nic Oscillator Derivation of Second Order Susceptib scillator model used for deriving the linear susceptibility can be er susceptibility. BUT, it does not agree in some details with the ever, since it is experimentally measured coefficients that are use rsion efficiencies etc., the anharmonic oscillator model is quite u measured values of susceptibility are used at frequencies far from equency. n the displacement is added to the potential, i.e. rmonic potential Anharmonic potential Nonlinear “force constant” property are called “no 1. Diagonalizing does not imply is diagonali 2. Indices in are interchangeable. ) ( ) ( 3 1 2 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( m m m m m k m j m i m ijk m i m i m ii m i m q V q V q q q k q q k q V ) ( ) ( ) ( ) ( ) ( ) ( m m m m q V q V ii k ijk k ijk k
Transcript
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.
)()( 3
1
2
1)( )()()()()()()()()()()()()( 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
Em
eqq
q
VF
12)(
)()(
Nonlinear “Driven” SHO Equation:
Equation cannot be solved exactly - use successive approximations
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(E2
1),( ..)e(E
2
1),( ..)e(E
2
1),( 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
Dm
eQ
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
k
qq
])()()(
)()(
)()()(
)()([
2)(
*
*
*
*
3
2)2(
bkcjai
bkcj
ckbjai
ckbj
eijkai
DDDDDDm
ekQ
! 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
DDDk
m
eNQeN
)()()(),;(
*30
3)2(
aibjcke
ijkcbaijk
DDDm
keN
)()()(),;(
30
3)2(
bkajcie
ijkbacijk
DDDm
keN
)()()(),;(
*30
3)2(
biajcke
ijkcabijk
DDDm
keN
Also
)()()(),;(
*30
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([2
1)2(P )()()2()()()2(
0 abxyiyx
bayxixyi eeee
)()(ˆˆ),;2()()(ˆˆ),;2(22
1)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
zq
xq
Origin of Nonlinearity in KDP (KH2PO4)
)()( })(E)({2
1)( *
aiitai
aitai
aii EeetE
)()(2
1)()(
2
1 )(
}2
1)(
2
1)({
2
1
})()({2
1
2
1)(
*
)(*)(
)(*)(
aaiaaii
taiai
taiai
taiai
taiaii
E
dtedte
dtedteE
tdedett
tdetEEdeEtE
tia
tti
titi
a ))()(2
1)(
2
1)(
)(2
1)( )()( :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(
)()()(),];[()(
)()(),];[(2
1
)(2
1)(
ddEEP
dtddEEe
dtetPP
kjijki
kjijkti
tiii
Total incident field at time t
)}()(E)()(E{
)}()(E)()(E{4
1{}{}
)(){}{},];[()(
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
)()(2
1)()(
2
1 )( *
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({P2
1)( 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(2
1 )(E)(E),;0(
2
1)0(P
)(E),;2( 2
1)(E)(E),;2(
2
1)2(P
ajaaijj
smi
akajaaijki
ajaaaijj
smi
akajaaaijkai
Exactly the same as we assumed before!
Smi Singleeigenmode input
Sum and Difference Susceptibilities
beams input with 2 different frequencies → 2 eigenmode input
)}()(E)()(E
)()(E)()(E{2
1)(
*
*
bbibbi
aaiaaiiE
})(E)(E)(E)({E2
1)( * 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(E4
1
..)()()()E(E4
1)()(
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 Startingt
PP
t
E
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{2
1 )(2
22
2
2
ccezz
ikkz
E tkzi
EE
E
neglect
zki(kpzki(kp pp ezk
i
z
zez
z
zik
)2
0)20 ),(
2
),(),(
),(2
P
EP
E
Assume CW (or long pulsed) fields, ..e),(2
1 ..)e
2
1 )-()-( cczPccz,Eωtzkippωtkzi 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 that
a specific spatial Fourier component of the perturbation polarization, i.e. at kp is explicitly
assumed. 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 zzkc
idz
zd vac(1)
21
)1(
vac2
2)1(
0 (0)e)( )(2
)( EP
EEE
E
vac)1(
vac)1(
2
1 )11(n calculatioExact kknk
dilute
Second Harmonic Coupled Wave Equations
zkkii
)(itzki
iNL)(
ipNLNLp ez
ki
dz
zdccetrP
)]2([)2(2
02
2)()2(2 ),()2(2
]2[)2,( ..
2
1),,(
P
EP
Example: SHG with 1 eigenmode input )]( ,[ vac nkk
)(2 )2,(2
1
..),(),;2(4
1)2,(
..])([2)2(
]2)(2[2)2(0
2
kkez
ccezzP
pcctzki
i
tzkijijj
)(i
P
E
zkkijijj
i ezn
ki
dz
zd )]2()(2[2)2(vac ),(),;2(2
1
)2(2
)2(),2(
E
E
]/)[(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
dz
zdee ),(),;2(
2
1
)2(2
2)2,( 1)(ˆ)(ˆ that Noting 2)2(*
E
E
)(ˆ)(ˆ),;2()2(ˆ )2()(2 )2(*)2( jjijjieff eeekkk
VNB: Also valid for circular polarization!
)(ˆ)(ˆ),;2()2(ˆ 2
1:Defining )2(*)2()2()2( jjijjieffijkijk eededd
kzieff ezd
cni
dz
zd ),(),;(-2)2(
)2,( 2)2(
EE
So far the depletion of the input beams has been neglected
),;2()2( effd ),2;()2( effd
2
Up-conversion Down-conversion
kzieff ezd
cniz
dz
d ),(),;2()(2
),2( 2SHG 2)2( EE
kzieff ezzd
cniz
dz
d ),()2,(),2;(
)(),( 2DFG *)2(
EEE
DFG: 2- )](2,2),2([ )](,),([ EE
kk
)(ˆ)2(ˆ),2;()(ˆ2
1 )2()(2 *)2(*)2( kjijkieff eeedkkk
kzieff ezzd
cni
dz
zd ),()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
dz
dezd
cniz
dz
d ),()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
dz
zd ),(),(),;(-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
izdz
dezzd
cniz
dz
d
ezzdcn
izdz
d
),()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
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