1. Janez Žabkar Advisers: dr. Marko Zgonik dr. Marko Marinček
PHASE MATCHING
Janez Žabkar
Advisers: dr. Marko Zgonik
dr. Marko Marinček

2. Introduction Motivation Basics of nonlinear optics
Birefringent phase matching
Quasi phase matching
Conclusion

3. Motivation An eye-safe laser
Problems with other laser sources (Er:glass – low repetition rates, diode lasers – small peak powers)
Recent progress in growing large nonlinear crystals enables efficient conversion
A basic condition for efficient nonlinear conversion is phase-matching

4. Nonlinear conversion – second harmonic generation

5. Nonlinear optics (1) The wave equation for a nonlinear medium is:
EM field of a strong laser beam causes polarization of material:
Putting in:
We get:
And using:
Nonlinear optical
coefficient:
d = ε0 χ / 2

6. Nonlinear optics (2)
The phase difference between the wave at ω3 and the waves at ω1, ω2 is:
With the non-depleted pump approximation and condition for conservation of energy:
We obtain:

7. Nonlinear optics (3) Hence the energy flow per unit area: =1 for ∆k=0

8. Birefringent phase matching (1)
type-I phase matching for SHG:
Polar diagram showing the dependance of refractive indices on propagation direction in a uniaxial, negative birefringent crystal for type-I phase matching.

9. Birefringent phase matching (2)
type-II phase matching for SHG:
Polar diagram showing the dependance of refractive indices on propagation direction in a uniaxial, negative birefringent crystal for type-II phase matching.

10. Poynting vector walk-off

11. Birefringent phase matching (3)
Dispersion in LiNbO3. The extraordinary refractive index can have any value between the curves.

12. Quasi phase matching for SHG
Isotropic, dispersive crystal
Fundamental field (ω1)
lc = π/∆k, coherence length
∆k=k2-2k1
SH polarization of the medium (ω2 = 2ω1)
SH field (ω2) radiated by SH polarization

13. Periodically poled crystal
Nonlinear optical
coefficient:
d = ε0 χ / 2
A schematic representation of periodically poled nonlinear crystal.

14. Performance of quasi phase matching
Recall:
growth of the SH field
For perfect birefringent PM (∆k=0) and d(z)=deff:
Nonzero elements of tensor d: d11 = - d12 = - d26
d14 = - d25
For ordinary polarization:
deff = d11 cos(θ) cos(3φ)
For extraordinary polarization:
deff = d11 cos2(θ) sin(3φ) +
d14 sin(θ) cos(θ)
Where deff is an effective nonlinear coefficient obtained from tensor d for a certain crystal, direction of propagation and polarization:
Nonzero elements of tensor d: d11 = - d12 = - d26
d14 = - d25
Example: QUARTZ

15. Performance of quasi phase matching
growth of the SH field lc
Since: →
perfect periodically poled structure
We get:
the difference to
birefringent PM
Second harmonic field:

16. Performance of quasi phase matching
birefingent PM
∆k=0
QPM
∆k≠0

17. Some benefits of QPM
The possibility of using largest nonlinear coefficients which couple waves of the same polarizations, e.g. in LiNbO3:
Noncritical phase matching with no Poynting vector walk-off for any collinear interaction within the transparency range
The ability of phase matching in isotropic materials, or in materials which possess too little / too much birefringence

18. Fabrication of a periodically poled crystal

19. Conclusion
Phase matching is necessary for efficient nonlinear conversion
Ideal birefringent PM: intensity has quadratic dependence on interaction length
QPM: smaller efficiency than birefringent PM (4/π2 factor in intensity)
Advantages of QPM (larger nonlinear coefficients,...)