Ch. 2 NONLINEAR SUSCEPTIBILITIES

Transcription

1 NONLINEAR OPTICS Ch. 2 NONLINEAR SUSCEPTIBILITIES Field notations Nonlinear susceptibility tensor : definition - 2 nd order NL susceptibility - 3 rd order NL susceptibility - n th order NL susceptibility Properties of the NL susceptibilities Contracted notation d eff Spatial symmetries Field notation We assume that the electric field vector can be expressed as a plane wave (or as a projection of plane waves, i.e through a Fourier transformation) : avec Purely REAL quantity Polrization state Notation : Similarly for the macroscopic polarization : Purely REAL quantity Notation : 2 1

2 Nonlinear susceptibility tensor - Definition Case of the nonlinear interaction of 2 1 and 2 in a 2 nd order NL medium : Classical anharmonic oscillator : scalar expression of the = z (all the dipoles are supposed identically oriented along the linear polarization state of the applied field ): E x y General description : the array of dipoles are oriented along the z 3 directions x,y et z + different oscillator parameters for each direction General relation : y x 3 x Nonlinear susceptibility tensor - Definition Case of the nonlinear interaction of 2 1 and 2 in a 2 nd order NL medium : General description : the array of dipoles are oriented along the z 3 directions x,y et z + different oscillator parameters for each direction General relation : y Vector / Tensor notation : Vector Tensor of rank 3 Vectors 4 2

3 Nonlinear susceptibility tensor - Definition 2 nd order NL susceptibility : = tensor of rank 3 It contains 9x 3 = 27 components Comment : Each tensor is defined for a set of frequencies. The value of the components of the tensor depends on the frequencies (in a general manner)!!! General expression of the 2nd order NL ploarization : Expression of the i th component : 5 Nonlinear susceptibility tensor - Definition 3 rd order NL susceptibility : = tensor of rank 4 81 components!!!! General expression of the 3rd order NL polarization : Expression of the i th component : N th order NL susceptibility just have fun!! 6 3

4 Properties of NL susceptibilities Nonlinear susceptibilities = Tensor Complete description of the waves interaction (3 waves in this case) requires the determination of : r P ( 1 ), r P ( 2 ), r P ( 3 ) r P ( 1 ), P r ( 2 ), P r ( 3 ) 12 tensors = 12 x 27 = 324 components!!! 7 Properties of NL susceptibilities Reality of the fields Intrinsic Permutation Symmetry * The quantities : and are numerically equal Consequence Lossless media Expression of NL is a purely real quantity Verification : in the case of the classical oscillator model discussed in ch1, since << 0 8 4

5 Properties of NL susceptibilities Degeneracy Factor Determination of P( ) : summation over field frequencies in interaction and for which = L Due to intrinsic permutation simplification occurs Example : Sum-Frequency generation Intrinsic permutation 9 Degeneracy Factor Properties of NL susceptibilities - 2nd order NL Polarization expression Degeneracy factor = Number of distinct permutation of the applied fields [(j, 1 ), (k, 2 )] 1 : only 1 distinct field (case of 2 generation with a linearly polarized field (x, ) ) 2 : number of distinct fields =2-3rd order NL Polarization expression 1 : number of distinct field =1 3 : number of distinct fields =2 6 : number of distinct fields =3 10 5

6 Properties of NL susceptibilities Degeneracy Factor - 2nd order NL Polarization expression Degeneracy factor = Number of distinct permutation of the applied fields [( j, 1 );(k, 2 )] = 1 : only 1 distinct field (case of 2 generation with a linearly polarized field (x, ) ) 2 : distinct fields (case where (j, 1 ) and (k, 2 ) are distinct) 11 Properties of NL susceptibilities Degeneracy Factor - 3rd order NL Polarization expression Degeneracy factor = Number of distinct permutation of the applied fields ( j, 1 );(k, 2 );(l, 3 ) [ ] = 1 : only 1 distinct field 3 : when 2 distinct fields 6 : all the fields are distinct 12 6

7 Properties of NL susceptibilities Kleinman Symmetry - Lossless Media Lossless media : no exchange of energy with the nonlinear medium (See Boyd, Ch1, section 1.5) Simultaneous permutations of the indices with the frequency arguments Far from any material resonance, NL does not depend on frequencies Consequence : Permutation of the indices without permuting frequencies + intrinsic permutation Full permutation of the indices, without permuting the frequencies 13 Contracted notation d eff When the Kleinman symmetry condition is valid Or For 2nd harmonic generation process Permutation symmetry of the last two indices Contraction notation of the last two indices d 11 d 12 L d 16 d il = d 21 L d 26 d 31 L d 36 Matrix with 6x3 components 14 7

8 Spatial Symmetries Spatial symmetry properties of the nonlinear material : reduction of the number of independent components Example : media inside which the directions x and y are similar (from th point of view of its NL response) (2) (2) zxx = zyy (for instance) Strong reduction of the numbers of independent components Important example : Centre-symmetric material 2nd order nonlinear susceptibility vanishes (i.e silica) (generalization : 2N th order ) =0 15 Spatial Symmetries 16 8

9 Spatial Symmetries EXAMPLE : KDP crystal Point group 42m - 3 nonzero coefficient, 2 numerically equal coefficents : 2 generation : Determination of r P (2 ) 17 9