Nonlinear Optics Lab. Hanyang Univ.
Chapter 8. Second-Harmonic Generation
and Parametric Oscillation
8.0 Introduction
Second-Harmonic generation :
Parametric Oscillation :
2
)( 321213
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 :
)()()()( 3)3(2)2()1( tEtEtEtP
)()()( )3()2()1( tPtPtP
: Linear susceptibility where, )1(
: 2nd order nonlinear susceptibility )2(
: 3rd order nonlinear susceptibility )3(
)2(P : 2nd order nonlinear polarization
)2(P : 3rd order nonlinear polarization
Maxwell’s wave equation :
2
2
2
2
2
22
t
P
t
E
c
nE
Source term : drives (new) wave
Nonlinear Optics Lab. Hanyang Univ.
Second order nonlinear effect
)()( 2)2()2( tEtP
Let’s us consider the optical field consisted of two distinct frequency components ;
c.c.)( 21
21 titi
eEeEtE
][2
]c.c.22[)(
*
22
*
11
)2(
)(*
21
)(
21
22
2
22
1
)2()2( 212121
EEEE
eEEeEEeEeEtPtitititi
(OR))(2)0(
)DFG(2)(
)SFG(2)(
)SHG()2(
)SHG()2(
*
22
*
11
)2(
*
21
)2(
21
21
)2(
21
2
2
)2(
2
2
1
)2(
1
EEEEP
EEP
EEP
EP
EP
: Second-harmonic generation
: Sum frequency generation
: Difference frequency generation
: 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) : )()( xVxV
Potential energy for the electric dipole can be described as
...42
)( 422
0 Bxm
xm
xV
Restoring force :
...32
0
mBxxm
x
VF
Equation of motion :
mteEBxxxx )/(2 32
0
Damping force
Restoring force
Coulomb force
Nonlinear Optics Lab. Hanyang Univ.
Purtubation expansion method :
c.c.)( 21
21 titi
eEeEtE
Assume,
)()( tEtE
)3()3()2()2()1( xxxx
Each term proportional to n should satisfy the equation separately
mteExxx )/(2 )1(2
0
)1()1(
02 )2(2
0
)2()2( xxx
02 )1(3)3(2
0
)3()3( Bxxxx
: Damped oscillator 0)2( x
Second order nonlinear effect in centrosymmetric media
can not occur !
Nonlinear Optics Lab. Hanyang Univ.
2) Noncentrosymmetric media (inversion anti-symmetric) : )()( xVxV
Potential energy for the electric dipole can be described as
...32
)( 322
0 Dxm
xm
xV
Restoring force :
...22
0
mDxxm
x
VF
Equation of motion :
mteEDxxxx )/(2 22
0
Damping force
Restoring force
Coulomb force
Nonlinear Optics Lab. Hanyang Univ.
Similarly,
c.c.)( 21
21 titi
eEeEtE
Assume,
)()( tEtE
)3()3()2()2()1( xxxx
Each term proportional to n should satisfy the equation separately
mteExxx )/(2 )1(2
0
)1()1(
0][2 2)1()2(2
0
)2()2( xDxxx
022 )2()1()3(2
0
)3()3( xDBxxxx
Solution :
ccexextxtiti
.)()()( 21
2
)1(
1
)1()1(
jj
j
j
j
ji
E
m
e
L
E
m
ex
2)()(
22
0
)1(
: Report
Nonlinear Optics Lab. Hanyang Univ.
Example) Solution for SHG
)(
)/(2
1
2
2
1
22)2(2
0
)2()2(1
L
EemeDxxx
ti
Put general solution as ti
extx 12
1
)2()2( )2()(
)()2(
)/()2(
1
2
1
2
1
2
1
)2(
LL
EmeDx
: Report
Similarly,
)()2(
)/()2(
2
2
2
2
2
2
2
)2(
LL
EmeDx
)()()(
)/(2)(
2121
21
2
21
)2(
LLL
EEmeDx
)()()(
)/(2)(
2121
*
21
2
21
)2(
LLL
EEmeDx
)()()0(
)/(2
)()()0(
)/(2)0(
22
*
22
2
11
*
11
2)2(
LLL
EEmeD
LLL
EEmeDx
Nonlinear Optics Lab. Hanyang Univ.
Susceptibility
)()( jj NexP
)()()()( 3)3(2)2()1()( tEtEtEPtPj
j Polarization :
)(
)/()(
2)1(
j
jL
meN
: linear susceptibility
2)1()1(
322
23)2( )]()[2(
)()2(
)/(),,2( jj
jj
jjjeN
mD
LL
ameN
: SHG
)()()(
)/(),,(
2121
23
2121
)2(
LLL
DmeN
)()()( 2
)1(
1
)1(
21
)1(
32
eN
mD
)()()(
)/(),,(
2121
23
2121
)2(
LLL
DmeN
: SFG
: DFG
: OR
)()()( 2
)1(
1
)1(
21
)1(
32
eN
mD
)()()0(
)/(),,0(
23)2(
jj
jjLLL
DmeN
)()()0( )1()1()1(
32 jjeN
mD
Nonlinear Optics Lab. Hanyang Univ.
<Miller’s rule> - empirical rule, 1964
)()()(
),,(
2
)1(
1
)1(
21
)1(
2121
)2(
32eN
mD 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=w0
2/d : roughly the same for all noncentrosymmetric solids.
44
0
2
3)2(
dm
e
(non-resonant case) : used in rough estimation of nonlinear coefficient.
2
0
22
0 2)( jjj iL 3/1 dN dD /2
0
6
0
2
0
233
2121
23
2121
)2( )/)(/)(/1(
)()()(
)/(),,(
dmed
LLL
DmeN
esu103 8
: 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 :
ti
mni
ti
mniimnmn ePePtP
)()()()(),r(
),()(),,()()(
)2(
mknjmnmn
jk nm
ijkmni EEP where,
2nd order nonlinear susceptibility tensor
# Full matrix form of )( mniP
)()(),,(
)()(),,(
)()(),,(
)()(),,()(
222222
)2(
121212
)2(
212121
)2(
111111
)2(
kj
jk
ijk
kj
jk
ijk
kj
jk
ijk
kj
jk
ijkmni
EE
EE
EE
EEP
2,1, mn
: SHG
: SHG
: SFG
: SFG
Nonlinear Optics Lab. Hanyang Univ.
Example 1. SHG
12
21
13
31
23
32
33
22
11
321312331313332323333322311
221212231213232223233222211
121112131113132123133122111
)2(
)2(
)2(
EE
EE
EE
EE
EE
EE
EE
EE
EE
P
P
P
nz
ny
nx
Example 2. SFG
.
)()(
.
...
.),,(.
...
.
)()(
.
...
.),,(.
...
)(
)(
)(
nkmjnmmnijk
mknjmnmnijk
mnz
mny
mnx
EE
EE
P
P
P
Nonlinear Optics Lab. Hanyang Univ.
Properties of the nonlinear susceptibility tensor
1) Reality of the fields
),r(),,r( tEtPi are real measurable quantities : *)()( mnimni PP
*
*
)()(
)()(
mkmk
njnj
EE
EE
*)2()2( ),,(),,( mnmnijkmnmnijk
2) Intrinsic permutation symmetry
),,(),,()( )2()2(
nmmnikjmnmnijkmniP
Nonlinear Optics Lab. Hanyang Univ.
4) Kleinman symmetry (nonresonant, is frequency independent)
3) Full permutation symmetry (lossless media : is real)
)(
*)()()(
321
)2(
321
)2(
321
)2(
213
)2(
jki
jkijkiijk
)()()(
)()()(
213
)2(
213
)2(
213
)2(
213
)2(
213
)2(
213
)2(
kjijikikj
kijjkiijk
intrinsic
: Indices can be freely permuted !
)()()(
)()()(
312
)2(
231
)2(
123
)2(
132
)2(
321
)2(
213
)2(
kjijikikj
kijjkiijk
If does not depend on the frequency,
Nonlinear Optics Lab. Hanyang Univ.
Define, 2nd order nonlinear tensor,
)2(
21
ijkijkd
)()(2)()(
mkn
jk nm
jijkmni EEdP
## If the Kleinman’s symmetry condition is valid, the last two indices can be simplified
to one index as follows ;
654321:
21,2113,3132,23332211:
l
jk
and,
363534333231
262524232221
161514131211
dddddd
dddddd
dddddd
dil : 18 elements
ijkd can be represented as the 3x6 matrix ;
Nonlinear Optics Lab. Hanyang Univ.
Again, by Kleinman symmetry (Indices can be freely permuted),
141323332415
121424232216
161514131211
dddddd
dddddd
dddddd
dil: Report
dil has only 10 independent elements :
Nonlinear Optics Lab. Hanyang Univ.
Example 1. SHG
)()(2
)()(2
)()(2
)(
)(
)(
2
)2(
)2(
)2(2
2
2
363534333231
262524232221
161514131211
yx
zx
zy
z
y
x
z
y
x
EE
EE
EE
E
E
E
dddddd
dddddd
dddddd
P
P
P
Example 2. SFG
)()()()(
)()()()(
)()()()(
)()(
)()(
)()(
4
)(
)(
)(
2121
2121
2121
21
21
21
363534333231
262524232221
161514131211
3
3
3
xyyx
xzzx
yzzy
zz
yy
xx
z
y
x
EEEE
EEEE
EEEE
EE
EE
EE
dddddd
dddddd
dddddd
P
P
P
: Report
Nonlinear Optics Lab. Hanyang Univ.
8.2 Formalism of Wave Propagation in Nonlinear Media
Maxwell equation
t
dih
t
he Ped 0 ei σ
Polarization : NL0 PeP e
Assume, the nonlinear polarization is parallel to the electric field, then
2
NL
2
2
22 ),(rPeee
t
t
tt
Total electric field propagating along the z-direction :
.].)([2
1),(e
.].)([2
1),(e
.].)([2
1),(e
)(
3
)(
)(
2
)(
)(
1
)(
333
222
111
ccezEtz
ccezEtz
ccezEtz
zkti
zkti
zkti
),(e),(e),(ee)()()( 221 tztztz
where,
213 and
Nonlinear Optics Lab. Hanyang Univ.
1 term
..
2
)()(eee
)()[(*
23
2
2
2
)(2
1
)(
1
)(2 2323
11
1 ccezEzE
td
tt
zkkti
..)(
)(2
)(
2
1 )(
1
2
1
)(11
)(
2
1
2
111111 ccezEkez
zEike
z
zE zktizktizkti
..)(
2)(2
1 )(111
2
111 cce
dz
zdEikzEk
zkti
2
1
2
11
)()(
dz
zEd
dz
zdEk (slow varying approximation)
...... Text
Nonlinear Optics Lab. Hanyang Univ.
zkkkieEEd
iE
dz
dE )(*
31
2
2*
2
2
2
*
2 231
22
zkkkieEEd
iE
dz
dE )(
21
3
33
3
33 321
22
zkkkieEEd
iE
dz
dE )(*
23
1
11
1
11 123
22
Similarly,
Nonlinear Optics Lab. Hanyang Univ.
8.3 Optical Second-Harmonic Generation
2, 21321
Neglecting the absorption ; 01,2,3
zkiezEdi
dz
dE )(2)()2(
)]([2
where, )()2(
13 22 kkkkk
Assume, the depletion of the input wave power due to the conversion is negligible
ki
ezEdilE
kli
1)]([)( 2)()2(
Nonlinear Optics Lab. Hanyang Univ.
Output intensity of 2nd harmonic wave :
2
22
4)(
2
22
0
2)2(2
)2/(
)2/(sin
2
1)(
2
1
lk
lklE
n
dlE
A
PI
Conversion efficiency :
A
P
lk
lk
n
ld
P
PSHG
2
2
3
2222/3
0
2
)2/(
)2/(sin2
Phase-matching in SHG
Maximum output @ )()2( 2;0 kkk : phase-matching condition
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)
If ,0k2
2
)2/(
)2/(sin
lk
lkI
: decreases with l
)()2( 2
22
kkklc
Nonlinear Optics Lab. Hanyang Univ.
Technique for phase-matching in anisotropic crystal
cnk /)(
nnkk 2)()2( 2So,
Example) Phase matching in a negative uniaxial crystal
)(
1sincos22
2
2
0
2
ee nnn
Nonlinear Optics Lab. Hanyang Univ.
# If
0
2 nne , there exists an angle m at which 0
2 )( nn m ,
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.
0
2 )( nn m 2
0
22
2
22
0
2
)(
1
)(
sin
)(
cos
nnn e
mm 22
0
22
22
0
2
02
)()(
)()(sin
nn
nn
e
m
Example (p. 289)
Experimental verification of phase-matching
])([2/ 0
2
nnc
llk e
)()(2
)()()2sin(
2)(
3
0
22
0
22
me
mn
nn
c
llk
Taylor series expansion )(2 en near
m
)(2 m : Report
2
2
2)]([
)]([sin)(
m
mP
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
22
4)(22
2)2(
)2/(
)2/(sin)()(
kl
kllrEdrE
Total power of fundamental beam with Gaussian beam profile :
20
2 /
0
)( )( r
eErE
42
1 2
02
0sectioncross
2)()(
EdxdyEP
Nonlinear Optics Lab. Hanyang Univ.
So, Conversion efficiency :
2
2
2
0
)(
3
2222/3
0
)(
)2(
)2/(
)2/(sin2
kl
kl
w
P
n
ld
P
P
: 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)
2
2)(
2
232/3
0focusingconfocal
)(
)2(
)2/(
)2/(sin2
kl
klP
n
ld
cP
P
nlw 2/2
0 2l (**)
Example (p. 292)
Nonlinear Optics Lab. Hanyang Univ.
Second-Harmonic Generation with a Depleted Input
Considering depletion of pump field, constant)(),( 21 zEzE
Define, 3,2,1 lEn
A l
l
ll
zki
zki
zki
eAAi
Adz
dA
eAAi
Adz
dA
eAAi
Adz
dA
)(
21333
)(*
31
*
22
*
2
)(
3
*
2111
22
22
22
(8.2-13) where,
)( 213
321
321
0
kkkk
nnnd
l
ll
SHG : 21 AA
a transparent medium : 0l , and perfect phase-matching case : Let’s consider 0k
*
131
2AAi
dz
dA 2
13
2Ai
dz
dA
Nonlinear Optics Lab. Hanyang Univ.
Define, 33 AiA
1
*
111 real] is)0([ real is )( AAAzA
2
13
131
2
1
2
1
Adz
Ad
AAdz
dA
0)(2
3
2
1 AAdz
d: Total energy conservation
Initial condition : )0(2
1
2
3
2
1 AAA
))0((2
1 2
3
2
13 AA
dz
Ad
])0(
2
1)tanh[0()( 113 zAAzA
# )0()(,)0( 1
'
31 AzAzA
: 100% conversion
[2N( photons) N(2 photons)]
Nonlinear Optics Lab. Hanyang Univ.
Conversion efficiency :
])0(2
1[tanh
)0(
)(1
2
2
1
2
3
)(
)2(
zAA
zA
P
PSHG
Nonlinear Optics Lab. Hanyang Univ.
8.4 Second-Harmonic generation Inside the Laser Resonator
# Second-harmonic power Pump beam power
# Laser intracavity power : )1/(~int RPP outra Efficient SHG
SH output power :
202 )( isopt LgAIP
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 : )()2( 2 kk
Nonlinear Optics Lab. Hanyang Univ.
8.6 Parametric Amplification
: )( 213213
# Special case : 1=2 (degenerate parametric amplification)
Analogous Systems :
- Classical oscillators
- Parasitic resonances in pipe organs(1883, L. Rayleigh) :
- RLC circuits
0)sin2( 2
02
2
vtdt
dv
dt
vdp
Example) RLC circuit
t
C
CCC po sin1
0
Nonlinear Optics Lab. Hanyang Univ.
0)sin1(1
00
2
2
vtC
C
LCdt
dv
dt
vdp
0CCAssuming
Put, ]cos[ tav
0)(
][)()(22
0 tititi Peieie
where,
00
2
0
0
2
0C
1
2
1
RC
C
LC
Steady-state solution :
00 or 0
) that (so 2
pp
2/frequency aat circuit 0 poscillatessly spontaneou
(degenerate parametric oscillation)
Phase matching Threshold condition
Nonlinear Optics Lab. Hanyang Univ.
Optical parametric Amplification
Polarization of 2nd order nonlinear crystal :
2
0 deep ε
)()()()( 0 tetptetd εε
de )1(0 ε
es
Ad
s
A
s
AC
)1(0 ε
tEe psin0
ts
AdE
s
AC p
sin
)1( 00
ε
Nonlinear Optics Lab. Hanyang Univ.
(8.2-13),
3,2,1 lEn
A l
l
ll
zki
zki
zki
eAAi
Adz
dA
eAAi
Adz
dA
eAAi
Adz
dA
)(
21333
)(*
31
*
22
*
2
)(
3
*
2111
22
1
22
1
22
1
3,2,1
321
321
213
l
nnnd
kkkk
l
ll
o
ε
ε
where,
0l (phase-matching), and also depletion of field due to When 0k,321 (lossless),
the conversion is negligible,
1
**
2*
21
2
2A
ig
dz
dAA
ig
dz
dA )0()0( 3
21
213 dE
nnAg
o
εwhere,
Nonlinear Optics Lab. Hanyang Univ.
Solution :
zg
iAzg
AzA
zg
iAzg
AzA
2sinh)0(
2cosh)0()(
2sinh)0(
2cosh)0()(
1
*
2
*
2
*
211
Qualitative understanding of parametric oscillation :
31
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 :
zg
iAzA
zg
AzA
2sinh)0()(
2cosh)0()(
1
*
2
11
0)0(2 A
z
)(zA
|)(| 1 zA
|)(| 2 zA
Photon flux :
2sinh)0()()()(
2cosh)0()()()(
2
12
*
22
2
11
*
11
gzAzAzAzN
gzAzAzAzN
AAN *
gz
gz
eA
eA
4
)0(
4
)0(
2
1
2
1
1
1
gz
gz
Nonlinear Optics Lab. Hanyang Univ.
8.7 Phase-Matching in Parametric Amplification
0k,(lossless)02,1
zki
zki
eAg
idz
dA
eAg
idz
dA
)(
1
*
2
)(*
21
2
2
zkis
zkis
emzA
emzA
)]2/([
2
*
2
)]2/([
11
)(
)(
Put, bkgs
22 )(2
1
zkiszkis
zkiszkis
ememzA
ememzA
)]2/([
2
)]2/([
2
*
2
)]2/([
1
)]2/([
11
)(
)(
Nonlinear Optics Lab. Hanyang Univ.
)0(2
),0(2
)0()(),0()(:0
1
0
*
2*
2
0
1
*
2
*
211
Ag
idz
dAA
gi
dz
dA
AzAAzAz
zz
General solution :
)sinh()0(
2)sinh(
2)cosh()0()(
)sinh()0(2
)sinh(2
)cosh()0()(
1
*
2
)2/(*
2
*
21
)2/(
1
bzAb
gibz
b
kibzAezA
bzAb
gibz
b
kibzAezA
zki
zki
possible isidler and signal theofgrowth sustained nok g Unless#
k offunction is t coefficienGain #
b
Nonlinear Optics Lab. Hanyang Univ.
Phase-Matching
Example) Phase-matching by using a negative uniaxial crystal
21
33
3
3
2
3
1
2/122
sincos)(
ee
e
m
e
mme nn
nnn
213
3
2
3
1213
nnnkkk
c
nk
: Report
Nonlinear Optics Lab. Hanyang Univ.
8.8 Parametric Oscillation
0lossbut depletion, no ,0 k
)0()( 33 AzA
1
*
22
*
2
*
2111
22
1
22
1
Ag
iAdz
dA
Ag
iAdz
dA
2,1
2,12,1
3
21
21
0
)0(
dE
nngwhere,
(8.8-1)
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), 0*
21 dz
dA
dz
dA
So,
022
022
1
*
22
1
*
211
AAg
i
Ag
iA
Condition for nontrivial solution :
0
2
2
2
2det
2
1
gi
gi
21
2 g : Threshold condition for parametric oscillation
absorption in crystal, reflections on the interfaces,
cavity loss(mirrors, diffraction, scattering), …
Nonlinear Optics Lab. Hanyang Univ.
If we choose to express the mode losses at 1 amd 2 by the quality factors, respectively,
Decay time (photon lifetime) of a cavity mode :
cQ
n
i
iii
Qtc
1(4.7-5)
Temporal decay rate :
n
c
)0(3
21
21
0
dEnn
g
21
2 g and
2121
3 1)(
Ed t
2
30
32
3 2nA
PE
Threshold pump intensity :
2
3
2
303
2
1E
n
A
P
Pump intensity :
Threshold pump intensity :
21
2
21
2
302
3
2
303
2
1)(
2
1
QQd
nE
n
A
Ptth
Nonlinear Optics Lab. Hanyang Univ.
Example) Absorption loss = 0
(4.7-5), (4.7-3) )1( i
iii
Rc
lnQ
: given by only the cavity mirror’s reflectivity
22
21
21321
2/3
03 )1)(1(
2
1
dl
RRnnn
A
P
t
Example (p. 311)
Nonlinear Optics Lab. Hanyang Univ.
8.9 Frequency Tuning in Parametric Oscillation
Phase-Matching condition : 221133213 nnnkkk
c
nk
If the phase matching condition is satisfied at the angle, =0
20201010303 nnn
00 iii nnn 0 iii 0
constant# 321 20102201103
21
And, we have
))(())(()( 2201201101103303 nnnnnn
Nonlinear Optics Lab. Hanyang Univ.
Neglecting the second order terms,
2010
220110331
202
101
nn
nnn
0
33
nn
2
2
22
1
1
11
20
10
nn
nn
(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)
)]/()/([)(
)/(
222011102010
331
nnnn
n
Parametric oscillation frequency with the angle :
Nonlinear Optics Lab. Hanyang Univ.
Example) Frequency tuning by using a negative uniaxial crystal
2
0
23
33
33
11)2sin(
2
nn
nn
e
2
220
1
1102010
0
2
0
2
3
303
1
)(
)2sin(11
2
133
nnnn
nnn
e
Nonlinear Optics Lab. Hanyang Univ.
8.11 Frequency Up-Conversion
321 : Sum Frequency Generation
213 kkk Phase-matching condition :
0,0,constant2 kA
13
31
2
2
Ag
idz
dA
Ag
idz
dA
Solution :
zg
iAzg
AzA
zg
iAzg
AzA
2sin)0(
2cos)0()(
2sin)0(
2cos)0()(
133
311
2
031
31 dEnn
g
where,
Nonlinear Optics Lab. Hanyang Univ.
0)0(3A
zg
AzA
zg
AzA
2sin)0()(
2cos)0()(
22
1
2
3
22
1
2
12
1
2
3
2
1 )0()()( AzAzA therefore
Power :
zg
PzP
zg
PzP
2sin)0()(
2cos)0()(
2
1
1
33
2
11
# Oscillating function with z (cf : parametric oscillation)
Nonlinear Optics Lab. Hanyang Univ.
Conversion efficiency :
l
g
P
lP
2sin
)0(
)( 2
1
3
1
3
4
22
1
3 lg
Typically, conversion efficiency is small
2
031
31 dEnn
g
A
P
nnn
dl
P
lP 2
2/3
0321
222
3
1
3
2)0(
)(
Example (p. 318)