Multi-wave MixingIn this lecture a selection of phenomena based on the mixing of two or more waves to produce anew wave with a different frequency, direction or polarization are discussed. This includesinteractions with non-optical normal modes in matter such as molecular vibrations which underthe appropriate conditions can be excited optically via nonlinear optics.In nonlinear optics “degenerate” means that all the beams are at the same frequency and“non-degenerate” identifies interactions between waves of different frequency. Since theinteractions occur usually between coherent waves, the key issue is wavevector matching. Because there is dispersion in refractive index with frequency, collinear wave-vector matched non-degenerate interactions are not trivial to achieve, especially in bulk media. Of course, thewaves can be non-collinear to achieve wavevector conservation in which case beam overlapreduces interaction efficiency.

Degenerate Four Wave Mixing (D4WM)Beam Geometry and Nonlinear Polarization

“P1” and “p2” are counter-propagating pump waves, “s” is the input signal, “c” is the conjugate

0cs2p1p kkkk

cs kk

1p2p kk

)()(E)-()(E)()(E)-()(E

)()(E)()(E)()(E)-()(E21)(

(c)*(c)(s)*(s)

*(p2)(p2)*(p1)(p1)

iiii

iiiiiE

321321321321)3(

0)3( )()()()(),,;(ˆ)( dddδEEEP lkjijkli

Each is the total field!Each total field is given by

General case for third order polarization, in frequency domain

Full 4-wave mixing process will contain 8x8x8 terms!But, will need wavevector conservation in interaction → reduces # of terms

1. Assume beams are co-polarized along the x-axis and treat as scalar problem

2. E(p1) and E(p2) (pump beams) are strong, and E(s) and E(c) are weak beams3. For the nonlinear polarization, only products containing both pump beams and either the signal or conjugate beam are important 4. Nonlinear polarization (products of 3 fields) need p1, p2 and c or s

rki*x

(xxxxxxxxxx

xxxxxxxx

xxxxxxxxx

e

s(c))2p)1(p1p2pcs

)3(2p1pcs

)3(

c1p2ps)3(

c2p1ps)3(

1pc2ps)3(

2pc1ps)3(

0s(3)

)},,;(ˆ),,;(ˆ

),,;(ˆ),,;(ˆ

),,;(ˆ),,;(ˆ{41)(P e.g.

EEE

).(Pfor resultsSimilar c(3) x

.].}2222

)}{()({}22

22)}{()([{41

)((c)*(p2))((s)*(p2))((c)*(p1))((s)*(p1)

21)((c)(p2))((s)(p2)

)((c)(p1))((s)(p1)21

spspspsp

spsp

spsp

ccee

ee

rkkirkkirkkirkki

rkkirkki

rkkirkki

EEEEEEEE

EEEE

EEEE

5. “grating model”: products of two fields create a gratingin space, and a third wave is deflected by the grating to form beams s or c. Initially the only 2 field products of interest are (p1 or p2) x (c or s).

)()( 21)3()1(

effective, xxxxxxxx EE

There are two kinds of time dependences present here correspondingto the first two inputs in 1. oscillates at 2 (requires electronic nonlinearity)2. DC in time

..)()( 21 cc ..)()( 21 cc

),,;(ˆ 321)3( xxxx),( 21

- Now form E(1)E(2) x E(3) subject to the following restrictions1. Terms can only multiply terms with so that the

output frequency is 2. The product of three different beams is required3. Because the pump beams are the “strong” beams, only products with two pump beams are

kept which generate signal or conjugate beams. Assume Kleinman limit.

)()( 21 )( 3

First term generates the conjugate and the second term the signal.

}{),,;(46)(P)()()( ss (c)*(s)(p2)(p1))3(

0(3)

321rki

xrki*

xxxxxxxx eeEEE EEEE

}{);(-2)(P ss (c)(s(p2)(p1)||2

20

2(3) rki*x

rki)*xxxx eencn

EEEE

D4WM Field Solutions

),'()()(),'('

),'()()(),'('

(c)*(p2)(p1)20

(s)

(s)*(p2)(p1)20

(c)

znnizdzd

znnizdzd

EEEE

EEEE

Using the SVEA in undepletedpump beams approximation

Simplifying in the undepleted pump approximation

)'(1)'('

)()(1 )'(1)'('

(c)*

4

(s)

(p2)(p1)20

4

(s)*

4

(c)

zizdzd

nnzizdzd

WM

WMWM

EE

EEEE

Applying the boundary conditions: 0),0( 0),'( (s)(c) EE L

Output of conjugate beam:

WM

Li4

(s)*(c) ' tan),0(),0(

EE

R>1 ? YES! Photons come out of pump beams!Need to include pump depletion.

WM

L

4

22(s)

2(c) 'tan|),0(||),0(|ty"Reflectivi"

EE

WM

LL

4

22(s)

2(s) 'sec|),0(||),'(|vity"Transmissi"

EE

Can get gain on both beams!

T and R and n"oscillatio" 2

4

WM

L'

Unphysical, need pump depletion

Manley Rowe Relation

)'((s)* )'((c)*4

1 )'((s)'

)'((s)*

)'((c)* )'((s)*4

1)'((c)'

)'((c)*

zxzWM

izdzdxz

zxzWM

izdzdxz

EEEE

EEEE

)'('

)'('

)()( zIdzdzI

dzd cs

Since signal travels along +z, and the conjugate travelsalong –z, both beams grow together at expense of the pump beams.

Can be shown easily when z z and allowing pump beam depletion

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

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

(p1)*(c)(s)20

(p2)

(p2)*(c)(s)20

(p1)

zzznnizdzd

zzznnizdzd

EEEE

EEEE

Note that the p1 and s, and the p2 and c beams travel in the same direction

)()( and )()( )c()2p()s()1p( zIdzdzI

dzdzI

dzdzI

dzd

Pump beam #1 depletes Pump beam #2 depletesSignal beam grows Conjugate beam grows

Wavevector Mismatch What if pump beams are misaligned, i.e. not exactly parallel?

cs2p1p kkkkk

Assume that z and z are essentially coincident

kzi

WM

kzi

WMeziz

dzdeziz

dzd )(1)( )(1)( (c)*

4

(s)(s)*

4

(c) EEEE

Form of solutions, subject to the usual boundary conditions 24

22 4/ with WMk 2

4max

242

2 2)2/(R ]

2[)(cos

)(sinR

WMWM kL

kL

L

Linear Absorption absorption of all 4 beams, no pump depletion approximation to signal and conjugate used

)(2(p2)(p2)2(p1)(p1) )0()( ;(0)e(z)

Lzzez

EEEE

)(2

)()( );(2

)()( (c)(s)*

4

(c)(s)(c)*

4

(s) zzizdzdzziz

dzd

WMWMEEEEEE

224

2

2(p2)(p1)20

4WM

]2/[

)()0( 1 Redefine

WM

L

eLnn

EE

224

2

)]sin(}2

{)cos([

)(sinRLL

L

WM

)()0(}4

);();({)(4R )2p()1p(

2vac

2||22

||22

vac LIIk

nLk

Complex )3(̂ But is in general a complex quantity, i.e. )3()3( i

index change absorption change

)3(̂

Both the real (n2) and imaginary (2) parts of contribute to D4WM signal)3(̂

Three Wave Mixing

Assume a thin isotropic medium. Co-polarized beams, x-polarized with small angles between input beams

)cos,(sin );cos,sin(),( 2p1p kkkkkk zy z

y

x

)1*(p2)2(p)2*(p2)1(p E][E and E][ELook at terms

rkki

rkki

ecnnz

ecnnz

)2()2(p*2)1(p||2

20

(s2)

)2()1(p*2)2(p||2

20

(s1)

2p1p

1p2p

])[;(-][)(P

])[;(-][)(P

EE

EE

zkkkiss

ss

ssssppppe

ki

zSVEA )(2)2,1(

2,1

20

)2,1(2,12,11,2

2(z)

PE

)cos,sin3(2p1p2 ][2

)(

)cos,sin3(1p2p2 ][2

)(

)2(p*2)1(p202s

)2(2s

(s2)

)1(p*2)2(p201s

)2(1s

(s1)

2s21p

1s1p2p

kkknneizdzd

kkknneizdzd

rkkki

rkkki

p

EEE

EEE

Assuming that the beams are much wider in the x-y plane than a wavelength, wavevector isconserved in the x-y plane. For the signal field which must be a solution to the wave equation,

)291()sin

291( ]sin91[sin9 2222222222 kkkkkkkkk zyz

222,2,1,1,1,2, 4 )

291(cos2 kkkkkkkkk szpzpzszpzpzz

).0(])0(2

)sinc;([)(

),0(])0(2

)sinc;([)( :equationsSVEA thegIntegratin

)2p(2)1p(2||2vac

)2s(

)1p(2)2p(2||2vac

)1s(

IILkLnkLI

IILkLnkLI

z

z

(s2)(s1) E and E can interact with the pump beamsagain to produce more output beams etc.

(p2)(p1) E and E

Called the Raman Nath limit of the interaction

Non-degenerate Wave Mixing

0}|)(|

)()(|)(|

)()(

|)(|)()(

|)(|)()({1)()()()(

4

444

3

333

2

222

1

1114321

kkn

kk

n

kkn

kkn

ckkkk

In the most general non-degenerate case with frequency inputs, in which and are the pump beams, then for frequency and wavevector conservation,

4321 and ,, 12 4321

A frequent case is one pump beam from which two photons at a time are used to generate twosignal beams at frequencies above and below the pump frequency which, for efficiency, requires

0 )()()(2 ;2 443311431 nnnk

..)(21 ; ..);(

21 ; ..)(

21 )(

3)4()(

3)3()(

1)1( 443311 cceEccerEcceE tzkitzkitzki EEE

Assuming(1) co-polarized beams, (2) the Kerr effect in the non-resonant regime,(3) cross-NLR due to the pump beam only,(4) and a weak signal (3) input,the signal and idler (conjugate) beamnonlinear polarizations are

The fields are written as:

,);()(

);()(2 );()(2 :Defining

)(2111||2

4,3

14,3vac4,3

111||21vac111||24,3

14,3vac4,3

1

ieInnnk

InkkInnnk

ziezizizdzd ),(),(),( :SVEA theinto ngsubstituti and 3,4

*4,34,34,34,3 EEE

})()]([),,;(~43

)(),,;(~4

62)(

])();(2[4,3

*213,4114,3

)3(0

3,414,3114,3)3(

14,3

)(

11vac11||2 zIknkixxxx

xxxxNL

e

Icn

EE

EP

, )(),( )(),( :onssubstituti theUsing 434433

zizi ezBzezBz EE

44344342

2

34334332

2)]([ ;)]([ B

dzdiBB

dzdB

dzdiBB

dzd

.2

)( 4)]([21

2)( 43

432

4343 gii

zz eBeBB 4,34,34,3 :form theof solutionsFor

The signal and idler grow exponentially for when g is real!!For imaginary g, the solutions are oscillatory

24343 )]([4

0)0,( ,0)0,( conditionsboundary For the 34 EE

.)sinh()0,(),(

,)]sinh(2

)())[cosh(0,(),(

2)(

34

4

2)(

4333

43

43

zi

zi

egzg

iz

egzg

izgz

*EE

EE

A number of simplifications can be made which give insights into the conditions for gain.Expanding Δk around for small . The sign of Δk is negative in the normal dispersion region and positivein the anomalous dispersion region. The condition for gain can now be written as

2124311 )()()()(2)( kkkkk

3114

.0]21);()(2[]

21);()(2[ 2

2111||21vac2

2111||21vac kInkkInk

0)]}()()(2[);({

]);()()(2[

23vac

4

13vac

3

11vac111||2

2111||24vac3vac

43

1

knnk

nnkInk

Inkknn

n

After some tedious manipulations valid for small Δ, the condition for gain becomes

Gain occurs for both signs of the GVD and the nonlinearity provided that the intensity exceeds the threshold value

.||21|||);(|)(2 2

2111||21vac kkInk

This means that the cross-phase nonlinear refraction due to the pump beam must exceed theindex detuning from the resonance for gain to occur.

Nonlinear Raman SpectroscopyUsually refers to the nonlinear optical excitation of vibrational or rotational modes. A minimum of two unique input beams are mixed together to produce the normal mode at the sum or difference frequency. Although any Raman-active mode will work, vibrational modes typically are very active in modulating the polarizability.

Note: Must include dissipative loss of the normal modes in Manley-Rowe relations

a

a

Degenerate Two Photon Vibrational Resonance

Optical coupling between two vibrational levels (inside the vibrationalmanifold of the electronic ground state)

0 r lity tensopolarizabi

kqk

ijkk

Lijij q

qα

Vibrational amplitude

e.g. the symmetric breathing mode in a methane molecule (CH4)

CH

H

H

H CH

H

H

H

..)(21),( )( ccetrE taraki

ainc

E

)()()(sdiscussion previous From )1()1(0 aaNLq

NL Eq

qp

EEqm

qqq aq

2)1(

02v

1v )]([

212 classical mechanics

)2(0

2)(2

02

.].)0(4|)(|

)2(4)([

21),(

ccqDm

eqDm

trq qatrki

qa

a aa

EE

Note: The molecular vibrations are not only drivenin time due to the field mixing, but also into aspatial pattern for the first term

qz

/ka

Real part gives 2 photon absorption; Imaginary part gives index change

21-v

2v

)3(2

0220

2

1-v

)()2()()(

32)(resonancenear 2PA For

a

aaq

aa

Iqcmn

Ndzd EE

)3(2

01-v

220

2max||22

||216

);( )();()(

qa

aaaaaa qcmnNII

dzd

)()()()2(

232

)( :);( associatedFor 21-v

2v

v)3(2

0220

2||2 aaa

aq

aa I

qcmnNi

dzdn

EE

21-v

2v

v)3(2

020

2||2

2vac

)()2(2

32);(

)()();()(

a

aq

aaaa

aaaaa

qcmnNn

I-nikdzd EE

0);( 2 0);( 2 ||2v||2v nn aa

1-v

22v

2)3(

2

0NL

44)(|)(|

8)(absorptionphoton woresonant tFor

aa

aaqa

iqmN

EEP

D(2a))2()]([ 2 )1(3)1()3(

aaaNL fff

21-v

2v

1-vv

2)3(

2

00 )()2(

24

)(|)(|16

)( SVEA

a

a

a

aaq

a

aa

iqcmn

Nidzd EEE

II

field strong - )(E

field strong - )(E

a

b

New fields generated at 2a-b and 2b-a (only one can be phase-matched at a time)

CARS – Coherent Anti-Stokes Raman Spectroscopy

Nonlinear Raman Spectroscopy

Nonlinear process drives the vibration at the difference frequency a-b between input fields

..)(..)(

21 )()(

inc ccecceE tbrbkib

tarakia

EE

I

field weak polarized- )(E

field strong polarized- )(E

jy

ix

b

a

Makes medium birefringent for beam “b” and changestransmission of medium “b”

RIKES – Raman Induced Kerr Effect Spectroscopy ( Raman Induced Birefringence)

RIKES jbNLnqn

ijnNLijq

n

ijnn

Liji Eq

qNPE

qqp

nn)()( )1()1(

0,loc0

)()()()(12:mechanics classical )1()1(0

2v

1v bjaiabq

n

ijnnnn EE

qmqqq

n

..)(

)()()()(

41),( ])()[(

*

*)1()1(

0 cceDqm

trq trkki

ba

bjaibaq

n

ijnn

ababn

EE

cceDq

xqm

NtrP trki

ba

bjaiq

n

ijnq

n

jinNLj

bbabNL

bbnn

.)(

)(|)(|8

),(

is at on polarizatinonlinear the,)]()([ define , Since

][*

2)3(

0

*

0

2)1()1()3(

EE

1. Index change produced at frequency b by beam of frequency a2. Nonlinear gain or loss induced in beam “b” by beam “a”3. One photon from beam “a” breaks up into a “b” photon and an optical phonon4. Propagation direction of beam “a” is arbitrary, only polarization important!

)()()(])[(

][][8

)(

21-v

2v

1-vv)3(

00v

220

bjaiba

baq

n

ijnq

n

jin

ab

bbj Ii

qqcmnnNi

dzd

SVEA

nn

EE

Imaginary part contribution → to nonlinear refractive index coefficient

21-v

2v

v)3(00

v20

2

b2

)(])[(][][

8);(

)()();(-)(

ba

baq

n

ijnq

n

jin

abab

bjaiab

bj

nn qqcmnnNn

Inc

idzd EE

21-v

2v

)3(20

v22

0

1-v

2

21-v

2v

)3(00

v22

0

-1v

)(])[(1][

4);(

)(])[(

)()(][

8)(

baq

n

jin

ab

bab

ba

bjaiq

n

ijnq

n

jin

ab

bbj

n

nn

qcmnnN

Iqqcmnn

Ndzd EE

Real part → contribution to nonlinear gain (or loss)

LII

ILI

LIeI

LI

aabb

bb

aabLI

b

b aab

)();()0,(

)0,(),(

)();(1)0,(),(Solving

2

2)();(2

Modulating the intensity of beam “a” modulates the transmission of beam “b”. Varying a - b through gives a resonance in the transmission! Assumed was a crystal. If medium is random, need to work in both lab and molecule frames of reference and then average over all orientations.

v

tiebrabrQabr ab )2(),E(),()2,(NLPfor Looking

CARS

For simplicity, assume two input co-polarized beams, b>a.

Coherent Anti-Stokes Raman Scattering (Spectroscopy)

rki

ab

aibiabbaq

n

iin

c

cabi

ab

aibiq

n

iinab

NLi

eDqmcn

Nidzd

DqmN

n

n

)()()()2()]()[(||

)(8)2(

)(

)()(||4

)2(

*2)1(2)1()1(2

00

*2)3(2

0

EEE

EEP

Field at is written as abc 2 ..)(21 ])([ cce tczcki

ci EThis process requires wavevector matching to be efficient,

)( bk

)( ak

)( ck

)( bk

Cannot get collinear wavevector matchingbecause of index dispersion in the visible

cab kkkk

2

since c >> b - a then angles are small relative to z-axis

)(2)()( 2 c babca kkknnn

21-v

2v

222)3(4

02v

240

42

222

)()]([

)()()2

(sinc][||

64 ),(

ab

abq

n

iin

abc

cc

IILk

qmcnnnLNLI

n

Differences between RIKES and CARS

Automatically wave-vector matched inisotropic medium. No new wave appears.

Have resonance at ||v ba

Requireswave-vector matching

Can also have CSRS (different wavevector matching conditions)Signal appears at )2( ba

When is tuned through , resonantenhancement in the signal occurs. Monolayersensitivity has been demonstrated. There is also a

)( ab

“background” contribution due to electronictransitions via:

v

),,];2[()3(bababxxxx

For comparable contributions ofbackground and resonance terms