Download ch 12-Non-Linear Optics

Non-Linear Optics
Chapter 12
Physics 208, Electro-optics
Peter Beyersdorf
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ch 12. 1
Non-Linear Optics
We’ve seen that an externally applied electric
field can alter the index of refraction of a
material. At sufficiently high intensities, the
electric field associated with a propagating wave
can have the same effect. This is the premise
behind the field of non-linear optics.
ch 12. 2
Non-Linear Response
Consider the response (material polarization) of a
material to an applied electric field (i.e. a propagating
wave)
Pi = !0 Xij Ej + 2dijk Ej Ek + 4Xijkl Ej Ek El + . . .
When the applied electric field is small compared to
the internal binding fields of the material, the response
is primarily linear
Pi ≈ !0 Xij Ej
At higher intensities the higher order terms in the
material polarization come into play. Like the electrooptic effect, centro-symmetric crystals do not have a
2nd order nonlinearity (i.e. dijk=0)
ch 12. 3
Non-Linear dijk Coefficients
In a non-linear material (dijk≠0) the material
polarization will have a component at twice the
optical frequency. From:
Pi (t) = 2dijk Ej (t)Ek (t)
the fields can
be written# in
complex form
!
#
!
Pi (t) = 2dijk
leading
to
!
"j eiω1 t + E
" ∗ e−iω1 t
E
j
2
"k eiω2 t + E
" ∗ e−iω2 t
E
k
2
#
1
∗
i(ω
−ω
)t
∗
−i(ω
−ω
)t
i(ω
+ω
)t
∗
∗
−i(ω
+ω
)t
1
2
1
2
"j E
" e 1 2 +E
" E
"
"j E
"k e 1 2 + E
" E
"
Pi (t) = dijk E
+E
k
j ke
j ke
2
giving
!j (ω1 )E
!k (ω2 )
P!i (ω1 + ω2 ) = dijk E
ch 12. 4
Non-Linear dijk Coefficients
The dijk tensor can be written in contracted form
so that



Px
d11
 Py  =  d21
d31
Pz
d12
d22
d32
d13
d23
d33
d14
d24
d34
d15
d25
d35
d16
d26
d36








Ex2
Ey2
Ez2
2Ez Ey
2Ez Ex
2Ex Ey








and has the same form constraints due to crystal
symmetry groups as the electrooptic tensor rijk
More specifically:
dijk = −
!ii !jj
rijk
4!0
ch 12. 5
Wave Equation in a Non-Linear Medium
Starting with the usual Maxwell’s equations
!
"
!
∂
D
∂
! + P!
! ×H
! = J! +
∇
= J! +
#0 E
∂t
∂t
!
"
∂
! ×E
! =−
!
∇
µ0 H
∂t
and writing the first of these equations in terms
of the linear and non-linear polarization
components
! + P!N L
P! = "0 χL E
where
we have
and
!
P!N L
"
i
= 2d!ijk Ej Ek
! " ∂ P!
∂
NL
! +
! ×H
! = σE
!+
∇
$E
∂t
∂t
∂ ! !"
!
!
∇×E =−
µ0 H
∂t
ch 12. 6
Wave Equation in a Non-Linear Medium
which can be combined to get
!
!
∂E
∂2E
∂2 !
∇ E = µ0 σ
+ µ0 $ 2 + µ0 2 PN L
∂t
∂t
∂t
2!
considering the one-dimensional case of
propagation in the z-direction with fields of the
form
!
"
1
E1i (z) ei(ω1 t−k1 z) + c.c.
2
"
1!
i(ω2 t−k2 z)
Ek (ω2 , z, t) =
E2k (z) e
+ c.c.
2
"
1!
i(ω3 t−k3 z)
Ej (ω3 , z, t) =
E3j (z) e
+ c.c.
2
Ei (ω1 , z, t) =
c.c. stands for
“complex conjugate”
where i,j, and k can be either the x or y
directions, and ω3=ω1+ω2.
ch 12. 7
Wave Equation in a Non-Linear Medium
computing the derivative with the plane-wave
solutions gives at frequency ωl
!
"
1
dE
(z)
li
∇2 Ei (ωl , z, t) = − kl2 Eli (z) + 2ikl
ei(ωl t−kl z) + c.c.
2
dz
for l=1,2,3 where Eli(z)=Ei(ωl) and we have
neglected terms of the form d2E/dz2 because we
assume the field amplitudes are varying slowly
compared to the optical period, i.e.
dEli
d2 Eli
kl !
dz
dz 2
ch 12. 8
Wave Equation in a Non-Linear Medium
!
The wave equation can thus be written as
"
!
"
#
$
k12
dE1i i(ω1 t−k1 z)
1
∂2
2
i(ω1 t−k1 z)
E1i + ik1
e
+ c.c = −iω1 µ0 σ + ω1 µ0 # E1i e
+ c.c − µ0 2 [PN L (z, t)]i
2
dz
2
∂t
!
which can be written using""
k12=ω12μ0ε and ω1=ω3-ω2 as
ik1
"
!
" j E"k
P"N L " =" 2d"!ijk E
i
,
dE1i −ik1 z
iω1 σµ0
∗ −i(k3 −k2)z
e
=−
E1i e−ik1 z + µ0 ω12 d"ijk E3j E2k
e
dz
2
giving components of E(ω1), E(ω2) and E(ω3) of
!
!
µ0
µ0 !
∗ −i(k3 −k2 −k1 )z
E1i − iω1
dijk E3j E2k
e
"1
"1
!
!
∗
dE2k
σ2 µ0 ∗
µ0 "
∗ −i(k1 −k3 +k2 )z
=−
E2k + iω2
dkij E1i E3j
e
dz
2
"2
"2
!
!
dE3j
σ3 µ0
µ0 !
=−
E3j − iω3
djik E1i E2k e−i(k1 +k2 −k3 )z
dz
2
"3
"3
dE1i
σ1
=−
dz
2
ch 12. 9
Wave Equation in a Non-Linear Medium
dE1i
σ1
=−
dz
2
!
µ0
E1i − iω1
"1
Absorption, i.e. for solution
E=E0e-αz/2 we have
α=σ
!
µ0
#
!
µ0 !
∗ −i(k3 −k2 −k1 )z
dijk E3j E2k
e
"1
Conversion to/from other
frequencies. Whether
conversion is from or to
this field depends on the
phase of this field relative
to E2 and E3. If Δkz , i.e.
(k3-k2-k1)z changes by π
the conversion switches
signs.
ch 12.10
Second Harmonic Generation
Consider case of ω1=ω2=ω3/2 and define
Δk k3-(k1+k2) where Δk=0 in the absence of
dispersion (i.e. dn/dλ=0).
Assume the “pump” wave at ω1 is not depleted
(dE1/dz=0) and the material is transparent at
frequency ω3 (i.e. σ=0) and solve for dE3/dz to
get
dE3j
σ3
=−
dz
2
!
µ0
E3j − iω3
"3
dE3j
= −iω
dz
!
!
µ0 !
djik E1i E2k e−i(k1 +k2 −k3 )z
"3
µ0 !
djik E1i E1k ei∆kz
"
ch 12. 11
Second Harmonic Generation
dE3j
= −ω
dz
!
µ0 !
djik E1i E1k ei∆kz
"
giving at the end of a crystal of length L
E3j (L) = −iω
or
!
µ0 !
ei∆kL − 1
d E1i E1k
" jik
i∆k
E3j (L) = −2iei∆kL/2 ω
!
µ0 !
sin(∆kL/2)
djik E1i E1k
"
∆k
or for the intensity of the sum frequency
component
1
!I(ω3 )" =
2
!
" ∗
E (L) E3j (L) = 2
µ0 3j
!
µ0 2 " " #2 2 2 2 sin2 12 ∆kL
ω dijk E1i E1k L "
#2
1
"
∆kL
2
! µ " 32 ω 2 d!2 L2
sin2 12 ∆kL
0
ijk
!Ij (2ω)" = 8
!Ii (ω1 )" !Ik (ω1 )" #
$2
1
"
n3
∆kL
2
ch 12. 12
Phase Matching
From the expression of the intensity of the sum
frequency wave
! µ " 32 ω 2 d!2 L2
sin2 12 ∆kL
0
ijk
!Ij (2ω)" = 8
!Ii (ω1 )" !Ik (ω1 )" #
$2
1
"
n3
∆kL
2
we can see that a long crystal benefits
frequency conversion efficiency, but the length is
limited by the sinc2(ΔkL/2) factor
unless Δk=0.
sin2 12 ∆kL
!1
"2
2 ∆kL
ch 12.13
Sinc2ΔkL/2 term
sin2 12 ∆kL
!1
"2
2 ∆kL
1
0.75
0.5
0.25
ΔkL
-10!
-8!
-6!
-4!
-2!
0
2!
4!
6!
8!
10!
2 1
sin
2
2 ∆kL
<
I(ω
)
>∝
L
!1
"2
3
Since
∆kL
2
1
2π
< I(ω3 ) >max ∝
L
=
2
at
max
(∆k)
∆k
Thus Δk should be minimized for maximum conversion efficiency
ch 12.14
Phase Matching
Physical interpretation of the Sinc2ΔkL/2 term is
that if Δk≠0 the pump wave at ω and the
second harmonic wave at 2ω will drift out of
phase over a distance lc/2=π/Δk causing the
conversion to cancel the second harmonic wave
rather than augment it.
In normally dispersive materials (dn/dω>0) Δk is
not zero without special efforts to arrange it to
be so.
In typical materials, lc≈100 μm
ch 12.15
Angle Phase Matching
In crystals the index of refraction
seen by one polarization can be
tuned by adjusting the angle, for
example in a uniaxial crystal
1
cos2 θ sin2 θ
+
=
n2e (θ)
n20
n2e
allowing the value of Δk to be
adjusted if the waves at ω1, ω2
and ω3 do not all have the same
polarization state. Depending on
the relative polarization states we
have different “types” of phase
matching
Scheme
Polarizations
E(ω1) E(ω1) E(ω2) E(ω3)
o
o
e
I
e
o
e
II (IIA)
e
e
III (IIB) o
e
e
e
IV
o
o
o
V
e
o
o
VI (IIB)
e
o
VII (IIA) o
e
e
o
VIII (I)
For ω1≤ω2<ω3
ch 12. 16
Angle Phase Matching
1
cos2 θ sin2 θ
+
=
n2e (θ)
n20
n2e
Most common non-linear crystal
are negative uniaxial and normally
Scheme
Polarizations
dispersive (dn/dω>0) therefore
E(ω1) E(ω1) E(ω2) E(ω3)
requiring type I, II or III phase
o
o
e
I
matching.
e
o
e
II (IIA)
What types of phase matching
e
e
III (IIB) o
would be useful in a positive
e
e
e
IV
uniaxial crystal with normal
o
o
o
V
dispersion?
e
o
o
VI (IIB)
e
o
VII (IIA) o
e
e
o
VIII (I)
For ω1≤ω2<ω3
ch 12.17
Type I and II phase matching
The most common angle phase matching is type I
and II:
Type I phase-matching has the sum frequency
wave E(ω3) with a different polarization than
the other two waves
Type II phase-matching has one of either E
(ω1) or E(ω2) with a different polarization
state than the other two waves.
ch 12.18
Type I Phase Matching
Technically this is type VIII, but it is
commonly referred to as type 1 since the
sum frequency wave is orthogonally
polarized to both pump waves
Normal shells can be used as a geometric tool to
determine proper phase matching angle. In a
positive uniaxial crystal with normal dispersion:
→
k
θm
z-axis
Normal shells
for ω1=ω2
requiring
Normal shells
for ω3=2ω1
1
cos2 θm
sin2 θm
= 2
+ 2
n2o (2ω)
no (ω)
ne (ω)
neω(θm)=n02ω
ch 12.19
Type I Phase Matching
Normal shells can be used as a geometric tool to
determine proper phase matching angle. In a
negative uniaxial crystal with normal dispersion:
→
k
θm
z-axis
Normal shells
for ω1=ω2
requiring
Normal shells
for ω3
1
cos2 θm
sin2 θm
= 2
+ 2
n2o (ω)
no (2ω)
ne (2ω)
noω(θm)=ne2ω
ch 12.20
Quasi Phase Matching
If the crystal domain polarity can be engineered to flip
signs every lc the polarity of the sum frequency wave
being generated can flip every time the sum frequency
wave drifts π out-of-phase with the driving waves.
Conversion efficiency (AU)
10
+z
+z
+z
7.5
Phase matched
Quasi-Phase matched
5
2.5
0
Not phase matched
1
2
Z/lc
3
ch 12.
Engineered QPM materials
Periodically Poled Lithium Niobate (PPLN)
Periodically Poled Lithium Tantalate (PPLT)
Orientation Patterned Gallium Arsenide (OpGaAs)
Lithium
niobate
crystal
Photoresist
Conducting gel
Constant
current
HV source
0.5–1mm
50mm
Poling method for PPLN
Phase contrast image of PPLN
ch 12.22
Momentum Conservation
Phase matching is a form of momentum
conservation (k1+k2=k3) that must be satisfied
along with energy conservation (ω1+ω2=ω3) in
non-linear processes. If we don’t require the
beams be collinear then phase matching can be
achieved by crossing beams
k1
k2
k1
k2
k1
k2 kg
k1
k2
k3
k3
k3
k3
collinear type I
phase matching
collinear type II
phase matching
Quasi phase
matching
non-collinear type
IV phase matching
ch 12.23
Example of SHG in KDP
Determine the type of phase matching to use for
second harmonic generation in KDP with a
fundamental wavelength of λ=694.3 nm, and
determine the phase matching angle using
" " ne(ω)=1.466"" " ne(2ω)=1.487
" " no(ω)=1.506"" " no(2ω)=1.534
ch 12.24
Example of SHG in KDP
→
k
With"ne(ω)=1.466"" " ne(2ω)=1.487
"
" no(ω)=1.506"" " no(2ω)=1.534
this is a negative uniaxial crystal
with normal dispersion. We can use
type I phase matching and require
θm
z-axis
ne2ω(θm)=n0ω
1
cos2 θm
sin2 θm
= 2
+ 2
n2o (ω)
no (2ω)
ne (2ω)
ch 12.25
References
Yariv & Yeh “Optical Waves in Crystals” chapter 12
ch 12.26