Phase Matching

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Phase Matching
Alex Filin
Everything you always wanted to know about it
but 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 nonlinearitymechanical analog
Nonlinear conditions Force
Linear conditions Force
Potential
Potential
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Origin of optical nonlinearityPolarization
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Origin of optical nonlinearity
Linear conditions
Nonlinear conditions

• Where P is polarization
• eo is free-space permittivity
• is susceptibility
• E is electric field

• Where
• (i) is nonlinear susceptibility
• of ith order

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Origin of optical nonlinearity

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

• All mixing phenomena,
• involving generation of
• sum and difference
• frequencies (SHG,
• parametric amplification)
• Pockels effect
• Optical rectification
• c(2) vanishes in media
• with inversion symmetry

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Second Harmonic Generation
Why does phase mismatching happen?
w
E(z)
2w
t1
z
w
E(z)
2w
t2
z
w
E(z)
2w
t3
z
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Second Harmonic Generation
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 q q0 and Or n2w
nw , but nw lkw/2p and n2w (l/2)k2w/2p So,
2kw k2w, or
Dk k(2w) - 2k(w) gt 0
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Second Harmonic Generation
One can show, that electric field
And Poynting vector
Because
In ideal case (Dk 0)
gt
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Second Harmonic Generation
In real case Dk 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
n2w nw
Dk k(2w) - 2k(w) gt 0
or
b) Correct crystal length to reach maximum SHG
power
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Coherent Anti-Stokes Raman Spectroscopy (CARS)
wP
wCARS
wP
wS
wRaman
2wP
wS
wCARS

• q1 and q2 correspond to wP
• q3 corresponds to wS
• wP wS wRaman is the Raman shift (Raman
  active vibrational mode)

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Coherent Anti-Stokes Raman Spectroscopy (CARS)
Intensity
Where

is the refractive index at frequency
is the intensity of i-th signal
Phase matching for BOXCARS
is the interaction length
kP1
kP2
kS
kCARS
After Maker and Terhune (1989)
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Principles of BOXCARS Method
Geometry of laser beams for BOXCARS
Phase matching for BOXCARS
Lens 2
kP2
kP1
wCARS
jP
jS
wPump
kS
kCARS
wStokes
f
For h ltlt f
h
d
or
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Phase Matching in fs-BOXCARS
h
f
I
f
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fs-CARS Theory
normalized CARS frequency and normalized Stokes
detuning
Where
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Phase Matching in fs-BOXCARS
ps-CARS
fs-BOXCARS
So far
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Phase Matching in fs-BOXCARS
Our results
Without G-correction
With G-correction
1.60
1.60
1.70
1.70
1.80
1.80
CARS Photon Energy, meV
CARS Photon Energy, meV
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Phase Matching in fs-BOXCARS
Comparison theory and experiment
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- experiment
- no correc.
- with correc.
0.5
Intensity, arb.units
0
2200
1000
1800
1400
Wavenumber, cm-1
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Conclusion

• Every nonlinear optical phenomenon requires its
  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 its results.