Nonlinear Optics- Lecture 5

3 Pages·0.11 MB·English

by  Fredrik Jonsson

Nonlinear Optics 5A5513 (2003)
Lecture notes
Lecture 5
In the previous lecture, the quantum mechanical origin of the linear and nonlinear susceptibilities
was discussed. In particular, a perturbation analysis of the density operator was performed, and
the resulting system of equations was solved recursively for the nth order density operator ρˆ (t)
n
in terms of ρˆn−1(t), where the zeroth order term (independent of the applied electric field of the
light) is given by the Boltzmann distribution at thermal equilibrium.
So far we have obtained the linear optical properties of the medium, in terms of the first order
susceptibility tensor (of rank-two), and we will now proceed with the next order of interaction,
giving the second order electric susceptibility tensor (of rank three).
The second order polarization density
For the second order interaction, the corresponding term in the perturbation series of the density
operator in the interaction picture becomes1
1 t
ρˆ′(t)= [Hˆ′(τ ),ρˆ′(τ )]dτ
2 i~Z I 1 1 1 1
−∞
1 t 1 τ1
= [Hˆ′(τ ), [Hˆ′(τ ),ρˆ ]dτ ]dτ (1)
i~Z I 1 i~Z I 2 0 2 1
−∞ −∞
1 t τ1
= [Hˆ′(τ ),[Hˆ′(τ ),ρˆ ]]dτ dτ
(i~)2 Z−∞Z−∞ I 1 I 2 0 2 1
In order to simplify the expression for the second order susceptibility, we will in the following
analysis make use of a generalization of the cyclic perturbation of the terms in the commutator
inside the trace, as
Tr{[Qˆ (τ ),[Qˆ (τ ),ρˆ ]]Qˆ }=Tr{ρˆ [[Qˆ ,Qˆ (τ )],Qˆ (τ )]}. (2)
α 1 β 2 0 µ 0 µ α 1 β 2
By inserting the expression for the second order term of the perturbation series for the density
operatorintothequantummechanicaltraceofthesecondorderelectricpolarizationdensityofthe
medium, one obtains
1
P(2)(r,t)= Tr[ρˆ (t)Qˆ ]
µ V 2 µ
1 1 t τ1
= Tr Uˆ (t) [Hˆ′(τ ),[Hˆ′(τ ),ρˆ ]]dτ dτ Uˆ (−t) Qˆ
V h(cid:16) 0 (i~)2 Z−∞Z−∞ I 1 I 2 0 2 1 0 (cid:17) µi
=ρˆ′2(t) (interaction picture)
| {z }
=ρˆ2(t) (Schr¨odinger picture)
| {z }
={E (τ ) and E (τ ) are classical fields (omit space dependence r)}
α 1 α 1
1 t τ1
= Tr Uˆ (t) [Qˆ (τ ),[Qˆ (τ ),ρˆ ]]E (τ )E (τ )dτ dτ Uˆ (−t)Qˆ
V(i~)2 n 0 Z−∞Z−∞ α 1 β 2 0 α 1 β 2 2 1 0 µo
={Pull out E (τ )E (τ ) and the integrals outside the trace}
α1 1 α2 2
1 ItshouldbenoticedthattheformgiveninEq.(1)notonlyappliestoanensembleofmolecules,
ofarbitrarycomposition,butalsotoanykindoflevelofapproximationfortheinteraction,suchas
the inclusion of magnetic dipolar interactions or electric quadrupolar interactions as well. These
interactions should (of course) be incorporated in the expression for the interaction Hamiltonian
Hˆ′(τ), here described in the interaction picture.
I
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Nonlinear Optics 5A5513 (2003) Lecture notes 5
1 t τ1
= Tr Uˆ (t)[Qˆ (τ ),[Qˆ (τ ),ρˆ ]]Uˆ (−t)Qˆ E (τ )E (τ )dτ dτ .
V(i~)2 Z−∞Z−∞ n 0 α 1 β 2 0 0 µo α 1 β 2 2 1
In analogy with the results as obtained for the first order (linear) optical properties, now express
the term E (τ )E (τ ) in the frequency domain, by using the Fourier identity
α1 1 α2 2
∞
E (τ )= E (ω )exp(−iωτ )dω,
αk k Z αk k k
−∞
which hence gives the second order polarization density expressed in terms of the electric field in
the frequency domain as
1 ∞ ∞ t τ1
P(2)(r,t)= Tr Uˆ (t)[Qˆ (τ ),[Qˆ (τ ),ρˆ ]]Uˆ (−t)Qˆ E (ω )E (ω )
µ V(i~)2 Z−∞Z−∞Z−∞Z−∞ n 0 α 1 β 2 0 0 µo α 1 β 2
×exp(−iω τ )exp(−iω τ )dτ dτ dω dω
1 1 2 2 2 1 2 1
={Use exp(−iωτ)=exp(−iωt)exp[−iω(τ −t)]}
1 ∞ ∞ t τ1
= Tr Uˆ (t)[Qˆ (τ ),[Qˆ (τ ),ρˆ ]]Uˆ (−t)Qˆ E (ω )E (ω )
V(i~)2 Z−∞Z−∞Z−∞Z−∞ n 0 α 1 β 2 0 0 µo α 1 β 2
×exp[−iω (τ −t)−iω (τ −t)]dτ dτ exp[−i(ω +ω )t]dω dω
1 1 2 2 2 1 1 2 2 1
=ωσ
∞ ∞ | {z }
=ε χ(2) (−ω ;ω ,ω )E (ω )E (ω )exp(−iω t)dω dω ,
0Z Z µαβ σ 1 2 α 1 β 2 σ 2 1
−∞ −∞
where the second order (quadratic) electric susceptibility is defined as
χ(2) (−ω ;ω ,ω )
µαβ σ 1 2
1 t τ1
= Tr Uˆ (t)[Qˆ (τ ),[Qˆ (τ ),ρˆ ]]Uˆ (−t)Qˆ
ε0V(i~)2 Z−∞Z−∞ n 0 α 1 β 2 0 0 µo
×exp[−iω (τ −t)−iω (τ −t)]dτ dτ
1 1 2 2 2 1
′
={Make use of Eq. (2) and take τ =τ −t}
1 1
=...
1 0 τ1′
= Tr{ρˆ [[Qˆ ,Qˆ (τ′)],Qˆ (τ′)]}exp[−i(ω τ′ +ω τ′)]dτ′dτ′.
ε0V(i~)2 Z−∞Z−∞ 0 µ α 1 β 2 1 1 2 2 2 1
This obtained expression for the second order electric susceptibility does not possess the property
of intrinsic permutation symmetry. However, by using the same arguments as discussed in the
analysis of the polarization response functions in lecture two, we can easily verify that this tensor
can be cast into a symmetric and antisymmetric part as
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χ(2) (−ω ;ω ,ω )= [χ(2) (−ω ;ω ,ω )+χ(2) (−ω ;ω ,ω )]
µαβ σ 1 2 2 µαβ σ 1 2 µβα σ 2 1
symmetric part
| 1 {z }
+ [χ(2) (−ω ;ω ,ω )−χ(2) (−ω ;ω ,ω )],
2 µαβ σ 1 2 µβα σ 2 1
antisymmetric part
| {z }
and since the antisymmetric part, again following the arguments for the second order polarization
response function, does not contribute to the polarization density, it is customary (in the Butcher
andCotterconventionaswellasallotherconventionsinnonlinearoptics)tocastthesecondorder
susceptibility into the form
χ(2) (−ω ;ω ,ω )
µαβ σ 1 2
1 1 0 τ1′
= S Tr{ρˆ [[Qˆ ,Qˆ (τ′)],Qˆ (τ′)]}exp[−i(ω τ′ +ω τ′)]dτ′dτ′,
ε0V(i~)22! Z−∞Z−∞ 0 µ α 1 β 2 1 1 2 2 2 1
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Lecture notes 5 Nonlinear Optics 5A5513 (2003)
where S, commonly called the symmetrizing operator, denotes that the expression that follows is
tobesummedoverthe2!=2possiblepairwisepermutationsof(α,ω )and(β,ω ),henceensuring
1 2
that the second order susceptibility possesses the intrinsic permutation symmetry,
χ(2) (−ω ;ω ,ω )=χ(2) (−ω ;ω ,ω ).
µαβ σ 1 2 µβα σ 2 1
Higher order polarization densities
The previously described principle of deriving the susceptibilities of first and second order are
straightforward to extend to the nth order interaction. In this case, we will make use of the
following generalization of Eq. (2),
Tr{[Qˆ (τ ),[Qˆ (τ ),...,[Qˆ (τ ),ρˆ ]]...]Qˆ }
α1 1 α2 2 αn n 0 µ
=Tr{ρˆ [...[[Qˆ ,Qˆ (τ )],Qˆ (τ )],...Qˆ (τ )]},
0 µ α1 1 α2 2 αn n
which, when applied in the evaluation of the expectation value of the electric dipole operator of
the ensemble, gives the nth order electric susceptibility as
χ(µnα)1···αn(−ωσ;ω1,...,ωn)
1 1 0 τ1 τn−1
= S ··· Tr{ρˆ [...[[Qˆ ,Qˆ (τ )],Qˆ (τ )],...Qˆ (τ )]}
ε0V(−i~)nn! Z−∞Z−∞ Z−∞ 0 µ α1 1 α2 2 αn n
×exp[−i(ω τ +ω τ +...+ω τ )]dτ ··· dτ dτ ,
1 1 2 2 n n n 2 1
wherenowthesymmetrizingoperatorSindicatesthattheexpressionfollowingitshouldbesummed
over all the n! pairwise permutations of (α ,ω ),...,(α ,ω ).
1 1 n n
It should be emphasized the symmetrizing operator S always implies summation over all the
n! pairwise permutations of (α ,ω ),...,(α ,ω ), irregardless of whether the permutations are
1 1 n n
distinct or not. This is due to that eventually occuring degenerate permutations are taken care of
in the degeneracy coefficient K(−ω ;ω ,...,ω ) in Butcher and Cotters convention, as described
σ 1 n
in lecture three and in additional notes that has been handed out in lecture four.
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