NONLINEAR OPTICS - Institut Optique
Transcript of NONLINEAR OPTICS - Institut Optique
NONLINEAR OPTICS
Ch. 1 INTRODUCTION TO NONLINEAR OPTICS • Nonlinear regime - Order of magnitude
• Origin of the nonlinearities
- Induced Dipole and Polarization
- Description of the classical anharmonic oscillator
model
- Linear polarization and susceptibility
- Nonlinear polarization and susceptibility
• Nonlinear interactions : a brief description
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Induced dipole and macroscopic polarization
A simplistic description of the interaction between a wave and an atom :
- Simplistic model for an atom =
Nucleus (charge +)
Electronic cloud (charge - )
- An applied static electric field acts on the electrons trajectories
E
Change in position of the center of mass of the electrons
€
! F = q
! E + ! v ×
! B ( )
∆x
Lorentz Force
ü Microscopic scale
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Induced dipole and macroscopic polarization
- With an applied EM fiels = temporal deformation of the electronic cloud
Abrev. : EM field means Electromagnetic Field
E
• time variation of the position of the center of mass : ∆x(time)
• the electronic cloud is linked with the nucleus, presence of a restoring
force (linear function of the displacement of the electrons)
• vibrating dipole ( induced dipole) :
time p
time
Induced Dipole
€
! p = q ×Δ
! x (t)
In a linear regime : linear relation between the displacement of the electrons and the restoring force
Induced dipole frequency ω = applied EM field frequency ω
The magnetic part of the Lorentz force is negligible
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Induced dipole and macroscopic polarization
ü Macroscopic scale
- LINEAR MEDIUM case made of N atoms (N identical dipoles)
= « macroscopic »
source term
E time
p time
Induced dipole
P time
Polarisation
Average
€
E(t) = Acos(ωt + kz)
P(t)∝ χ (1)Acos(ωt + kz)
Case of a z propagative EM wave with freq. ω
Source term @ ω
€
! P (t) = N ×
! p = ε0χ
(1)! E (t)
Def. : LINEAR susceptibility of the medium
The polarization vector P(t) acts as a source term in the wave equation
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Classical anharmonic oscillator model
• Classical anharmonic oscillator - Description
Induced dipole (microscopic quantity) :
Nucleus (fixed)
Electron Electronic cloud
Polarization (MACROscopic quantity) : x(z,t) ??
• Equation of motion
E time
p time
Induced Dipole
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Classical anharmonic oscillator model
- Two applied EM wave with frequencies ω1 and ω2 :
E time
p time
Induced Dipole
LINEAR CASE : « LINEAR » response of the vibrating dipole
Vibrating dipoles with frequencies = ω1 and ω2
A material with a collection of N dipoles in a linear
vibrating regime = LINEAR MEDIUM
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Classical anharmonic oscillator model
- Two applied EM wave with frequencies ω1 and ω2 :
E time
p time
NONLINEAR CASE : « NONLINEAR » dependence of the restoring force with the displacement of the center of mass
Induced Dipole
Vibrating dipoles with frequencies = ω1 and ω2 + harmonics
A material with a collection of N dipoles in a nonlinear
vibrating regime = NONLINEAR MEDIUM
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Classical anharmonic oscillator model
• Equation of motion
Solution : perturbation method, taking into account
Driven Coulomb force Restoring force Damping term
⇤20x � �x2 � ⇥x3
x(1) � x(2) � x(3)
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Linear polarization
Harmonic oscillator : equation of motion
Driven solution
Induced dipole
Linear polarizability (microscopic quantity)
Macroscopic Polarization Linear susceptibility
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Linear polarization
Harmonic oscillator : equation of motion
Dispersion Absorption (or amplification)
Lorentzian line shape
Linear Susceptibility
�(1)(⇥) = ��(⇥) + ı���(⇥)
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2nd Order Nonlinear Polarization
Anharmonic oscillator : equation of motion
Driven solution
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2nd Order Nonlinear Polarization
Anharmonic oscillator : equation of motion
Nonlinear Polarization
2nd order Nonlinear Susceptibility
Optical RECTIFICATION
(induces static electric field)
2nd Harmonic Generation
creation of an electric field with
a 2ω frequency component
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2nd Order Nonlinear Polarization
Anharmonic oscillator : equation of motion Conclusion & Comments
• The macroscopic polarization induced inside the material is then given by the sum :
With : (Complex amplitudes)
• Phase mismatching between the polarization component @ 2ω and the free propagative wave @ 2ω :
wavevector related to P(2ω) ≠ wavevector of E(2ω) 2k(ω) ≠ k(2ω)
• Strong enhancement of the nonlinear susceptibility is expected once ω or 2ω (or both) is close to a material transition (@ω0)
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3rd Order Nonlinear Polarization
Anharmonic oscillator : equation of motion Case of a centro-symmetric material
Driven solution
Linear and Nonlinear Polarization
3rd Harmonic generation
Optical Kerr Effect
x(z, t) = �x(1)(z, t) + �2x(2)(z, t) + �3x(3)(z, t) x(2) = 0
⇤(3)(⌅1,⌅2,⌅3) =�N�e4
⇥0m3D(⌅1 + ⌅2 + ⌅3)D(⌅1)D(⌅2)D(⌅3)
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3rd Order Nonlinear Polarization
Conclusion & Comments • The macroscopic polarization induced inside the material is then given by the sum :
With : (Complex amplitudes)
• Phase mismatching between the polarization component @ 3ω and the free propagative wave @ 3ω :
3k(ω) ≠ k(3ω)
• Strong enhancement of the nonlinear susceptibility is expected once ω or 3ω (or both) is close to a material transition (@ω0)
⇤(3)(⌅1,⌅2,⌅3) =�N�e4
⇥0m3D(⌅1 + ⌅2 + ⌅3)D(⌅1)D(⌅2)D(⌅3)
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Nonlinear interactions
ü A short review :
- LINEAR MEDIUM case made of N atomes (N identical dipoles)
= « macroscopic »
source term
E time
p time
Induced dipole
P time
Polarisation
Average
€
E(t) = Acos(ωt + kz)
P(t)∝ χ (1)Acos(ωt + kz)
Case of a z propagative EM wave with freq. ω
Source term @ ω
€
! P (t) = N ×
! p = ε0χ
(1)! E (t)
Def. : LINEAR susceptibility of the medium
The polarization vector P(t) acts as a source term in the wave equation
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Nonlinear interactions
ü A short review : - Case of a NONLINEAR MEDIUM : collection of N identical atomes The nonlinear response of the medium can be expressed as (we have assumed that the medium have an instantaneous response - Case of a lossless and a dispersionless medium ) :
€
P(t) = ε0χ(1)E(t) +ε0χ
(2)E(t)E(t) +ε0χ(3)E(t)E(t)E(t)
= ε0χ(1)E(t) +ε0χ
(2)E 2(t) +ε0χ(3)E 3(t) +!
Considering a z propagative EM wave @ ω
€
E(t) = Acos(ωt + kz)
Linear response
€
P(t)∝ χ (1)Acos(ωt + kz)
Source term @ ω
1st Order
2nd Order + 3rd Order Nonlinear Response
€
P(t)∝ χ (2)A2 cos2(ωt + kz)2
∝ χ (2)A2 cos(2ωt +2kz) + χ (2)A2
2nd Order
Source term @ 2ω !!!
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Nonlinear interactions
ü A short review :
- Case of a NONLINEAR MEDIUM : collection of N identical atomes
The nonlinear response of the medium can be expressed as :
€
P(t) = ε0χ(1)E(t) +ε0χ
(2)E(t)E(t) +ε0χ(3)E(t)E(t)E(t)
= ε0χ(1)E(t) +ε0χ
(2)E 2(t) +ε0χ(3)E 3(t) +!
Considering a z propagative EM wave @ ω
€
E(t) = Acos(ωt + kz)
Linear response
€
P(t)∝ χ (1)Acos(ωt + kz)
Source term @ ω
1st Order
2nd Order + 3rd Order Nonlinear Response
€
PNL (t)∝ χ (3)A3 cos3(ωt + kz)
∝ χ (3)A3 cos(3ωt + 3kz) + χ (3)A2Acos(ωt + kz) +!
3rd Order
Source term @ 3ω Third harmonic generation
Source term @ ω Optical Kerr effect
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Example : Optical Kerr Effect
ü Variation of the refractive index
€
P(t) = ε0χ(1)E(t) +ε0χ
(3)E 3(t)
Considering a z propagative EM wave @ ω
€
E(t) = Acos(ωt + kz)
I ∝ A2
Kerr effect induces a variation of the refractive index directly proportional to the wave intensity.
€
P(t) = PL (t) +PNL (t)∝ χ (1)Acos(ωt + kz) + χ (3)A2Acos(ωt + kz)
€
P(t)∝ χ (1) + χ (3)A2( )Acos(ωt + kz) NONLINEAR Regime
€
PL (t)∝ χ (1) Acos(ωt + kz) LINEAR Regime
Wave intensity
Directly related to the refractive index
Case of a lossless medium
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Nonlinear interactions
⇒ 2nd order nonlinearities
• optical rectification : induces a static electric field • 2nd harmonic generation : generate a EM wave @ 2ω • Sum and Difference frequency generation : generate a EM wave @ ω1± ω2, fluorescence, amplification and parametric oscillation
⇒ 3rd order nonlinearities • Optical Kerr effect : nonlinear refractive index change • Solitons • Self-phase modulation • Four-wave mixing : nonlinear interaction of 4 degenerates or non-degenerates waves • Raman scattering • Brillouin scattering
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E
time p
time
Ind
uc
ed
dip
oe
€
E1(t) = A1 cos(ω1t + k1z)E2(t) = A2 cos(ω2t + k2z)E3(t) = A3 cos(ω3t + k3z)
Nonlinear interaction of 3 waves with 3 different frequencies
€
PNL (t)∝ χ (3)E1(t)E2(t)E3(t)
Frequency components of the nonlinear polarization :
ω1+ ω2+ ω3 ; ω1+ ω2 - ω3 ; ω1 - ω2+ ω3 ; ω1 - ω2 - ω3 ;
ω3
ω2
ω1
ω3 ω2 ω1
Example : Four-wave mixing
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Example : Source term @ ω4 = ω1+ ω2 - ω3
€
PNL (t)∝ χ (3)A3 cos (ω1 +ω2 −ω3) t + (k1 + k2 − k3)z( )
ω3
ω2
ω1
Example : Four-wave mixing
z
z
Assumption : low dispersive medium / The refractive indices @ ω1, ω2, ω3, ω4 are equals ⇒ the polarization @ ω4 radiates in phase with a z propagative
field @ ω4 = « phase matching condition »
€
E4 (t) = A4 cos(ω4 t + k4z)
Once the phase matching condition is fulfilled, the in-phase array of
dipoles contributes to efficiently generate a propagative field @ ω4 €
k4 = k1 + k2 − k3
€
k4 = k1 = k2 = k3
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ω3
ω2
ω1
€
PNL (t)∝ χ (3)A3 cos (ω1 +ω2 −ω3) t + (k1 + k2 − k3)z( )
€
E4 (t) = A4 cos(ω4 t + k4z)
Assumption : dispersive medium / The refractive indices @ ω1, ω2, ω3, ω4 are no more equals ⇒ the radiating polarization @ ω4 is no more in phase
with the z propagative field @ ω4 = « NO phase matching condition »
The EM field @ ω4 can not be efficiently generated because of a strong phase mismatch between the source term and
the propagative wave @ ω4
Example : Four-wave mixing
€
k1 + k2 − k3 ≠ k4
Example : Source term @ ω4 = ω1+ ω2 - ω3
z
z