Chapter 8. Second-Harmonic Generation and Parametric...

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


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


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 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


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 !


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


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


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


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


<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


Qualitative understanding of Second order nonlinear effect

in a noncentrosymmetric media


2 component


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


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


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


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,


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 ;


Again, by Kleinman symmetry (Indices can be freely permuted),

141323332415

121424232216

161514131211

dddddd

dddddd

dddddd

dil: Report

dil has only 10 independent elements :


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


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


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


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,


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(


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


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


# 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


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


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)


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


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)]



])0(2

1[tanh

)0(

)(1

2

2

1

2

3

)(

)2(

zAA

zA

P

PSHG


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


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


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


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


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


(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,


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]


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


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

)(

)(


)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


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


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)


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), …


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)(

QQ

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


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)


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


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 :


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


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,


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)



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)

Chapter 8. Second-Harmonic Generation and Parametric...

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