Nonlinear Optics.pdf
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NONLINEARNONLINEAR OPTICSOPTICS
Ch. 2 NONLINEAR SUSCEPTIBILITIES
• Field notations
•Nonlinear susceptibility tensor : definition
- 2nd order NL susceptibility
- 3rd order NL susceptibility
- nth order NL susceptibility
• Properties of the NL susceptibilities
• Contracted notation deff
• Spatial symmetries
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Field notationField notation
We assume that the electric field vector can be expressed as aplane wave (or as a projection of plane waves, i.e through aFourier transformation) :
Purely REAL quantityPolrization state
Notation :
Similarly for the macroscopic polarization :
Notation :
avec
Purely REAL quantity
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Nonlinear susceptibility tensor - DefinitionNonlinear susceptibility tensor - Definition
Case of the nonlinear interaction of 2 waves @ 1 and 2 in a 2nd
order NL medium :
•Classical anharmonic oscillator : sca lar sca lar expression of thepolarization @=1+2
(al l the d ip oles a re sup p osed id en t ic a lly orien ted a lon g the linea r
p olar iza t ion sta te o f the a p p lied f ield ) :
• General description : the array of dipoles are oriented along the
3 direc tions x,y et z + different oscillator parameters for eachdirection
x
y
z
xy
z
E
General relation :
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Nonlinear susceptibility tensor - DefinitionNonlinear susceptibility tensor - Definition
Case of the nonlinear interaction of 2 waves @ 1 and 2 in a 2nd
order NL medium :
• General description : the array of dipoles are oriented along the3 direc tions x,y et z + different oscillator parameters for each
direction
x
y
z
General relation :
Vector / Tensor notation :
VectorsVector Tensor of rank 3
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Nonlinear susceptibility tensor - DefinitionNonlinear susceptibility tensor - Definition
2nd order NL susceptibility :
= tensor of rank 3
It contains 9x 3 = 27 components
Comm en t : Ea c h te nsor is d efined for a set of freque nc ies.
The va lue o f the c om po nents of the tensor dep end s on the frequenc ies
( in a gene ra l m a nner) !!!
• General expression of the 2nd order NL ploarization :
Expression of the i t h component :
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Nonlinear susceptibility tensor - DefinitionNonlinear susceptibility tensor - Definition
Nth order NL susceptibility
… just have fun !!
3rd order NL susceptibility :
= tensor of rank 4
81 components !!!!
• General expression of the 3rd order NL polarization :
Expression of the i t h component :
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Properties of NL susceptibilitiesProperties of NL susceptibilities
No nlinea r susce p tib i li ties = Tensor
r
P( 1),
r
P( 2),
r
P( 3)
...
...
12 tensors = 12 x 27 = 324 components !!!
r
P( 1),r
P( 2),r
P( 3)
...
...
...
...
...
...
Complete description of the waves interaction (3 waves in thiscase) requires the determination of :
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Properties of NL susceptibilitiesProperties of NL susceptibilities
• Rea lity of the field s
*
• Intrinsic Perm utation Sym metry
The quantities :
and are numerically equal
Consequence
• Lossless m ed ia
Verification : in the case of the classica l oscillator model discussedin ch1, since <<
0
Expression of NL is a purely real quantity
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Properties of NL susceptibilitiesProperties of NL susceptibilities
• Degene rac y Fac tor
Determination of P( ) : summation over field frequencies in interaction and
for which =
1+
2 +
3 +L
Due to intrinsic permutation simplification occurs
Exam p le : Sum-Frequency generation
Intrinsic permutation
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Properties of NL susceptibilitiesProperties of NL susceptibilities
• Dege ne rac y Fac to r
- 2nd order NL Polarization expression
Degeneracy factor = Number of distinct permutation of theapplied fields [(j, 1), (k, 2)]
1 : only 1 distinct field (case of 2 generation with alinearly 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
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Properties of NL susceptibilitiesProperties of NL susceptibilities
Degeneracy fac tor
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)
= Number of distinct permutation of the applied fields
=
( j , 1);(k , 2)[ ]
• Deg enerac y Fac tor
- 2nd order NL Polarization expression
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Properties of NL susceptibilitiesProperties of NL susceptibilities
1 : only 1 distinct field
3 : when 2 distinct fields
6 : all the fields are distinct
= Number of distinct permutation of the applied fields
=
( j , 1);(k ,
2);(l,
3)[ ]
Degeneracy fac tor
• Deg enerac y Fac tor
- 3rd order NL Polarization expression
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Properties of NL susceptibilitiesProperties of NL susceptibilities
• Kleinm a n Sym m etry - Lo ssless Med ia
Lossless media : no exchange of energy with the nonlinear medium
(See Boyd, Ch1, sec tion 1.5)
Far from any material resonance, NL does not depend on frequencies
Consequence :
+ intrinsic permutation
Full permutation of the indices, without permuting the
frequencies
Simultaneous permutations of the indiceswith the frequency arguments
Permutation of the indices without permuting frequencies
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Contracted notationContracted notation d d e ff e ff
When the Kleinman symmetry condition is validOr
For 2nd harmonic generation process
Permutation symmetry of the last two indices
d il=
d 11
d 12L d
16
d 21
L d 26
d 31
L d 36
Matrix with 6x3 components
Contraction notation of the last two indices
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Spatial SymmetriesSpatial Symmetries
Spatial symmetry properties of the nonlinear material : reductionof the number of independent components
• Example : media inside which the direc tions x and y are similar(from th point of view of its NL response)
zxx
(2 )=
zyy
(2 )(for instance)
Strong reduction of
the numbers of
independent
components
• Important example : Centre-symmetric material
2nd order nonlinear susceptibility vanishes(i.e silica...)
=0
(generalization : 2Nth order )
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Spatial SymmetriesSpatial Symmetries
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Spatial SymmetriesSpatial Symmetries
EXAMPLE : KDP c rysta l
2 generation : Determination ofr
P(2 )
Point group 42m - 3 nonzero coefficient, 2 numericallyequal coefficents :