Chapter 8. Second-Harmonic Generation
and Parametric Oscillation
8.0 Introduction
Second-Harmonic generation :    2
Parametric Oscillation : 3  1 2 (1 2 3 )
Reference :
R.W. Boyd, Nonlinear Optics,
Chapter 1. The nonlinear Optical Susceptibility
Nonlinear Optics Lab.
Hanyang Univ.
The Nonlinear Optical Susceptibility
General form of induced polarization :
P(t )  (1) E(t ) ( 2) E 2 (t ) (3) E 3 (t )
P(1) (t )P( 2) (t )P(3) (t )
where,
 (1) : Linear susceptibility
 ( 2) : 2nd order nonlinear susceptibility
 (3) : 3rd order nonlinear susceptibility
P( 2) : 2nd order nonlinear polarization
P( 2) : 3rd order nonlinear polarization
Maxwell’s wave equation :
2 E
n  E  P
 2
2
2
c t
t
2
2
2
Source term : drives (new) wave
Nonlinear Optics Lab.
Hanyang Univ.
Second order nonlinear effect
P( 2) (t )  ( 2) E 2 (t )
Let’s us consider the optical field consisted of two distinct frequency components ;
E(t )E1ei1t E2ei2t c.c.
 P ( 2) (t ) ( 2) [ E12e 2i1t  E22e 2i2t 2 E1E2e i (1 2 )t 2 E1E2*e i (1 2 )t c.c.]
2  ( 2) [ E1E1*  E2 E2* ]
P (21 )  ( 2 ) E12 (SHG )
P (2 2 ) 
( 2)
2
2
: Second-harmonic generation
E (SHG )
P (1  2 )2  ( 2 ) E1 E2 (SFG ) : Sum frequency generation
P (1  2 )2  ( 2 ) E1 E2* (DFG) : Difference frequency generation
P (0)2  ( 2 ) ( E1 E1*  E2 E2* ) (OR) : Optical rectification
# Typically, no more than one of these frequency component will be generated
 Phase matching !
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Susceptibility and Polarization
1) Centrosymmetric media (inversion symmetric) : V ( x) V ( x)
Potential energy for the electric dipole can be described as
m
m
V ( x)  02 x 2  Bx 4 ...
2
4
Restoring force :
F 
V
m 02 xmBx 3 ...
x
Equation of motion :
x2x02 xBx3 eE(t )/m
Damping force
Coulomb force
Restoring force
Nonlinear Optics Lab.
Hanyang Univ.
Purtubation expansion method :
Assume,
E(t )E1ei1t E2ei2t c.c.
E (t )E (t )
xx (1) ( 2) x ( 2) (3) x (3) 
Each term proportional to n should satisfy the equation separately
2
 x(1) 2x (1) 0 x(1) eE(t )/m
x(2) 2x ( 2) 02 x(2) 0 : Damped oscillator   x ( 2) 0
x(3) 2x (3) 0 2 x(3) Bx (1) 0
3
Second order nonlinear effect in centrosymmetric media
can not occur !
Nonlinear Optics Lab.
Hanyang Univ.
2) Noncentrosymmetric media (inversion anti-symmetric) : V ( x) V ( x)
Potential energy for the electric dipole can be described as
m
m
V ( x)  02 x 2  Dx 3 ...
2
3
Restoring force :
F 
V
m 02 xmDx 2 ...
x
Equation of motion :
x2x02 xDx2 eE(t )/m
Damping force
Coulomb force
Restoring force
Nonlinear Optics Lab.
Hanyang Univ.
Similarly,
Assume,
E(t )E1ei1t E2ei2t c.c.
E (t )E (t )
xx (1) ( 2) x ( 2) (3) x (3) 
Each term proportional to n should satisfy the equation separately
2
 x(1) 2x (1) 0 x(1) eE(t )/m
x(2) 2x ( 2) 0 2 x( 2) D[ x(1) ]2 0
x(3) 2x (3) 0 2 x(3) 2DBx(1) x(2) 0
Solution :
x(1) (t )x(1) (1 )ei1t  x(1) (2 )ei2t c.c
Ej
e Ej
e
(1)
x ( j )

m L( j ) m 02  2j 2i j
Nonlinear Optics Lab.
: Report
Hanyang Univ.
Example) Solution for SHG
2 2i1t 2
D
(
e
/
m
)
e
E1
x( 2) 2x ( 2) 02 x ( 2) 
L2 (1 )
Put general solution as
x( 2) (t )x( 2) (21 )e2i1t

Similarly,
D(e/m) 2 E12
x (21 )
L(21 ) L2 (1 )
( 2)
D(e/m) 2 E22
x (22 )
L(22 ) L2 (2 )
( 2)
2D(e/m) 2 E1E2
x (1  2 )
L(1  2 ) L(1 ) L( 2 )
( 2)
2D(e/m) 2 E1E2*
x (1  2 )
L(1  2 ) L(1 ) L( 2 )
( 2)
2D(e/m) 2 E1E1* 2D(e/m) 2 E2 E2*
x (0)

L(0) L(1 ) L(1 ) L(0) L( 2 ) L( 2 )
( 2)
Nonlinear Optics Lab.
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: Report
Susceptibility
Polarization : P(t ) P( j )  (1) E(t ) ( 2) E 2 (t ) (3) E 3 (t )
j
P ( j )  Nex( j )

N (e 2 /m)
 ( j )
L( j )
(1)
: linear susceptibility
N (e3 /m2 )a
mD (1)
(1)
2
: SHG
 (2 j , j , j )


(
2

)[

(

)]
j
j
2
2 3
L(2 j ) L ( j ) N e
( 2)
mD
N (e3 /m2 ) D
 2 3  (1) (1  2 )  (1) (1 )  (1) ( 2 ) : SFG
 (1 2 ,1,2 )
L(1 2 ) L(1 ) L(2 ) N e
( 2)
N (e3 /m2 ) D
mD
 (1 2 ,1,2 )
 2 3  (1) (1  2 )  (1) (1 )  (1) ( 2 ) : DFG
L(1 2 ) L(1 ) L(2 ) N e
( 2)

( 2)
N (e3 /m2 ) D
mD
(0, j , j )
 2 3  (1) (0)  (1) ( j )  (1) (  j ) : OR
L(0) L( j ) L( j ) N e
Nonlinear Optics Lab.
Hanyang Univ.
<Miller’s rule> - empirical rule, 1964
 ( 2) (1  2 , 1 , 2 )
mD

 (1) (1  2 )  (1) (1 )  (1) (2 ) N 2 e 3
is nearly constant for all noncentrosymmetric crystals.
# N ~ 1023 cm-3 for all condensed matter
# Linear and nonlinear contribution to the restoring force would be comparable when the displacement
is approximately equal to the size of the atom (~order of lattice constant d) :
m02d=mDd  D=w02/d : roughly the same for all noncentrosymmetric solids.
 
( 2)
e3
m 2 04 d 4
(non-resonant case) : used in rough estimation of nonlinear coefficient.
L( j )02  2j 2i j 02
  ( 2) (1 2 ,1,2 )
N1/d 3
D02 /d
(1/d 3 )(e3 /m2 )(02 /d )
N (e3 /m2 ) D
8


3

10
esu
6
L(1  2 ) L(1 ) L(2 )
0
: good agreement with
the measured values
Nonlinear Optics Lab.
Hanyang Univ.
Qualitative understanding of Second order nonlinear effect
in a noncentrosymmetric media
Nonlinear Optics Lab.
Hanyang Univ.
2 component
Nonlinear Optics Lab.
Hanyang Univ.
General expression of nonlinear polarization and
Nonlinear susceptibility tensor
General expression of 2nd order nonlinear polarization :
Pi (r, t )Pi (n m )ei (n m )t Pi (n m )ei (n m )t
where,
( 2)
Pi (n m ) ijk
(n m ,n ,m ) E j (n ) Ek (m ), n, m1, 2
jk ( nm )
2nd order nonlinear susceptibility tensor
# Full matrix form of Pi (n m )
( 2)
Pi ( n  m ) ijk
(1 1 ,1 ,1 )E j (1 ) Ek (1 ) : SHG
jk
( 2)
 ijk
(1  2 ,1 , 2 )E j (1 ) Ek ( 2 ) : SFG
jk
( 2)
 ijk
( 2 1 , 2 ,1 )E j ( 2 ) Ek (1 ) : SFG
jk
( 2)
 ijk
( 2  2 , 2 , 2 )E j ( 2 ) Ek ( 2 ) : SHG
jk
Nonlinear Optics Lab.
Hanyang Univ.
Example 1. SHG
 Px (2 n )   111

 
P
(
2

)
 y
n   211
 P (2 )   
n   311
 z
122
 222
 322
133
 233
 333
123
 223
 323
132
 232
 332
113
 213
 313
131 112
 231  212
 331  312
Example 2. SFG
 E1 E1 


 E2 E2 
E E 
 3 3
121  E2 E3 

 221  E3 E2 
 321  E1 E3 


E
E
 3 1
 E1 E2 


E
E
 2 1
.
.
.
 Px ( n  m )   .


 


 Py ( n  m )  .  ijk ( n  m , n , m ) . E j ( n ) Ek ( m ) 
 P (  )   .

.
.
.
 z n m  

.
.
.
.




 .  ijk ( n  m , m , n ) . E j ( m ) Ek ( n ) 
.

.
.
.


Nonlinear Optics Lab.
Hanyang Univ.
Properties of the nonlinear susceptibility tensor
1) Reality of the fields
Pi (r, t ), E(r, t ) are real measurable quantities :
Pi (n  m )  Pi (n  m )*
E j (n )  E j (n )*
Ek (m )  Ek (m )*
( 2)
( 2)
 ijk
(n  m ,n ,m )  ijk
(n  m , n , m )*
2) Intrinsic permutation symmetry
( 2)
( 2)
Pi (n m )  ijk
(n m ,n ,m )ijk
(n m ,m ,n )
Nonlinear Optics Lab.
Hanyang Univ.
3) Full permutation symmetry (lossless media :  is real)
( 2)
 ijk
(3 1  2 )   (jki2) (1  2 3 )  (jki2) (1  2 3 )*
  (jki2) (1  2 3 )
4) Kleinman symmetry (nonresonant,  is frequency independent)
( 2)
( 2)
 ijk
(3 1  2 )   (jki2) (3 1  2 )  kij
(3 1  2 )
intrinsic
( 2)
( 2)
  ikj
(3 1  2 )  (jik2) (3 1  2 )  kji
(3 1  2 )
: Indices can be freely permuted !
Nonlinear Optics Lab.
Hanyang Univ.
( 2)
Define, 2nd order nonlinear tensor, dijk  12 ijk
Pi (n  m )   2dijk E j (n ) Ek (m )
jk ( nm )
## If the Kleinman’s symmetry condition is valid, the last two indices can be simplified
to one index as follows ;
jk : 11 22 33 23,32 31,13 12,21
l : 1 2 3
4
5
6
and,
d ijk can be represented as the 3x6 matrix ;
d11 d12 d13 d14 d15 d16 
d il  d 21 d 22 d 23 d 24 d 25 d 26 
d 31 d 32 d 33 d 34 d 35 d 36 
: 18 elements
Nonlinear Optics Lab.
Hanyang Univ.
Again, by Kleinman symmetry (Indices can be freely permuted),
dil has only 10 independent elements :
d11 d12 d13 d14 d15 d16 
d il  d16 d 22 d 23 d 24 d14 d12 
d15 d 24 d 33 d 23 d13 d14 
: Report
Nonlinear Optics Lab.
Hanyang Univ.
Example 1. SHG
 Px (2 )   d11

 
 Py (2 ) 2 d 21
 P (2 )   d
 z
  31
d12
d 22
d 32
d13
d 23
d 33
d14
d 24
d 34
d15
d 25
d 35
Example 2. SFG
 Px ( 3 )   d11

 
 Py ( 3 ) 4 d 21
 P ( )   d
 z 3   31
d12
d13
d14
d15
d 22
d 23
d 24
d 25
d 32
d 33
d 34
d 35
 E x ( ) 2 


2
E
(

)


y
d16 

2

E
(

)
z

d 26 

2 E y ( ) E z ( ) 

d 36 

2
E
(

)
E
(

)
z
 x

 2 E ( ) E ( ) 
y
 x

: Report
E x (1 ) E x ( 2 )




E
(

)
E
(

)


y
1
y
2
d16 

E z (1 ) E z ( 2 )


d 26 
 E y (1 ) E z ( 2 ) E z (1 ) E y ( 2 ) 
d 36 

E
(

)
E
(

)

E
(

)
E
(

)
x
1
z
2
z
1
x
2


 E ( ) E ( ) E ( ) E ( ) 
y
1
x
2 
 x 1 y 2
Nonlinear Optics Lab.
Hanyang Univ.
8.2 Formalism of Wave Propagation in Nonlinear Media
Maxwell equation
d
h i
t
e 
h
t
d 0eP
i σ e
Polarization : P 0  eePNL
Assume, the nonlinear polarization is parallel to the electric field, then
e
 2e  2 PNL (r,t )
 e   2  
t
t
t 2
2
Total electric field propagating along the z-direction : ee(1 ) ( z,t )e(2 ) ( z,t )e(2 ) ( z,t )
1
2
1
e ( 2 ) ( z ,t ) [ E2 ( z )ei ( 2t k2 z ) c.c.]
2
1
e (3 ) ( z ,t ) [ E3 ( z )ei (3t k3 z ) c.c.]
2
where, e (1 ) ( z ,t ) [ E1 ( z )e i (1t k1 z ) c.c.]
Nonlinear Optics Lab.
and
3 1 2
Hanyang Univ.
1 term
2 (1 )
e

e(1 )
 2e(1 )
 2  E3 ( z ) E2* ( z ) i[(3  2 )t( k3 k2 ) z
 1
 1


d
e

c
.
c
.


t
t 2
t 2 
2


1   2 E1 ( z ) i (1t k1z )
E1 ( z ) i (1t k1z ) 2
i (1t k1 z ) 
 
e
2ik1
e
k1 E1 ( z )e
 c.c.
2
2  z
z


1
dE ( z ) 
 k12 E1 ( z )2ik1 1 ei (1tk1z ) c.c.
2
dz 
dE1 ( z )
d 2 E1 ( z )
k1

dz
dz2
(slow varying approximation)
...... Text
Nonlinear Optics Lab.
Hanyang Univ.

dE1  1 
i1 

E1 
d E3 E2*e i ( k3 k2 k1 ) z
dz
2 1
2 1
*
dE
  * i 2 
2
Similarly,
 2
E2 
d E1E3*e i ( k1k3 k2 ) z
dz
2 2
2 2
dE3
3 
i3 

E3 
d E1 E2e i ( k1k2 k3 ) z
dz
2 3
2 3
Nonlinear Optics Lab.
Hanyang Univ.
8.3 Optical Second-Harmonic Generation
1 2  , 3 1 2 2
Neglecting the absorption ;  1,2,30
dE( 2 )
i 


d [ E ( ) ( z )]2 ei ( k ) z
dz
2 
where,
k k3 2k1 k (2 ) 2k ( )
Assume, the depletion of the input wave power due to the conversion is negligible
 E
( 2 )
ikl

1
( )
2e
(l ) i
d [ E ( z )]

ik
Nonlinear Optics Lab.
Hanyang Univ.
Output intensity of 2nd harmonic wave :
P2 1  ( 2 ) 2 1    2 d 2 ( ) 4 2 sin 2 (k l /2)
I 
E (l ) 
E
l
2
A 2 
2  0 n
(k l /2) 2
Conversion efficiency :
3/ 2


P2

 2 d 2l 2 sin 2 (k l /2) P
 SHG  2 
P   0 
n3
(k l /2) 2 A
Phase-matching in SHG
Maximum output @ k 0 ; k ( 2 ) 2k ( )
: phase-matching condition
sin 2 (k l /2)
If k 0, I 
: decreases with l
(k l /2) 2
Coherence length : measure of the maximum crystal length that is useful in producing the SHG
(separation between the main peak and the first zero of sinc function)
lc 
2
2
 ( 2 )
k k 2k ( )
Nonlinear Optics Lab.
Hanyang Univ.
Technique for phase-matching in anisotropic crystal
k ( )  n /c
So, k ( 2 ) 2k ( )  n2 n
Example) Phase matching in a negative uniaxial crystal
cos2 sin 2
1


n02
ne2 ne2 ( )
Nonlinear Optics Lab.
Hanyang Univ.
# If ne2  n0, there exists an angle m at which n2 ( m )n0,
so, if the fundamental beam is launched along m as an ordinary ray,
the SH beam will be generated along the same direction as an extraordinary ray.
2

n ( m )n0
cos2 m sin 2 m
1



(n02 ) 2 (ne2 ) 2 (n0 ) 2
(n0 ) 2 (n02 ) 2
 sin  m  2 2
(ne ) (n02 ) 2
2
Example (p. 289)
Experimental verification of phase-matching
l
k l /2 [ne2 ( )n0 ]
c
Taylor series expansion n ( ) near   m
2
e
sin 2 [  (  m )]
 P2 ( ) 
[  (  m )]2
(ne2 ) 2 (n02 ) 2
2l
 k ( )l 
sin(2 m )
(  m )
 3
c
2(n0 )
: Report
2 (  m )
Nonlinear Optics Lab.
Hanyang Univ.
Nonlinear Optics Lab.
Hanyang Univ.
Second-Harmonic Generation with Focused Gaussian Beams
If z0>>l, the intensity of the incident beam is nearly independent of z within the crystal
2
4

sin
(kl/2)
 E ( 2 ) (r )   2 d 2 E ( ) (r ) l 2

(kl/2) 2
2
E
( )
(r )  E0e
 r 2 / 02
Total power of fundamental beam with Gaussian beam profile :
P
( )
1 
 2  02 
( ) 2


E
dxdy 
E0 

cross
section
2 
  4 
Nonlinear Optics Lab.
Hanyang Univ.
So, Conversion efficiency :
3/ 2
P ( 2 )     2 d 2l 2  P ( ) sin 2 (kl/2)
 2 
2 
( )
3
2
P

n

w
(

kl
/
2
)
0
0
 


: identical to (8.3-5) for the plane wave case
(*) P(2) can be increased by decreasing w0
until z0 becomes comparable to l
# It is reasonable to focus the beam until l=2z0 (confocal focusing)
w02 l/2n
P ( 2 )
   ( )
P
l 2 (**)
3/ 2
2     3d 2l ( ) sin 2 (kl/2)
  
P
2
c   0 
n
(kl/2) 2
confocalfocusing
Example (p. 292)
Nonlinear Optics Lab.
Hanyang Univ.
Second-Harmonic Generation with a Depleted Input
Considering depletion of pump field, E1 ( z), E2 ( z)  constant
Define, Al 
nl
l
El
l  1,2,3
(8.2-13)  dA1   1 A1  i A2* A3e i ( k ) z
dz
2
2
dA2*

i
  2 A2*  A1 A3*e i ( k ) z
dz
2
2
dA3

i
  3 A3  A1 A2 e i ( k ) z
dz
2
2

l
where,  l   l
   1 23


 0  n1n2 n3
k  k3  (k1  k 2 )
  d 
SHG : A1  A2
Let’s consider a transparent medium :  l 0 , and perfect phase-matching case : k 0

dA1

i A3 A1*
dz
2
dA3

i A12
dz
2
Nonlinear Optics Lab.
Hanyang Univ.
A1 ( z) is real [ A1 (0) is real]  A1*  A1
Define, A3 iA3
dA1
1
  A3 A1
dz
2
dA3 1 2
 A1
dz 2

d 2
2
( A1  A3 )0 : Total energy conservation
dz
Initial condition : A12  A32  A12 (0)
dA3 1
2
  ( A12 (0)  A3 )
dz 2

1
A3 ( z ) A1 (0)tanh[ A1 (0) z ]
2
# A1 (0) z,
A3' ( z)A1 (0)
: 100% conversion
[2N( photons)  N(2 photons)]
Nonlinear Optics Lab.
Hanyang Univ.
Conversion efficiency :
 SHG 
( 2 )
A3 ( z )
2
P
2 1


tanh
[ A1 (0) z ]
2
P ( )
2
A1 (0)
Nonlinear Optics Lab.
Hanyang Univ.
8.4 Second-Harmonic generation Inside the Laser Resonator
# Second-harmonic power  Pump beam power
# Laser intracavity power : Pintra ~ Pout /(1R)  Efficient SHG
SH output power :

( P2 ) opt  I s A g 0  Li

2
Nonlinear Optics Lab.
Hanyang Univ.
8.5 Photon Model of SHG
Annihilation of two Photons at  and a simultanous creation of a photon at 2
- Energy :  =2
- Momentum : k ( 2 ) 2k ( )
Nonlinear Optics Lab.
Hanyang Univ.
8.6 Parametric Amplification
: 3 1 2 (3 1 2 )
# Special case : 1=2 (degenerate parametric amplification)
Analogous Systems :
- Classical oscillators
d 2v
dv
- Parasitic resonances in pipe organs(1883, L. Rayleigh) : 2    (02  2 sin  pt )v  0
dt
dt
- RLC circuits
Example) RLC circuit
 C

C  Co 1 
sin  p t 
C0


Nonlinear Optics Lab.
Hanyang Univ.
Assuming CC0
d 2v
dv
1
C



(
1

sin  pt )v  0
2
dt
dt LC0
C0
Put,
v  a cos[t   ]
(02   2 )ei (t  )  iei (t  )  iei[P  t  ]  0
02 C
1
1
where,  


LC0
2C0
RC0
2
0
Steady-state solution :
 p  2 (so that p     )
  0   0 or 
  0
 circuit spontaneously oscillatesat a frequency0  p /2
(degenerate parametric oscillation)
Phase matching
Threshold condition
Nonlinear Optics Lab.
Hanyang Univ.
Optical parametric Amplification
Polarization of 2nd order nonlinear crystal :
p  ε0 e  de2
d (t )  ε0e(t )  p(t )  εe(t )
ε 0 (1 )de
 C
A ε 0 (1  ) A Ad
s

s

s
e
eE0sin pt
C
ε 0 (1   ) A AdE 0

sin  p t
s
s
Nonlinear Optics Lab.
Hanyang Univ.
(8.2-13), Al 

nl
l
El
l  1,2,3
dA1
1
i
  1 A1  A2* A3e i ( k ) z
dz
2
2
dA2*
1
i
   2 A2*  A1 A3*ei ( k ) z
dz
2
2
dA3
1
i
   3 A3  A1 A2ei ( k ) z
dz
2
2
where, k  k3  k1  k 2 
   123

ε
 o  n1n2 n3
  d 
l   l

l  1,2,3
εl
When 1 2 3 ,  l 0 (lossless), k 0 (phase-matching), and also depletion of field due to
the conversion is negligible,
dA1
ig
  A2*
dz
2
dA2* ig *

A1
dz
2
   12

dE3 (0)
ε
n
n
 o 1 2
where, g  A3 (0)  
Nonlinear Optics Lab.
Hanyang Univ.
Solution :
g
g
z  iA2* (0) sinh z
2
2
g
g
A2* ( z )  A2* (0) cosh z  iA1 (0) sinh z
2
2
A1 ( z )  A1 (0) cosh
Qualitative understanding of parametric oscillation :
3
1
2
# Initially 1(or 2) is generated by two photon spontaneous fluorescence
or by cavity resonance
# 2(or 1) wave increases by difference frequency generation
between 3 and 1(or 2)
# 1(or 2) wave also increases by difference frequency generation
between 3 and 1(or 2)
# 2(or 1) wave : Signal [A(0)=0]
# 2(or 1) wave : Idler [A(0)>0]
Nonlinear Optics Lab.
Hanyang Univ.
Initial condition : A2 (0)0

g
A1 ( z ) A1 (0)cosh z
2
g
A2* ( z )iA1 (0)sinh z
2
A(z )
| A1 ( z )|
| A2 ( z )|
z
Photon flux : N  A*A
gz 
gz1
2
gz
2
N 2 ( z ) A2* ( z ) A2 ( z ) A1 (0) sinh

gz1
2
N1 ( z ) A1* ( z ) A1 ( z ) A1 (0) cosh
2
A1 (0)
2
e gz
4
A1 (0)
4
2
e gz
Nonlinear Optics Lab.
Hanyang Univ.
8.7 Phase-Matching in Parametric Amplification
1,2 0 (lossless), k0
dA1
g * i ( k ) z
 i A2 e
dz
2
dA2*
g
 i A1ei ( k ) z
dz
2
Put, A1 ( z )  m1e[ s i ( k / 2 )] z
A2* ( z )  m2 e[ s i ( k / 2 )] z
 s 
1 2
g (k ) 2 b
2
A1 ( z )  m1 e[ s i ( k / 2)] z  m1 e[ s i ( k / 2)] z
A2* ( z )  m2 e[ s i ( k / 2)] z  m2 e[ s i ( k / 2)] z
Nonlinear Optics Lab.
Hanyang Univ.
z 0 : A1 ( z ) A1 (0), A2* ( z ) A2* (0)
dA1
dz
g *
dA2*
i A2 (0),
2
dz
z 0
g
i A1 (0)
2
z 0
General solution :
ik 
g *


A1 ( z )ei ( k / 2) z  A1 (0) cosh(bz) 
sinh(bz)  i
A2 (0) sinh(bz)
2b

 2b
ik 
g


A2* ( z )e i ( k / 2) z  A2* (0) cosh(bz) 
sinh(bz)  i
A1 (0) sinh(bz)
2b

 2b
# Gain coefficient b is functionof k
# Unlessg  k no sustainedgrowth of thesignal and idler is possible
Nonlinear Optics Lab.
Hanyang Univ.
Phase-Matching
k3 k1 k2  n3 
k
n
c
1   2 
n  n
3
3
1
2
Example) Phase-matching by using a negative uniaxial crystal
 cos   sin 
ne ( m ) 3 m    3 m 
 ne   ne 
2
3
2



1/ 2

1   2 
ne  ne
3
3
1
Nonlinear Optics Lab.
2
: Report
Hanyang Univ.
8.8 Parametric Oscillation
k 0, no depletion,but loss0
A3 ( z ) A3 (0)
dA1
1
g
  1 A1  i A2*
dz
2
2
(8.8-1)
dA2*
1
g
   2 A2*  i A1
dz
2
2
  1 2

dE3 (0)

n
n
 0 1 2
where, g  
1, 2  1, 2

1, 2
Nonlinear Optics Lab.
Hanyang Univ.
Even though Eq. (8.8-1) describe traveling-wave parametric interaction, it is still valid if we
Think of propagation inside a cavity as a folded optical path.
If the parametric gain is equal to the cavity loss (threshold gain),
So,
dA1 dA2*

0
dz
dz
1
g
 1 A1  i A2*  0
2
2
g

i A1  2 A2*  0
2
2
absorption in crystal, reflections on the interfaces,
cavity loss(mirrors, diffraction, scattering), …
Condition for nontrivial solution :

det
1
i
g
2
2
0
g
2
i

2
2
 g 2 1 2
: Threshold condition for parametric oscillation
Nonlinear Optics Lab.
Hanyang Univ.
If we choose to express the mode losses at 1 amd 2 by the quality factors, respectively,
1 Q
tc  
 
c
Temporal decay rate :  
n
Decay time (photon lifetime) of a cavity mode :
g 1 2 and
2
   
g    1 2 dE3 (0)
  0  n1n2

(4.7-5)
 i 
i ni
Qi c
d ( E3 )t
1

1 2
Q1Q2
Threshold pump intensity :
P

Pump intensity : E  2 3
A  0 n32
2
3
P3 1  0 n32 2


E3
A 2 
2
2
P

n

n
1
1
1 2


2
3
0
3
0
3
Threshold pump intensity :   
(
E
)

th
3 t
2  d 2Q1Q2
 A 2 
Nonlinear Optics Lab.
Hanyang Univ.
Example) Absorption loss = 0
(4.7-5), (4.7-3)  Qi 
 P3  1   0 
    
 A t 2   
3/ 2
i ni l
c(1 Ri )
: given by only the cavity mirror’s reflectivity
n1n2 n3 (1  R1 )(1  R2 )
12l 2 d 2
Example (p. 311)
Nonlinear Optics Lab.
Hanyang Univ.
8.9 Frequency Tuning in Parametric Oscillation
Phase-Matching condition :
k3 k1 k2  3n3 1n1 2 n2
k
n
c
If the phase matching condition is satisfied at the angle, =0
3n30  10 n10  20 n20
 0  0   ni ni 0 ni
i i 0 i
# 1 2 3 constant  3 10 1 20 2 10 20
 1 2
And, we have
3 (n30 n3 )(10 1 )(n10 n1 )(20 1 )(n20 n2 )
Nonlinear Optics Lab.
Hanyang Univ.
Neglecting the second order terms,
1 1 10
2 20
3n3  10 n1  20 n2

n10  n20
n3 
n3

 0
n1 
n1
1
1 
10
n2 
n2
2
(3 is a fixed frequency, and if we use an extraordinary ray for the pump)
(If we use ordinary rays for the signal and idler)
2
20
Parametric oscillation frequency with the angle :
3 (n3 /  )
1

 (n10  n20 )  [10 (n1 / 1 )  20 (n2 / 2 )]
Nonlinear Optics Lab.
Hanyang Univ.
Example) Frequency tuning by using a negative uniaxial crystal
 1
n3
n33
  sin(2 )  3

2
 ne
2
2
  1  
     
3
  n0  
2
2

 1  
1
3  1 
 3n30  3   3  sin(2 0 )
 ne   n0  
1 2




n
n 
(n10 n20 )10 1  20 2 
 2 
 1
Nonlinear Optics Lab.
Hanyang Univ.
8.11 Frequency Up-Conversion
1  2  3 : Sum Frequency Generation
Phase-matching condition :
k 3  k1  k 2
A2 constant,  0, k 0

dA1
g
i A3
dz
2
dA3
g
i A1
dz
2
Solution :
g 
g 
A1 ( z )  A1 (0) cos z   iA3 (0) sin  z 
2 
2 
g 
g 
A3 ( z )  A3 (0) cos z   iA1 (0) sin  z 
2 
2 
where, g 
13   
  dE2
n1n3   0 
Nonlinear Optics Lab.
Hanyang Univ.
A3 (0)0 
2
2
g 
A1 ( z )  A1 (0) cos2  z 
2 
2
2
g 
A3 ( z )  A1 (0) sin 2  z 
2 
therefore A1 ( z )  A3 ( z )  A1 (0)
2
2
2
Power :
g
P1 ( z )  P1 (0) cos2 
2

z


g 
P3 ( z )  3 P1 (0) sin 2  z 
1
2 
# Oscillating function with z (cf : parametric oscillation)
Nonlinear Optics Lab.
Hanyang Univ.
Conversion efficiency :
2 2


g
l 
P3 (l ) 3
g


3
2


 sin  l  
1  4 
P1 (0) 1
2 
Typically, conversion efficiency is small
13   
  dE2
g
n1n3   0 

3/ 2
P3 (l )  l d     P2 
   

P1 (0) 2n1n2 n3   0   A 
2 2
3
2
Example (p. 318)
Nonlinear Optics Lab.
Hanyang Univ.