PHASE MATCHING
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PHASE MATCHING
Janez Žabkar
Advisers: dr. Marko Zgonik dr. Marko Marinček
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Introduction
• Motivation
• Basics of nonlinear optics
• Birefringent phase matching
• Quasi phase matching
• Conclusion
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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
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Nonlinear conversion – second harmonic generation
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Nonlinear optics (1)
• EM field of a strong laser beam causes polarization of material:
• The wave equation for a nonlinear medium is:
• And using:• We get:
• Putting in:
Nonlinear opticalcoefficient:d = ε0 χ / 2
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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:
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Nonlinear optics (3)
• Hence the energy flow per unit area: =1 for ∆k=0
∆k=0
∆k≠0
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Birefringent phase matching (1)
Polar diagram showing the dependance of refractive indices on propagation direction in a uniaxial, negative birefringent crystal for type-I phase matching.
type-I phase matching for SHG:
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Birefringent phase matching (2)
Polar diagram showing the dependance of refractive indices on propagation direction in a uniaxial, negative birefringent crystal for type-II phase matching.
type-II phase matching for SHG:
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Poynting vector walk-off
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Birefringent phase matching (3)
Dispersion in LiNbO3. The extraordinary refractive index can have any value between the curves.
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Quasi phase matching for SHG
Fundamental field (ω1)
SH polarization of the medium (ω2 = 2ω1)
SH field (ω2) radiated by
SH polarization
Isotropic, dispersive crystal lc = π/∆k, coherence length ∆k=k2-
2k1
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Periodically poled crystal
A schematic representation of periodically poled nonlinear crystal.
Nonlinear opticalcoefficient:d = ε0 χ / 2
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Performance of quasi phase matching
Recall: growth of the SH field
For perfect birefringent PM (∆k=0) and d(z)=deff:
Where deff is an effective nonlinear coefficient obtained from tensor d for a certain crystal, direction of propagation and polarization:
Example: QUARTZNonzero elements of tensor d: d11 = - d12 = - d26
d14 = - d25
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)
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Performance of quasi phase matching
perfect periodically poled structurelc
growth of the SH field
We get:
Second harmonic field:
the difference to birefringent PM
Since: →
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Performance of quasi phase matching
∆k=0
∆k≠0
QPM
birefingent PM
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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
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Fabrication of a periodically poled crystal
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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,...)