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Phase MatchingAlex Filin
Everything you always wanted to know about itbut were afraid to ask
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Outline
• Introduction: Origin of Optical Nonlinearity
• Phase Matching in SHG • Phase Matching in CARS• Conclusion
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Origin of optical nonlinearity:mechanical analog
Linear conditions
Force:kxxF )(
Potential:2
21)( kxxU
Nonlinear conditions
Force:3)( xkxxF
Potential:42
41
21)( xkxxU
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Origin of optical nonlinearity:Polarization
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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
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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
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Second Harmonic GenerationWhy does phase mismatching happen?
E(z)
E(z)
E(z)
z
z
z
t1
t2
t3
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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
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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
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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
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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
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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
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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
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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
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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
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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:
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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
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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
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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.