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
Induced dipole and macroscopic polarization ü Microscopic scale A simplistic description of the interaction between a wave and an atom : - Simplistic model for an atom = Electronic cloud (charge - ) Nucleus (charge +) - An applied static electric field acts on the electrons trajectories E x Change in position of the center of mass of the electrons! F = q E! + v! B! Lorentz Force ( ) 2
Induced dipole and macroscopic polarization The magnetic part of the Lorentz force is negligible - With an applied EM fiels = temporal deformation of the electronic cloud E p Induced Dipole variation of the position of the center of mass : x() 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) : Abrev. : EM field means Electromagnetic Field! 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 3
Induced dipole and macroscopic polarization ü Macroscopic scale - LINEAR MEDIUM case made of N atoms (N identical dipoles) E Case of a z propagative EM wave with freq. ω E(t) = Acos(ωt + kz) P(t) χ (1) Acos(ωt + kz) Source term @ ω p Induced dipole Average P Polarisation = «macroscopic» source term! P (t) = N! p = ε 0 χ (1)! E (t) Def. : LINEAR susceptibility of the medium The polarization vector P(t) acts as a source term in the wave equation 4
Classical anharmonic oscillator model Classical anharmonic oscillator - Description Electron Electronic cloud E Nucleus (fixed) p Induced Dipole Induced dipole (microscopic quantity) : Polarization (MACROscopic quantity) : x(z,t)?? Equation of motion 5
Classical anharmonic oscillator model - Two applied EM wave with frequencies ω 1 and ω 2 : LINEAR CASE : «LINEAR» response of the vibrating dipole E p Induced Dipole Vibrating dipoles with frequencies = ω 1 and ω 2 A material with a collection of N dipoles in a linear vibrating regime = LINEAR MEDIUM 6
Classical anharmonic oscillator model - Two applied EM wave with frequencies ω 1 and ω 2 : NONLINEAR CASE : «NONLINEAR» dependence of the restoring force with the displacement of the center of mass E p Induced Dipole Vibrating dipoles with frequencies = ω 1 and ω 2 + harmonics A material with a collection of N dipoles in a nonlinear vibrating regime = NONLINEAR MEDIUM 7
Classical anharmonic oscillator model Equation of motion Damping term Restoring force Driven Coulomb force Solution : perturbation method, taking into account 2 0x x 2 x 3 x (1) x (2) x (3) 8
Linear polarization Harmonic oscillator : equation of motion Driven solution Induced dipole Linear polarizability (microscopic quantity) Macroscopic Polarization Linear susceptibility 9
Linear polarization Harmonic oscillator : equation of motion Linear Susceptibility (1) ( ) = ( )+ı ( ) Dispersion Absorption (or amplification) Lorentzian line shape 10
2nd Order Nonlinear Polarization Anharmonic oscillator : equation of motion Driven solution 11
2nd Order Nonlinear Polarization Anharmonic oscillator : equation of motion Nonlinear Polarization Optical RECTIFICATION (induces static electric field) 2nd Harmonic Generation creation of an electric field with a 2ω frequency component 2nd order Nonlinear Susceptibility 12
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 (@ω 0 ) 13
3rd Order Nonlinear Polarization Anharmonic oscillator : equation of motion Case of a centrosymmetric material Driven solution x(z,t) = x (1) (z,t)+ 2 x (2) (z,t)+ 3 x (3) (z,t) x (2) =0 Linear and Nonlinear Polarization 3rd Harmonic generation Optical Kerr Effect (3) ( 1, 2, 3 )= N e 4 0 m 3 D( 1 + 2 + 3 )D( 1 )D( 2 )D( 3 ) 14
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 (@ω 0 ) (3) ( 1, 2, 3 )= N e 4 0 m 3 D( 1 + 2 + 3 )D( 1 )D( 2 )D( 3 ) 15
Nonlinear interactions ü A short review : - LINEAR MEDIUM case made of N atomes (N identical dipoles) E Case of a z propagative EM wave with freq. ω E(t) = Acos(ωt + kz) P(t) χ (1) Acos(ωt + kz) Source term @ ω p Induced dipole Average P Polarisation = «macroscopic» source term! P (t) = N! p = ε 0 χ (1)! E (t) Def. : LINEAR susceptibility of the medium The polarization vector P(t) acts as a source term in the wave equation 16
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) = ε 0 χ (1) E(t) +ε 0 χ (2) E(t)E(t) +ε 0 χ (3) E(t)E(t)E(t) 1st Order = ε 0 χ (1) E(t) +ε 0 χ (2) E 2 (t) +ε 0 χ (3) E 3 (t) +! Linear response P(t) χ (1) Acos(ωt + kz) Source term @ ω 2nd Order + 3rd Order Nonlinear Response Considering a z propagative EM wave @ ω E(t) = Acos(ωt + kz) 2nd Order P(t) χ (2) A 2 cos 2 (ωt + kz) 2 χ (2) A 2 cos(2ωt + 2kz) + χ (2) A 2 Source term @ 2ω!!! 17
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) = ε 0 χ (1) E(t) +ε 0 χ (2) E(t)E(t) +ε 0 χ (3) E(t)E(t)E(t) = ε 0 χ (1) E(t) +ε 0 χ (2) E 2 (t) +ε 0 χ (3) E 3 (t) +! Linear response 2nd Order + 3rd Order Nonlinear Response 1st Order P(t) χ (1) Acos(ωt + kz) Considering a z propagative EM wave @ ω E(t) = Acos(ωt + kz) P NL (t) χ (3) A 3 cos 3 (ωt + kz) 3rd Order χ (3) A 3 cos(3ωt + 3kz) + χ (3) A 2 Acos(ωt + kz) +! Source term @ ω Source term @ 3ω Third harmonic generation Source term @ ω Optical Kerr effect 18
Example : Optical Kerr Effect ü Variation of the refractive index P(t) Case of a lossless medium Directly related to the refractive index Considering a z propagative EM wave @ ω E(t) = Acos(ωt + kz) I A 2 = ε 0 χ (1) E(t) +ε 0 χ (3) E 3 (t) P(t) = P L (t) + P NL (t) χ (1) Acos(ωt + kz) + χ (3) A 2 Acos(ωt + kz) P(t) ( χ (1) + χ (3) A 2 )Acos(ωt + kz) NONLINEAR Regime P L (t) χ (1) Acos(ωt + kz) LINEAR Regime Wave intensity Kerr effect induces a variation of the refractive index directly proportional to the wave intensity. 19
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 @ ω 1 ± ω 2, 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 nondegenerates waves Raman scattering Brillouin scattering 20
Example : Four-wave mixing Nonlinear interaction of 3 waves with 3 different frequencies E 1 (t) = A 1 cos(ω 1 t + k 1 z) E 2 (t) = A 2 cos(ω 2 t + k 2 z) E 3 (t) = A 3 cos(ω 3 t + k 3 z) E ω 3 ω 2 p P NL (t) χ (3) E 1 (t)e 2 (t)e 3 (t) Frequency components of the nonlinear polarization : ω 1 + ω 2 + ω 3 ; ω 1 + ω 2 - ω 3 ; ω 1 - ω 2 + ω 3 ; ω 1 - ω 2 - ω 3 ; Induced dipoe ω 1 ω 3 ω 2 ω 1 21
Example : Four-wave mixing Example : Source term @ ω 4 = ω 1 + ω 2 - ω 3 Assumption : low dispersive medium / The refractive indices @ ω 1, ω 2, ω 3, ω 4 are equals the polarization @ ω 4 radiates in phase with a z propagative field @ ω 4 = «phase matching condition» k 4 = k 1 = k 2 = k 3 ω 3 z ω 2 P NL (t) χ (3) A 3 cos ((ω 1 +ω 2 ω 3 ) t + (k 1 + k 2 k 3 )z) ω 1 E 4 (t) = A 4 cos(ω 4 t + k 4 z) k 4 = k 1 + k 2 k 3 z Once the phase matching condition is fulfilled, the in-phase array of dipoles contributes to efficiently generate a propagative field @ ω 4 22
Example : Four-wave mixing Example : Source term @ ω 4 = ω 1 + ω 2 - ω 3 Assumption : dispersive medium / The refractive indices @ ω 1, ω 2, ω 3, ω 4 are no more equals the radiating polarization @ ω 4 is no more in phase with the z propagative field @ ω 4 = «NO phase matching condition» z ω 3 ω 2 P NL (t) χ (3) A 3 cos ((ω 1 +ω 2 ω 3 ) t + (k 1 + k 2 k 3 )z) ω 1 E 4 (t) = A 4 cos(ω 4 t + k 4 z) k 1 + k 2 k 3 k 4 The EM field @ ω 4 can not be efficiently generated because of a strong phase mismatch between the source term and z the propagative wave @ ω 4 23