nonlinear optics.pdf

7/18/2019 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

2N. Dubreuil - N ON LINEAR O PTICS

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 :



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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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 :



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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7N. Dubreuil - N ONLINEAR O PTICS

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 :



• 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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• 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



• 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





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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



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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• 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


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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EXAMPLE : KDP c rysta l

2 generation : Determination ofr

P(2 )

Point group 42m - 3 nonzero coefficient, 2 numericallyequal coefficents :

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