PHASE MATCHING

Janez Žabkar

Advisers: dr. Marko Zgonik dr. Marko Marinček

Introduction

• Motivation

• Basics of nonlinear optics

• Birefringent phase matching

• Quasi phase matching

• Conclusion

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

Nonlinear conversion – second harmonic generation

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

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:

Nonlinear optics (3)

• Hence the energy flow per unit area: =1 for ∆k=0

∆k=0

∆k≠0

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:

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:

Poynting vector walk-off

Birefringent phase matching (3)

Dispersion in LiNbO3. The extraordinary refractive index can have any value between the curves.

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

Periodically poled crystal

A schematic representation of periodically poled nonlinear crystal.

Nonlinear opticalcoefficient:d = ε0 χ / 2

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)

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: →

Performance of quasi phase matching

∆k=0

∆k≠0

QPM

birefingent PM

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

Fabrication of a periodically poled crystal

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,...)