Phase MatchingAlex Filin

Everything you always wanted to know about itbut were afraid to ask

Outline

• Introduction: Origin of Optical Nonlinearity

• Phase Matching in SHG • Phase Matching in CARS• Conclusion

Origin of optical nonlinearity:mechanical analog

Linear conditions

Force:kxxF )(

Potential:2

21)( kxxU

Nonlinear conditions

Force:3)( xkxxF

Potential:42

41

21)( xkxxU

Origin of optical nonlinearity:Polarization

Origin of optical nonlinearity

...)( 3)3(2)2()1(0 EEEP

Linear conditions Nonlinear conditions

EP 0Where P is polarizationo is free-space permittivity is susceptibilityE is electric field

Where(i) is nonlinear susceptibilityof ith order

Origin of optical nonlinearity

...)( 3)3(2)2()1(0 EEEP

• All mixing phenomena,involving generation of sum and differencefrequencies (SHG, parametric amplification)

• Pockels’ effect• Optical rectification

(2) vanishes in media with inversion symmetry

• Third Harmonic Generation • Kerr effect• All types of FWM phenomena,

including CARS

Second Harmonic GenerationWhy does phase mismatching happen?

E(z)

E(z)

E(z)

z

z

z

t1

t2

t3

Second Harmonic Generation

)(sin

)(cos

),(1

2

2

2

2

2

eoe nnn

In an uniaxial crystal

where ne and no are indexes of refraction for extraordinary and ordinary rays, respectively, is angle between k and optic axis of the crystal Phase matching conditions: = and Or n2 = n , but n = k/2 and n2 = (/2)k/2

So, 2k= k, or

)(),( oe nn

k = k(2) - 2k() => 0

Second Harmonic Generation

kkzezE kzi

)2/sin(),( 2/2

One can show, that electric field

And Poynting vector

2

2

2 )()2/(sin),(

kkzzS

2)2/sin(lim

0

zkkz

k

Because

=>

In ideal case (k = 0)

22

2

),(

),(

zzS

zzE

Second Harmonic GenerationIn real case k never is equal to 0,So, SHG power oscillates with z

Finally, phase matching for SHG requires 2 conditions:

a) Correct angle between k and crystal axis to reach

k = k(2) - 2k() => 0

n2 = nor

b) Correct crystal length to reach maximum SHG power

Coherent Anti-Stokes Raman Spectroscopy (CARS)

P

P

S

CARS

• q1 and q2 correspond to P• q3 corresponds to S•P –S = Raman is the Raman shift (Raman active vibrational mode)

Raman

Laser P

Laser S

2P-S

q1

q2

q3

Sample

2P = S + CARS

Coherent Anti-Stokes Raman Spectroscopy (CARS)

2

2222)3(

420

2

2

)2()2(sin

LkLkLII

cnnnI SP

CARSSP

CARSCARS

Intensity:

After Maker and Terhune (1989)

Where:

in i

iIis the refractive index at frequency

is the intensity of i-th signal

L is the interaction length

CARSSP kkkk

2kCARS

kS

kP1 kP2

Phase matching for BOXCARS

kCARSkS

kP1 kP2

Geometry of laser beams for BOXCARS Phase matching for BOXCARS

Principles of BOXCARS Method

Lens 2

CARSPump

Stokes

Mask Lens 1

PS

f

h

d

|kP1| |kS| |kCARS|2 = +

2Pump=Stokes + CARS

12sinsin Stokes

PumpPS

For h << f

12)( Stokes

PumpStokes dh

or

Phase Matching in fs-BOXCARS

12)( S

PS dh

)( 11 Sh

)( 22 Sh

h

I

h

r0

r0

f

f

2

0

20

0 2))()((exp)()(

rhhII

fs-CARS: Theory

222

21 1 12 2 2

1

2

21 Erf,

C S C

CARS C S

iS CI e B Aie

2 ;S P S R PC S

21 ;2

S SP

P P

FWHMFWHM

2

4ln 2P

P

FWHM

Where: normalized CARS frequency and normalized Stokes detuning

Phase Matching in fs-BOXCARS

2

222

)2()2(sin

LkLkLIAII SPCARS

ps-CARS:

fs-BOXCARS:

fr

dr

hrGII

S

S

C

CSCCARS

REALCARS

000 ,,,,),(

2

0

22

0

2

00

32exp,,,

rh

rd

dr

hrG

S

S

C

C

S

S

C

C

So far:

1.60 1.70 1.801.60

1.70 1.80 80 40 0 -40-80

Stokes Detuning, meVCARS Photon Energy, meV

CARS Photon Energy, meV80 40 0 -40

-80

Stokes Detuning, meV

Phase Matching in fs-BOXCARSOur results

Without G-correction With G-correction

1000 18001400 2200Wavenumber, cm-1

0

0.5

1In

tens

ity, a

rb.u

nits

- experiment- no correc. - with correc.

Phase Matching in fs-BOXCARSComparison theory and experiment

Conclusion

• Every nonlinear optical phenomenon requires it’s own unique approach to understand the phase matching conditions

• Understanding of phase matching is crucially important to run a nonlinear optical experiment correctly and for interpretation of it’s results.