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