Anharmonic Oscillator Derivation of Second Order Susceptibilities
The harmonic oscillator model used for deriving the linear susceptibility can be extended tothe second order susceptibility. BUT, it does not agree in some details with the quantumtreatment. However, since it is experimentally measured coefficients that are used to calculateexpected conversion efficiencies etc., the anharmonic oscillator model is quite useful. It failsprimarily when measured values of susceptibility are used at frequencies far from themeasurement frequency.A term cubic in the displacement is added to the potential, i.e.
)()( 31
21)( )()()()()()()()()()()()()( mmmmm
km
jm
im
ijkm
im
im
iim
im qVqVqqqkqqkqV
Harmonic potentialAnharmonic potential
Nonlinear “force constant”
Materials with the property are called “non-centrosymmetric”. )()( )()()()( mmmm qVqV
1. Diagonalizing does not imply is diagonalized!2. Indices in are interchangeable.
iikijkk
ijkk
Nonlinear Displacement (Type 1)
kjijke
ie
iiimi
mm
i qqkm
Emeqq
qVF 12
)(
)()(
Nonlinear “Driven” SHO Equation:
Equation cannot be solved exactly - use successive approximations
|)0(||,)2(||)(| )0()2()( )2()2()1()2()2()1(iiiiiii qqqqqqq
(1) (2) (1) Solve for neglecting terms
(2) Use solutions for to evaluate
(3) Solve for etc.
)1(iq
)1(iq )1()1(
kjijk qqk)2(
iq
ijkk
..)]()()()([41 ..)()(
41
])()([21])()([
21)()(
)()(*)(*)(2)()(
)(*)()(*)()()(
ccQQQQcceQQ
eQeQeQeQqq
mk
mj
mk
mj
timk
mj
timk
timj
timk
timj
mk
mj
Harmonic generation DC response
..)()()2(
)(41.)2(
21)2(
1) (Type axis crystal a along polarized beaminput single a with (SHG) Generation Harmonic Second
2)()()(
2
3
2)(2)()( cce
DDDmekcceQq ti
mj
mj
mi
j
e
mijj
timi
mi
Nonlinear Second Harmonic Polarization: Type 1
Similarly for the DC term )0()(miq
.)()()0(
),;0(ˆ),;0(ˆ
)()()],;0(ˆ),;0(ˆ[2
])()()0(
)()(
)()()0(
)()([
2)0(P
.].)()()()([41)()(
)(*)()(
)(
230
3)2()2(
*)2()2(0
)(*)()(
*
*)()()(
*)(
3
3)2(
)()(*)(*)()()(
mm
jm
jm
i
mijj
ie
ijjijjijj
jjijjijj
mj
mj
mi
jjm
jm
jm
i
jj
m
mijj
ei
mj
mj
mj
mj
mj
mj
DDD
k
m
keN
DDDDDDk
meN
ccQQQQqq
,)()2(
),;2(ˆ
)(),;2(ˆ2)()2(
)(
2)2()2(P
2)()(
)(
30
3)2(
2)2(02)()(
2)(
3
3)()2(
mm
jm
i
mijj
eijj
jijjmj
mi
j
m
mijj
em
mii
DD
k
mNe
DDk
meNQeN
0
inputs! separate as treatedare )E(-)(E and )E( optics,nonlinear In :VNB *
Properties of (2)
kj
i
ikjijk
or or 2 ,at enhanced
resonantly is (2)
),;2(),;2( (1)
)2(
)2()2(
)2(
i2/iNon-resonant
222 )( ;)( ;)2( 2 , caseresonant -non (3) kkjjiii DDD
symmetry"Kleinman " ~~~~~~
indices all changeinter can ),;2(~
)2()2()2()2()2()2(
222330
3)2(
jkiikjkijkjijikijk
kjie
ijkijk em
keN
(2) calculatereally t can' knowt don' )4( ijkk
)()()2(),;2( :Rewrite )5( )1()1()1()2( kkjjiiijkijk
s)dielectric(many 41 t)coefficien s(Miller' NLOearly 32
20
eN
kijkijk
Nonlinear (Type 2) Displacements (1)Type 2 refers to 2 different eigenmodes (different polarizations,different frequencies) mixed inside a crystal. Usually refers to twoorthogonally polarized fundamental beams, or to differentfrequencies of arbitrary polarization. The usual implementationof the first case is:
x
yE
450
..)e(E21),( ..)e(E
21),( ..)e(E
21),( cctrEcctrEcctrE t-i
cct-i
bbt-i
aacba
c=a+b b=c-a a=c-b
Most general case
- Sum frequency generation- Type 2 SHG for a=b orthogonally polarized beams
Difference frequency generation
.)(
)()( )(
()(
)(
()( )(
)()(
))()(
))(
cm
ke
ckc
mk
bm
je
bjb
mj
am
ie
aia
mi
ntsdisplacemelinear
DmeQ
Dm
eQ
Dm
eQ
Nonlinear Second Harmonic Polarization: Type 2
a=c-b
c.c.)]()()()([4
)()(
)()1()1(*)1()1(*
)2(2)2(
ticjbkckbj
e
ijk
aiiai
bceQQQQm
kqq
])()()(
)()(
)()()(
)()([
2)( *
*
*
*
3
2)2(
bkcjai
bkcj
ckbjai
ckbj
eijkai
DDDDDDmekQ
! and ,over is and
ofeach for summation and sincefirst the toidentical is termsecond The
zyxkj
kk ijkikj
)()(),;(
)()()(
)()(2
2)()(P
*)2(0
*
*
3
3)2()2(
ckbjcbaijk
ckbjai
bjckijk
eaiai DDD
kmeNQeN
)()()(),;( *3
0
3)2(
aibjcke
ijkcbaijk DDDm
keN
)()()(),;( 3
0
3)2(
bkajcie
ijkbacijk DDDm
keN
)()()(),;( *3
0
3)2(
biajcke
ijkcabijk DDDm
keN
Also
)()()(),;( *3
0
3)2(
aibjcke
ijkcbaijk DDDm
keN
)(Ε)(Ε),;()(P )2(0 bkajbacijkci
)()(),;()(P *)2(0 ckajcabijkbi
)()(),,;()(P *)2(0 ckbjcbaijkai
e.g. 2 fundamental beams polarized along orthogonal eigenmode axes, e.g. x and y
)]()(ˆˆ),;2()()(ˆˆ),;2([21)2(P )()()2()()()2(
0 abxyiyx
bayxixyi eeee
)()(ˆˆ),;2()()(ˆˆ),;2(221)2(P
),;(-2),;2(but
)()()2(0
)()()2(0
)2()2(
bayxixy
bayxixyi
iyxixy
eeee
)(ˆ ye
)(ˆ xe )2(ˆ ye
Type II
)(Eˆ)(Ee)(Ee)(EˆEˆ)(Eˆ
)](Ee)(Eˆ[)](Ee)(Eˆ[)(E)(E(a)(b)
y(b)
y(a)(b)22(a)22
(b)y
(a)(b)y
(a)totaltotal
xxyx
xx
eeee
ee
)()(),;2(2
)2(P )2(0)2( kjijki
)(E )(E ,, totalktotalj
Type I Type II
Type 2 Second Harmonic Generatiion
PO bonds give nonlinearity Non-resonant case
i
2,
2,
/)(
/)(
iezyz
iexyx
meQ
meQ
-
-
-
--
1 complete cycle of optical field 2 cycles of polarization to incident field
y
x
PO4 forms tetrahedron
Applied Field Electron trajectory
induced dipole
-yq
zqxq
Origin of Nonlinearity in KDP (KH2PO4)
)()( })(E)({21)( *
aiitai
aitai
aii EeetE
)()(21)()(
21 )(
}21)(
21)({
21
})()({21
21)(
*
)(*)(
)(*)(
aaiaaii
taiai
taiai
taiai
taiaii
E
dtedte
dtedteE
tdedett
tdetEEdeEtE
tia
tti
titi
a ))()(21)(
21)(
)(21)( )()( :defns
Identifies frequency forexpansion in time domain
Fourier component of field in frequency domain
Integral Formulation of Susceptibilities
Total incident field at time t
Second Order Susceptibility
tdtdtEtEtttttP kjijki )()(),()( )2(0
)2(
21)21(
212121)2(
0)2(
21])[2][1()2(
)21(210
21)21()2(
210
22
211
1)2(
0)2(
)()(),];[()(
{} integral evaluate ; :Define
}),({ x
)()(
),()()(
)()(),()(
ddeEEtP
tddtddtttt
ddtdtdetttt
eEE
ddtdtdettttEE
tdtddeEdeEtttttP
tikjijki
ttttiijk
tikj
ttiijkkj
tik
tijijki
2121212121)2(
0)2(
21212121)2()21(
0
)2()2(
)()()(),];[()(
)()(),];[(21
)(21)(
ddEEP
dtddEEe
dtetPP
kjijki
kjijkti
tiii
Total incident field at time t
)}()(E)()(E{
)}()(E)()(E{41{}{}
)(){}{},];[()(
2*
2
1*
1
21212121)2(
0)2(
aajaaj
aaiaai
ijki ddP
e.g. Single Input Fundamental for SHG
2121212121)2(
0)2( )()()(),];[()( ddEEP kjijki
)()(21)()(
21 )( *
aaiaaiiE Substituting:
2121212121
)2(0
)2( )()()(),];[()( ddEEP kjijki
Ensures energy conservation, 21
General Result for Total Input Fields )( and )( 21 EE
)}()(E)()(E)}{()(E)()(E{{}{}
)(){}{},];[(4
)(
2*
21*
1
21212121)2(0)2(
aakaakaajaaj
ijki ddP
Tedious but straight-forward
.}.)()0(P..)()2({P21)( BUT
(DC) .}.)()(E)(E),;0(
(SHG) ..)2()(E)(E),;2({4
)(
)2()2()2(
*k
)2(
)2(0)2(
ccccP
cc
ccP
iaaii
aajaaijk
aakajaaaijki
2)2(0
*)2(0
)2(
2)2(0
)2(0
)2(
|)(E|),;0(21 )(E)(E),;0(
21)0(P
)(E),;2( 21)(E)(E),;2(
21)2(P
ajaaijjsmi
akajaaijki
ajaaaijjsmi
akajaaaijkai
Exactly the same as we assumed before!
Smi Singleeigenmode input
Sum and Difference Susceptibilitiesbeams input with 2 different frequencies → 2 eigenmode input
)}()(E)()(E
)()(E)()(E{21)(
*
*
bbibbi
aaiaaiiE
})(E)(E)(E)({E21)( * ti
b*i
tibi
tiai
tiaii bbaa eeeetE
2121212121)2(
0)2( )()()(),];[()( ddEEP kjijki
Get SHG (2a and 2b) and DC like before Focus here on a b.
..)()()()E(E41
..)()()()E(E41)()(
21*
21
cc
ccEE
babkaj
babkajbkaj
)(E)(E),;()(P
)(E)(E),;()(P
*)2(0
)2(
)2(0
)2(
bkajbabaijkbai
bkajbabaijkbai
Slowly Varying Envelope Approximation (SVEA)
A simple method is needed to find the fields generated by the nonlinear polarizations
2
2
02
2
22 )(1 :point Starting
tPP
tE
cE
piii
i
where is a weak perturbation like . pP )2(P
However, it is not always possible to solve the wave equation in matter with arbitrarypolarization source terms. We will now develop a formalism in which the fields generated byperturbations can be easily calculated, provided that the perturbations are weak. It is calledthe Slowly Varying Envelope Approximation SVEA, sometimes called the Slowly VaryingPhase and Amplitude Approximation. It involves performing an integral instead of solving adifferential equation which can always be done numerically.
Assume that the complex amplitude of a generated wave varies slowly with z, i.e. is small over a wavelength.
E
z /E
..}2{21 )(
2
22
2
2cce
zzikk
zE tkzi
EEE
neglect
zki(kpzki(kp pp ezk
izzez
zzik )
20)2
0 ),(2
),(),(),(2
PEPE
Assume CW (or long pulsed) fields, ..e),(21 ..)e
21 )-()-( cczPccz,E ωtzkippωtkzi p P(E
..)(2
e)(e)e(2
)(
..)(2
e)(e)e(2
)(
2
2)-()-()-(2
2
2
2
2)-()-()-(
2
2
2
cctt
tt
itt
tP
cctt
tt
itt
tE
Tωtzki
TωtzkiωtzkiTT
ωtkziωtkziωtkzi
ppp
PPP
EEE
Aside: In more general case (short pulses) with pT PPP
This is a very useful result. It has been used for other small perturbations such as theacousto-optic effect, electro-optic effect, scattering by molecular vibrations etc. Note thata specific spatial Fourier component of the perturbation polarization, i.e. at kp is explicitlyassumed. This approach is equivalent to first order perturbation theory in quantum mechanics.
e.g. application to linear optics, dilute gas, i.e. 1>>(1) (gas density) zkip zz
kci
dzzd vac
(1)21
)1(
vac2
2)1(
0 (0)e)( )(2
)( EP
EEEE
vac)1(
vac)1(
21 )11(n calculatioExact kknk
dilute
Second Harmonic Coupled Wave Equations
zkkii
)(itzki
iNL)(
ipNLNLp ez
ki
dzzd
ccetrP )]2([)2(2
02
2)()2(2 ),()2(2]2[)2,(
..21),,(
PEP
Example: SHG with 1 eigenmode input )]( ,[ vac nkk
)(2 )2,(21
..),(),;2(41)2,(
..])([2)2(
]2)(2[2)2(0
2
kkez
ccezzP
pcctzki
i
tzkijijj
)(i
P
E
zkkijijj
i ezn
kidz
zd )]2()(2[2)2(vac ),(),;2(21
)2(2)2(),2(
EE
]/)[(2 ];/2)[2()2()2()( Note vac cnkcnknk pNL
orsunit vect - ˆ with ),(ˆ),( ; )2,(ˆ)2,( ijjii ezezzez EEEE multiply both sides by )2(ˆ* e
kzieffaiai ez
cni
dzzdee ),(),;2(
21
)2(22)2,( 1)(ˆ)(ˆ that Noting 2)2(* EE
)(ˆ)(ˆ),;2()2(ˆ )2()(2 )2(*)2( jjijjieff eeekkk
VNB: Also valid for circular polarization!
)(ˆ)(ˆ),;2()2(ˆ 21:Defining )2(*)2()2()2( jjijjieffijkijk eededd
kzieff ezd
cni
dzzd ),(),;(-2
)2()2,( 2)2(
EE
So far the depletion of the input beams has been neglected
),;2()2( effd ),2;()2( effd2
Up-conversion Down-conversion
kzieff ezd
cniz
dzd ),(),;2(
)(2),2( 2SHG 2)2(
EE
kzieff ezzd
cniz
dzd ),()2,(),2;(
)(),( 2DFG *)2(
EEE
DFG: 2- )](2,2),2([ )](,),([ EE
kk
)(ˆ)2(ˆ),2;()(ˆ21 )2()(2 *)2(*)2( kjijkieff eeedkkk
kzieff ezzd
cni
dzzd ),()2,(),-;2(-
)(),( *)2(
EEE
Both processes optimized simultaneously for wave-vector matching!
),;2(~),2;(ˆ),2;(ˆ )2()2(2223
0
3 )2(
ijkijk
kjie
ijksymmetryKleinmanijk m
keN
kzieff
kzieff ezzd
cniz
dzdezd
cniz
dzd ),()2,(~
)(),( ;),(~
)(2),2( *)2(2)2(
EEEEE
Therefore in the limit of Kleinman symmetry, all deff are equivalent (equal)!! It corresponds to being far off-resonance, i.e. non-resonant.
Recall:
Example: SHG with 2 eigenmode (polarization) inputs )],(),,[( ba kk
)(ˆ)(ˆ),;2()2(ˆ )]2()()([ )2(*)2( bk
ajijkieffba eededkkkk
kzibaeff ezzd
cni
dzzd ),(),(),;(-2
)2(2)2,( )2( EEE
2
)(E a
)(E b
),;2()2( effd ),2;()2( effd
Up-conversion Down-conversion
)(E a
)(E b
)(E b
)(E a
kziaeff
bb
kzibeff
aa
kzibaeff
ezzdcn
izdzdezzd
cniz
dzd
ezzdcn
izdzd
),()2,(~)(
),( ;),()2,(~)(
),(
),(),(~)(2
2),2( symmetry Kleinman
*)2(*)2(
)2(
EEEEEE
EEE
All 3 processes optimized simultaneously
E.g. Sum (SFG) and Difference (DFG) Frequency Generation bac
zkkibkajbacijkciba
baezzz ][)2(0
)2(c ),(),(),;(
21),( :)(SFG EEP
)(ˆ)(ˆ),;()(ˆ21 )2(*)2(
bkajbacijkcieffcba eeedkkkk
kzibabaceff
c
cc ezzdcn
idzzd ),(),(),;(-
)()(),( )2( EEE
)(ˆ)(ˆ),;()(ˆ21 *)2(*)2(
bkajbacijkbaieffcba eeedkkkk
kzibabaceff
c
cc ezzdcn
idzzd ),(),(),-;(-
)(),( *)2(
EEE
zkkibkajbacijkciba
baezzz ][*)2(0
)2(c ),(),(),;(
21),( :)(DFG EEP