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Non-Linear Optics
and
Generation of Tunable Pulses
Introduction to Optical Parametric Oscillators
Claudio Perego,
Dipartimento di Fisica
Università Degli Studi Milano-Bicocca
November, 9th 2010
UltraShort and Intense Laser
Technology and Metrology
UltraShort and Intense Laser
Technology and Metrology (USIL)
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Outline
•Non-linear Optics Basis
•Second order effects
Second Harmonic Generation
Sum/Difference Frequency
Generation
Optical Parametric Amplification
•Optical Parametric Oscillators
Basic Idea
Properties
•Summary
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Non-Linear Optics (1)
Bounded electron
•The bound electron oscillates
along the potential well as the
EM field is present
V(x)
•At low field intensities
the potential is almost harmonic
x
•At high intensities
the potential is anharmonic
Gauss Law
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Non-Linear Optics (2)
Polarization Vector:
Linear response:
Non-linear response:
…by observation:
c(2) effects: second harmonic generation, difference-sum frequency
generation,… (non-centrosymmetric materials)
c(3) effects: optical Kerr effect, four wave mixing, stimulated light
scattering, two-photon absorption (all materials)
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Non-Linear Optics (3)
EM Wave Equation:
Field Decomposition:
An equation for each frequency component is deduced:
Where:
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Non-Linear Optics (4)
We Assume:
•Plane Waves propagating along the z-axis
•Scalar field Approximation
•Slowly Varying Envelope Approximation
•t-independent amplitude
The harmonic component are simplified:
Equation for the n-th harmonic amplitude
In which:
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2nd Order Effects
Incident Field
Non-linear Medium (2nd Order )
Evaluating the 2nd Order Polarization:
SHG
SFG+DFG
OR
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2nd Harmonic Generation (1)
2 photons at the same w entering the medium
In the photonic interaction we
must take into account:


Phase Matching
From these laws we get
Which lead to
The velocity of the 2nd harmonic wave should coincide with that of
the fundamental wave.
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2nd Harmonic Generation (3)
Using EM field equations:
Fundamental wave
Harmonic Response
The dynamics of the 2nd harmonic amplitude can be solved giving:
Phase matching provides the best conversion efficiency
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2nd Harmonic Generation (3)
How to reach the phase matching condition?
•Normal dispersion
, phase matching impossible.
•We can use birefringence, negative uniaxial crystals
Optical Axis
Light
Incidence
The crystal can be oriented such
that :
Phase matching is achieved
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Sum/Difference Frequency
We can also consider the 2 photonic processes:
Sum Frequency Generation
Difference Frequency Generation
Using EM field equations: 3 Wave Mixing
Same equations but different initial conditions
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Sum Frequency Generation (SFG)
•The problem is treated considering a strong pump (wp) and a
weak signal (ws) as input fields.
•Up-conversion to wsum
•The pump is not affected by
interaction:
Undepleted Pump
Solving the EM equation we find:
The up-conversion oscillates along the propagation direction
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Difference Frequency Generation (DFG)
•In this case a wave with the difference of the input frequencies is
generated: the idler wave (wi)
•The pump has the highest
frequency
•Undepleted Pump
Solving the EM equation we find (
)
The signal is amplified along z, with gain
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Return Current
Alfvén Limit
An electron beam in a conductor is limited by the self-generated
magnetic field.
This introduces a maximum possible current
The strong charge separation created by the electron beam produces
a background current in the opposite direction
The beam can propagate but…
Weibel
Instability
This anisotropic configuration is
physically unstable
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Outline
•Inertial Confinement Fusion
Laser Driven Fusion
Fast Ignition
Electron Transport
•The Weibel Instability
Physics of Weibel Instability
Analytical Study
Experimental Study
Numerical Study
•Summary
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Weibel Instability Basics
Condition: velocity distribution anisotropy
Linear Phase
•Counterstreaming particles
•Perturbation B-field
•Lorentz Force
•Filaments Creation
•B-field Amplification
Non-Linear Phase
Return current
Fast beam
•Biot-Savart interaction
•Merging of the filaments
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Linear Phase
Growth Rate
Weibel, 1959
•Vlasov + Maxwell equations
•Perturbative approach
•Fourier Space + Semplifications
•Velocity space anisotropy
Anisotropic Maxwellian
Dispersion Relation
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In Fast Ignition…
Silva et al. 2002
Analytical model of the purely
transverse Weibel Instability.
Parametric study relevant for fast
ignition conditions.
No
instablility
The tranverse temperature of the
beam strongly suppresses the
instability
Problem Solved?
•Longitudinal modes play a role in the real case
•Non-linear phase
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Experimental Study
Tatarakis et al. 2003
Experiment
•Shadowgraphic study of fast
electron propagation in low
density plasma.
•Beam/plasma density ratio
comparable to fast ignition
case.
Results
Evidence of electron beam filamentation due
to Weibel Instability
Fine scale filamentation for early times
Larger filaments at later times
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Non-Linear
Phase
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Particle In Cell
Physical System
Electrons and ions
governed by Vlasov and Maxwell equations
Numerical System
sampled by numerical particles density functions
PIC Cycle
Current and charge
calculation on grid
Numerical particles
motion integration:
Spatial
Grid
Interpolation of Lorentz
Force on particles
Field evaluation by
Maxwell equations
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PIC Results
Karmakar et al. 2009
2D and 3D PIC Simulations:
•electron beam + counterstreaming
background plasma
•Density ratio relevant for Fast
Ignition
•Role of tranverse temperature
.
does not suppress the
filamentation in 3D
Non-linear interaction between
Weibel and 2-stream instabilities

Limitations
•Unable to simulate ICF densities
•Ideal initial conditions of the electron beam
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Outline
•Inertial Confinement Fusion
Laser Driven Fusion
Fast Ignition
Electron Transport
•The Weibel Instability
Physics of Weibel Instability
Analytical Study
Experimental Study
Numerical Study
•Summary
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Summary
•Fast Ignition approach to ICF
•Electron beam propagation
•Possible developement of Weibel instability
in the ICF plasma
•Several studies: analytic modeling,
experiments, PIC simulations
A lot of interesting work to be done…
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The End
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