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P. Piot, PHYS 630 – Fall 2008
Nonlinear optics• To date we considered dielectric media characterized by a linear
relation between polarization and E-fied
• Now we consider media characterized by a nonlinear relationbetween E and P
!
P = "0#E
!
P = a1E +1
2a2E
2 +1
6a3E
3 + ...
= "0 #E + # (2)E 2 + # (3)E 3 + ...( )= "0#E + 2dE 2 + 4# (3)E 3 + ...
2nd order 3rd order
P. Piot, PHYS 630 – Fall 2008
Nonlinear wave equation I• From Maxwell’s equation and now considering
• We obtain a nonlinear wave equation
where P is usually written as
!
"2E #
1
c0
2$t
2E = µ
0$t
2P
!
P = "0#E + P
NL
!
PNL
= 2dE2
+ 4" (3)E 3+ ...
!
D = "0E + P
P. Piot, PHYS 630 – Fall 2008
Nonlinear wave equation II• Using
• The nonlinear wave equation can be rewritten
with
• Born approximation consist in adopting an iterative solution for theNL wave equation
!
"2E #
1
c2$t
2E = S
S = µ0$t
2PNL
!
c = c0/n
n2
=1+ "
“source”
radiationS E
S(E)
P. Piot, PHYS 630 – Fall 2008
• Assume higher order than second order are negligible
• Take
• Corresponding polarization is
Second harmonic generation (SHG) I
!
PNL
= 2dE2
!
E =1
2E(")ei"t + c.c.( )
!
E = 2d1
4E(")ei"t + c.c.( ) E *
(")e# i"t + c.c.( )
= dEE * + d EEe2i"t + c.c.( )
dcoptical rectification
2ω2nd harm. generation
E
P
= +
P. Piot, PHYS 630 – Fall 2008
• Use to frequency-double lasers
Second harmonic generation (SHG) II
P.A. Franken, et al, Physical Review Letters 7, p. 118 (1961)
P. Piot, PHYS 630 – Fall 2008
Second harmonic generation (SHG) III
• The actually published results…
Input beamThe second harmonic
Note that the very weak spot due to the second harmonic is missing.It was removed by an overzealous Physical Review Letters editor,who thought it was a speck of dirt.
P. Piot, PHYS 630 – Fall 2008
• Consider a plane wave propagating along z, and write the paraxial Hemholtzequation
• Can be approximated as
where Δk=k(2ω)-k(ω) so finally
need Δk=0 to achieve maximumE(2ω) electric field (this was nottaken care in the 1961 experiment)
• This is referred to as “phase matching”
Phase matching I
!
"#
2E(2$) % 2ik&
zE(2$)[ ]e%k(2$ )z = µ
0&t
2dE
2($)( )e%k($ )z
!
"zE(2#) = AdE
2(#)e$kz
!
E(2")#sin$kz
$kz
Poor phase matching
Good phase matching
P. Piot, PHYS 630 – Fall 2008
Phase matching II
which will only be satisfied when:
Unfortunately, dispersion prevents this from ever happening!
So we’re creating light at ωsig = 2ω.
0
2 2 ( )polk k nc
!!= =
0 0
(2 )( ) (2 )
sig
sig sigk n nc c
! !! != =
sig polk k=
! 2!Frequency
Refra
ctive
in
dex
And the k-vector of the polarization is:
The phase-matching condition is:
The k-vector of the second-harmonic is:
!
n(2") = n(")
P. Piot, PHYS 630 – Fall 2008
Phase matching III
We can now satisfy the phase-matching condition.
Use the extraordinary polarizationfor ω and the ordinary for 2ω.
Birefringent materials have different refractive indices for different polarizations. Ordinary and extraordinary refractive indicescan be different by up to ~0.1 for SHG crystals.
! 2!FrequencyRe
fract
ive in
dex
!
ne
on
ne depends on propagation angle, so we can tune for a given ω.Some crystals have ne < no, so the opposite polarizations work.
!
no(2") = n
e(")
P. Piot, PHYS 630 – Fall 2008
Phase matching IV
0
ˆ2 cos
2 ( )cos
pol
pol
k k k k z
k nc
!
"" !
#= + =
$ =
r r r
2(2 )sig
o
k nc
!!=
ˆˆcos sink k z k x! != "r
ˆˆcos sink k z k x! !" = +r
z
x
But: So the phase-matchingcondition becomes:
θ
θ
!
n(2") = n(")cos#
•Phase matching via non-colinear overlap oftwo ω beams
P. Piot, PHYS 630 – Fall 2008
Application of noncollinear phase matching
•Noncollinear phasematching can be usedto measure the durationof ultra-short pulse