]
Frequency :  ,  
Wave vector : k1 [( ) 2 /c 2 k 2 ]1/2 ,
k1 [( ) 2 /c 2 k 2 ]1/2
Propagation in x>L :
 E0 J1 (1 )e
i[(   ) t  k1 ( x  L )  kz 0 ]

1
 1
E0 J1 (1 )ei[(  )t k1 ( x  L ) kz 0 ]
Diffraction angles :
K


k n
K

sin  1    
k
n
sin 1 
Nonlinear Optics Lab.
Hanyang Univ.
: +1 order
: -1 order
m-th order diffractive wave :
# Frequency :  m

sin


m
# Diffraction angle :
m
n
# L
2
1
!
2 2
: The restriction on length of medium is
severe at higher frequency
 Diffraction efficiency reduction
J 0 (1 )0, 1 
2
nL2.405, 5.520, 8.654,

1  1.85  J1 (1 )  0.582 : First order diffraction maximum
Nonlinear Optics Lab.
Hanyang Univ.
Bragg diffraction
In this regime, we can no longer consider the refractive index perturbation to act as a thin
phase grating. We should consider the propagation equation of light ;
2
 e  0 2    sin( t  kz)e
t

2
 2e
2
 ( 0 2 )e   0  sin( t kz) 2
t
t
2
converting to 2-D problem ( x : acoustic, z : light) :
2
i ( t kz )  e
 2e  2e
 2e 1
i ( t kz )


0 2   0  (e
e
) 2
x 2 z 2
t 2i
t
Total e-field :
e  e0  e  e
where, e0  A0 ( x,z )ei (tkr )  A0 ( x,z )ei (txz )
e  A( x,z )ei (tk  r )  A ( x,z )ei[ )t  x  z ]
Nonlinear Optics Lab.
Hanyang Univ.
Let,  20  k 2   2   2
(  ) 2 0  k2   2   2
And, slow varying approximation ;
2 A
A  2 A
A
 k
, 2  k
2
x
x z
z
A 
 A
2i 0   0 ei (t xz )
z 
 x
A 
 A
2i       ei[(  )t   x  z )]
z 
 x
A 
 A
2i       ei[(  ) t   x  z )]
z 
 x
1
  0  ei ( t kz ) e i ( t kz )  A0 ei[(t xz )] ( ) 2 A ei[(  ) t   x  z )] ( ) 2 A ei[(  )t   x  z )]
2i



Nonlinear Optics Lab.
Hanyang Univ.


A 
1
 A
2i 0   0 ei (xz )   2  0   A e i[  x(   k ) z ]  A e i[  x(   k ) z ]
z 
2i
 x
A 
1
 A
2i       e i (  x  z )   2  0 A0 e i[x(  k ) z ]
z 
2i
 x
Nonlinear Optics Lab.

Hanyang Univ.
(1) Small Bragg angle diffraction
A
 0 
0
z
 2i
dA0 i (x  z )
e
dx

 2i 

1 2
 0   A e i[  x (    K ) z ]  A e i[  x (    K ) z ]
2i

dA i (  x    z )
e
dx

     K
&
1 2
 0 A0e i[x  (   k ) z ]
2i
    K : Bragg condition
Nonlinear Optics Lab.
Hanyang Univ.
These equations have a solution for A when only    ,
The solutions for +=, -= are independent each other, so let 
~
dA0 ~ ix
kA e
dx
~
dA
~
kA0 e ix
dx
  0 
where,  
~
~
d 2 A0
dA0 2 ~

i ( )
 A0 0
dx 2
dx
2
4( )1/2
   
solution :
~
ix ( 12   )
ix ( 12   )
A0 ( x)C0e
 D0 e

where,   k
1/ 2
~ 

A0 
 A0
 2 0 
~ 
 A
A  
 
 2 0 
1/ 2

2
 14 ( ) 2

1/ 2

~
ix ( 1   )
ix ( 1   )  ix
A ( x) C e 2
 D e 2
e
 i     C
D  i    D

where, C 

Nonlinear Optics Lab.
1
2
0
1
2
0
Hanyang Univ.
Diffraction efficiency, h
~
~
A0 ( x0)1, A ( x0)0 (initial condition)
 C0  12  41  , D0  12  41 
0-th and 1-st diffraction powers :
2
~
P0 ( x) A0 ( x) cos 2 (x) 
   sin
2
~
P ( x) A ( x)  
2

2
 sin
2
2
2
(x)
(x)
i) P0 ( x) P ( x)1 2  2  14 ( ) 2 
ii) Maximum transfer :     0 ;   
P0 ( x)cos 2 (x)
P ( x)sin 2 (x)
Diffration efficiency :
h  p ( L)sin 2 (L)
h 1L 2 ,3 2 ,
Nonlinear Optics Lab.
Hanyang Univ.
  
1
0
1

2 
n 

 
  0 n 4 pS : scalar expression

 n3 pS 
 
 
    ncos B 
4c cos B 
c

Acoustic intensity : I a 
 
1 3 2
 va S (Text p. 483)
2

( M 2 I a )1/2
2cos B
M 2 n 6 p 2 / va3 : Figure of Merit
where,
Diffraction efficiency :


( M 2 I a )1/ 2 L 
 2 cos  B

h  sin 2 

Nonlinear Optics Lab.
Hanyang Univ.
Acoustic intensity for maximum efficiency :
I am 
2 cos 2  B
Diffraction figure of merit
of the material relative to water
2M 2 L2
Acoustic power for maximum efficiency
(LH cross-section, maximum impedance matching case) :
pam  I a LH 
2 cos 2 B  H 
2M 2
1

 
M2
L
Nonlinear Optics Lab.
Hanyang Univ.
(2) Large Bragg angle diffraction
  /2 
A
0
x

A0 i (xz ) 1 2
e
   0  A e i[  x(   K ) z ]  A e i[  x(   K ) z ]
z
4
A
1
   e i (  x  z )   2  0 A0 e i[x(  K ) z ]
z
4

x-dependent term 
Nonlinear Optics Lab.
  
Hanyang Univ.

These equations have a solution for A when only    K ,
The two solutions are independent each other, so let
1) K Co-directional coupling)
~
A0  ~ i (  ) z
 A e
z

~
A
 ~
   A0 e i (  ) z
z

Solutions :
~
iz ( 12   )
iz ( 12   )
A0 ( z )C0e
 D0e
 2  0   2 
1
where,  

4(    )1/2 4c 2  0 (    )1/2
1/ 2
~ 

A0 
 A0
2

0


1/ 2
~ 

A  
A
2 0  

      k

where,   

2
 ( )
1
4

2 1/ 2

~
iz ( 1   )
iz ( 1   )  iz
A ( z ) C e 2
 D e 2
e
     C
D  i    D

where, C  i

Nonlinear Optics Lab.
1
2
0
1
2
0
Hanyang Univ.
Diffraction efficiency, h
~
~
A0 ( z  0)  1 , A ( z  0)  0
 C0  12  41  , D0  12  41 
0-th and 1-st diffraction powers :
2
2
~


2


P0 ( z )  A0 ( z )  cos (z )  
sin 2 (z )

2 

2
2
~

P ( x)  A ( z ) 
sin 2 (z )

 
Diffraction efficiency :
 
h    sin 2 (L)
 
2

1
1    / 4 
2
2

sin {L 1 
2
  2
4 2

1/ 2
}
Nonlinear Optics Lab.
Hanyang Univ.
2) K Counter-directional coupling)
~
A0
~
 A ei (  ) z
z
~
A
~
 A0 e i (  ) z
z
Solutions :
where,    K


~
iz ( 12   )
i (  ) z
gz
 gz
A0 ( z )e
P1e Q1e  D0 e

~
1
A ( z ) e i (  ) z/2 P1 g  2i  e gz Q1 g  2i  e gz
 

where, g    ( )
2
1
4


2 1/ 2
Nonlinear Optics Lab.
Hanyang Univ.
~
~
A0 ( z  0)  1 , A ( z  D)  0

~ 2
P0 ( D) A0 
g2
2
g 2 cosh 2 gD 12   sinh 2 gD
~ 2
P ( D) A 
g    sinh
2
1
2
2
2
gD
2
g 2 cosh 2 gD 12   sinh 2 gD
# Application : DBR reflector
Nonlinear Optics Lab.
Hanyang Univ.
Surface Acousto-Optics
: Diffraction effect through a thin film surface or wave guide
: High intensity localized on the interface  enhancing the diffraction efficiency
: 1967, Ippen et al. (first experimental demonstration in a quartz)
Surface undulation profile by the acoustic wave :
xasin Kz,
K 2

Wave vectors of reflected and transmitted light waves :
k r  k cos  r x̂  sin  r ẑ 
k t  nk  cos t x̂  sin t ẑ 
Electric field on the surface :
E  E 0exp ik  cosx  sin z
Nonlinear Optics Lab.
Hanyang Univ.
1) Reflection wave
E r  c  Eeikr re -ikr r dzd
 E r cE0 eikx(cos cos r )i ( ksin  ) zik r r dz d
e
i sin

 J l ( )eil
l 


E r cE0  J l ( )eilKzi(ksin  ) zik r r dz d
l 
where,
 ka(cos cos r )

cE0  J l ( )2 (  lK ksin )e ik r r d
l 
Nonlinear Optics Lab.
Hanyang Univ.
k r rkcos r xksin r z

E r 2cE0  J l ( )e ikcos r xi ( ksin lk ) z
l 
where,  k sin r k sin lK  k (sin  r sin )lK
l 0,1,2,
Diffraction angle :
sin  r  sin  
l

 r    2kacos , the amplitude of reflected wave :

E r 2 cE0  J l (2kacos )e ikcosxi ( ksin lk ) z
l 
Nonlinear Optics Lab.
Hanyang Univ.
If a0, E r rE 0
 c r

2
 E r rE0  J l (2kacos )e ikcosxi ( ksin lk ) z
l 

rE 0 J l (2kacos )e i ( klkẑ )r
l 
Diffraction efficiency of l-th order :
hl  r rJ l2 (2ka cos  )
2
Nonlinear Optics Lab.
Hanyang Univ.
2) Transmitted wave
Similarly,
E t cE0 eikx(cos ncos t )i ( ksin  ) zik r r dz d
where,  nksin t
 E t tE0  J l kacos nkacos t einkcos t xi ( ksin lK ) z
where,
nksin t ksin lK
Diffraction efficiency of l-th order :
hl 
n cos  t 2 2
t J l (ka cos   nka cos  t )
cos 
Nonlinear Optics Lab.
Hanyang Univ.