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.
