nonlinear optics lab. hanyang univ. chapter 8. second-harmonic generation and parametric oscillation...

Nonlinear Optics LabNonlinear Optics Lab. . Hanyang Hanyang Univ.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 susceptibilitywhere,)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.)( 2121 titi eEeEtE

][2

]c.c.22[)(*22

*11

)2(

)(*21

)(21

222

221

)2()2( 212121

EEEE

eEEeEEeEeEtP titititi

(OR))(2)0(

)DFG(2)(

)SFG(2)(

)SHG()2(

)SHG()2(

*22

*11

)2(

*21

)2(21

21)2(

21

22

)2(2

21

)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

)( 4220 Bx

mx

mxV

Restoring force :

...320

mBxxmx

VF

Equation of motion :

mteEBxxxx )/(2 320

Damping force

Restoring force

Coulomb force


Purtubation expansion method :


Assume,

)()( tEtE

)3()3()2()2()1( xxxx

Each term proportional to n should satisfy the equation separately

mteExxx )/(2 )1(20

)1()1(

02 )2(20

)2()2( xxx

02 )1(3)3(20

)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

)( 3220 Dx

mx

mxV

Restoring force :

...220

mDxxmx

VF

Equation of motion :

mteEDxxxx )/(2 220

Damping force

Restoring force

Coulomb force


Similarly,


Assume,

)()( tEtE

)3()3()2()2()1( xxxx

Each term proportional to n should satisfy the equation separately

mteExxx )/(2 )1(20

)1()1(

0][2 2)1()2(20

)2()2( xDxxx

022 )2()1()3(20

)3()3( xDBxxxx

Solution :

ccexextx titi .)()()( 212

)1(1

)1()1(

jj

j

j

jj i

E

m

e

L

E

m

ex

2)()(

220

)1(

: Report


Example) Solution for SHG

)(

)/(2

12

21

22)2(2

0)2()2(

1

L

EemeDxxx

ti

Put general solution as tiextx 121

)2()2( )2()(

)()2(

)/()2(

12

1

21

2

1)2(

LL

EmeDx

: Report

Similarly,

)()2(

)/()2(

22

2

22

2

2)2(

LL

EmeDx

)()()(

)/(2)(

2121

212

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(

jj L

meN

: linear susceptibility

2)1()1(322

23)2( )]()[2(

)()2(

)/(),,2( jj

jjjjj eN

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(

jjjj LLL

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) : m0

2d=mDd D=w02/d : roughly the same for all noncentrosymmetric solids.

440

2

3)2(

dm

e

(non-resonant case) : used in rough estimation of nonlinear coefficient.

20

220 2)( jjj iL 3/1 dN dD /2

0

60

20

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

mniti

mniimnmn ePePtP )()( )()(),r(

),()(),,()()(

)2(mknjmnmn

jk nmijkmni EEP where,

2nd order nonlinear susceptibility tensor

# Full matrix form of )( mniP

)()(),,(

)()(),,(

)()(),,(

)()(),,()(

222222)2(

121212)2(

212121)2(

111111)2(

kjjk

ijk

kjjk

ijk

kjjk

ijk

kjjk

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(nmmnijkmnmnijkmniP


4) Kleinman symmetry (nonresonant, is frequency independent)

)()()(

)()()(

213)2(

213)2(

213)2(

213)2(

213)2(

213)2(

kjijikikj

kijjkiijk

intrinsic

3) Full permutation symmetry (lossless media : is real)

)(

*)()()(

321)2(

321)2(

321)2(

213)2(

jki

jkijkiijk

: Indices can be freely permuted !


Define, 2nd order nonlinear tensor, )2(

21

ijkijkd

)()(2)()(

mknjk 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

ijkdcan 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

d

iht

he Ped 0 ei σ

Polarization :NL0 PeP e

Assume, the nonlinear polarization is parallel to the electric field, then

2NL

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

ttzkkti

..)(

)(2

)(

2

1 )(1

21

)(11

)(2

12

111111 ccezEkez

zEike

z

zE zktizktizkti

..)(

2)(2

1 )(111

21

11 ccedz

zdEikzEk zkti

21

21

1

)()(

dz

zEd

dz

zdEk (slow varying approximation)

......Text


zkkkieEEdi

Edz

dE )(*31

2

2*2

2

2*2 231

22

zkkkieEEdi

Edz

dE )(21

3

33

3

33 321

22

zkkkieEEdi

Edz

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

224)(

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

kkk

lc


Technique for phase-matching in anisotropic crystal

cnk /)( nnkk 2)()2( 2So,

Example) Phase matching in a negative uniaxial crystal

)(

1sincos22

2

20

2

ee nnn


# If

02 nne , there exists an angle m at which

02 )( 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.

02 )( nn m

20

22

2

220

2

)(

1

)(

sin

)(

cos

nnn e

mm 22

022

220

202

)()(

)()(sin

nn

nn

em

Example (p. 289)

Experimental verification of phase-matching

])([2/ 02

nnc

llk e

)()(2

)()()2sin(

2)(

30

220

22

me

m n

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

224)(222)2(

)2/(

)2/(sin)()(

kl

kllrEdrE

Total power of fundamental beam with Gaussian beam profile :

20

2 /0

)( )( reErE

42

1 202

0sectioncross

2)()(

EdxdyEP


So, Conversion efficiency :

2

2

20

)(

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 2/20 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 ll

ll

zki

zki

zki

eAAi

Adz

dA

eAAi

Adz

dA

eAAi

Adz

dA

)(213

33

)(*31

*2

2*2

)(3

*21

11

22

22

22

(8.2-13) where,

)( 213

321

321

0

kkkk

nnnd

lll

SHG : 21 AA

a transparent medium : 0l, and perfect phase-matching case :Let’s consider 0k

*13

1

2AAi

dz

dA 2

13

2Ai

dz

dA


Define, 33 AiA

1*111 real] is)0([ real is )( AAAzA

21

3

131

2

12

1

Adz

Ad

AAdz

dA

0)( 2

32

1 AAdz

d: Total energy conservation

Initial condition : )0(21

23

21 AAA

))0((2

1 23

21

3 AAdz

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

22

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( 202

2

vtdt

dv

dt

vdp

Example) RLC circuit

t

C

CCC po sin1

0


0)sin1(1

002

2

vtC

C

LCdt

dv

dt

vdp

0CCAssuming

Put, ]cos[ tav

0)( ][)()(220 tititi Peieie

where,00

20

0

20 C

1

2

1

RC

C

LC

Steady-state solution :

00 or 0

) that (so 2

pp2/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 ll

ll

zki

zki

zki

eAAi

Adz

dA

eAAi

Adz

dA

eAAi

Adz

dA

)(2133

3

)(*31

*22

*2

)(3

*211

1

22

122

1

22

1

3,2,1

321

321

213

l

nnnd

kkkk

lll

o

ε

ε

where,

0l (phase-matching), and also depletion of field due toWhen 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

)(*2

1

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*

20

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

2/122sincos

)(

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

*211

1

22

1

22

1

Ag

iAdz

dA

Ag

iAdz

dA

2,12,12,1

321

21

0

)0(

dE

nngwhere,

(8.8-1)


Even though Eq. (8.8-1) describe traveling-wave parametric interaction, it is still valid if weThink 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

*2

21

*211

AAg

i

Ag

iA

Condition for nontrivial solution :

0

2

2

2

2det2

1

gi

gi

212 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

Q

tc

1

(4.7-5)

Temporal decay rate :n

c

)0(321

21

0

dEnn

g

21

2 g and2121

3 1)(

QQ

Ed t

230

323 2

nA

PE

Threshold pump intensity :

23

2303

2

1E

n

A

P

Pump intensity :

Threshold pump intensity :

212

212302

3

2303

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

2221

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

22

22

11

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

2333

33

11)2sin(

2 nn

nn

e

2

220

1

1102010

0

2

0

2

3303

1

)(

)2sin(11

21

33

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

2031

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()(

21

1

33

211

# 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

2031

31 dEnn

g

A

P

nnn

dl

P

lP 2

2/3

0321

2223

1

3

2)0(

)(

Example (p. 318)

nonlinear optics lab. hanyang univ. chapter 8. second-harmonic generation and parametric oscillation...

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