Nonlinear Susceptibility and Polarization
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Transcript Nonlinear Susceptibility and Polarization
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 )E1ei1t E2ei2t c.c.
P ( 2) (t ) ( 2) [ E12e 2i1t E22e 2i2t 2 E1E2e i (1 2 )t 2 E1E2*e i (1 2 )t c.c.]
2 ( 2) [ E1E1* E2 E2* ]
P (21 ) ( 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 xmBx 3 ...
x
Equation of motion :
x2x02 xBx3 eE(t )/m
Damping force
Coulomb force
Restoring force
Nonlinear Optics Lab.
Hanyang Univ.
Purtubation expansion method :
Assume,
E(t )E1ei1t E2ei2t c.c.
E (t )E (t )
xx (1) ( 2) x ( 2) (3) x (3)
Each term proportional to n should satisfy the equation separately
2
x(1) 2x (1) 0 x(1) eE(t )/m
x(2) 2x ( 2) 02 x(2) 0 : Damped oscillator x ( 2) 0
x(3) 2x (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 xmDx 2 ...
x
Equation of motion :
x2x02 xDx2 eE(t )/m
Damping force
Coulomb force
Restoring force
Nonlinear Optics Lab.
Hanyang Univ.
Similarly,
Assume,
E(t )E1ei1t E2ei2t c.c.
E (t )E (t )
xx (1) ( 2) x ( 2) (3) x (3)
Each term proportional to n should satisfy the equation separately
2
x(1) 2x (1) 0 x(1) eE(t )/m
x(2) 2x ( 2) 0 2 x( 2) D[ x(1) ]2 0
x(3) 2x (3) 0 2 x(3) 2DBx(1) x(2) 0
Solution :
x(1) (t )x(1) (1 )ei1t x(1) (2 )ei2t 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 2i1t 2
D
(
e
/
m
)
e
E1
x( 2) 2x ( 2) 02 x ( 2)
L2 (1 )
Put general solution as
x( 2) (t )x( 2) (21 )e2i1t
Similarly,
D(e/m) 2 E12
x (21 )
L(21 ) L2 (1 )
( 2)
D(e/m) 2 E22
x (22 )
L(22 ) 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.
Hanyang Univ.
: 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) :
m02d=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 )
N1/d 3
D02 /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 )ei (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, m1, 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 0eP
i σ e
Polarization : P 0 eePNL
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 : ee(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 (1tk1z ) c.c.
2
dz
dE1 ( z )
d 2 E1 ( z )
k1
dz
dz2
(slow varying approximation)
...... Text
Nonlinear Optics Lab.
Hanyang Univ.
dE1 1
i1
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 ( k1k3 k2 ) z
dz
2 2
2 2
dE3
3
i3
E3
d E1 E2e i ( k1k2 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 ; 1,2,30
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 )
ikl
1
( )
2e
(l ) i
d [ E ( z )]
ik
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
2l
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/2n
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 23
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 A32 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 /(1R) 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 CC0
d 2v
dv
1
C
(
1
sin pt )v 0
2
dt
dt LC0
C0
Put,
v a cos[t ]
(02 2 )ei (t ) iei (t ) iei[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 frequency0 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
eE0sin 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
123
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
12
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
gz1
2
gz
2
N 2 ( z ) A2* ( z ) A2 ( z ) A1 (0) sinh
gz1
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), k0
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 :
ik
g *
A1 ( z )ei ( k / 2) z A1 (0) cosh(bz)
sinh(bz) i
A2 (0) sinh(bz)
2b
2b
ik
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 n3
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 loss0
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 )
12l 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
13
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
13
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.