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Second-order Χ(2) nonlinear optical materials; crystal classes
and their symmetries and nonlinear optical tensors;
electrooptic effect ;
Lecture 7
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Anisotropic linear media
tensorvector vector
or
In nonmagnetic and transparent materials, 𝜒𝑖𝑗 = 𝜒𝑗𝑖, i.e. the 𝜒 tensor is real and symmetric.
It is possible to diagonalise the tensor by choosing the appropriate coordinate axes, leaving only 𝜒%%, 𝜒'' and 𝜒((. This gives :
Linear susceptibility is a tensor
In an anisotropic medium, such as a crystal, the polarisation field P is not necessarily aligned with the electric field of the light E. In a physical picture, this can be thought of as the dipoles induced in the medium by the electric field having certain preferred directions, related to the physical structure of the crystal.
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2
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Anisotropic linear media
The refractive index is: 𝑛 = 1 + 𝜒
hence 𝑛%% = 1 + 𝜒%%
𝑛'' = 1 + 𝜒''
𝑛(( = 1 + 𝜒((
Waves with different polarizations will see different refractive indices and travel at different speeds.
This phenomenon is known as birefringenceand occurs in some crystals such as calcite and quartz.
If 𝜒%% = 𝜒'' ≠ 𝜒((, the crystal is known as uniaxial.
If 𝜒%% ≠ 𝜒'' ≠ 𝜒(( the crystal is called biaxial.
or 𝜀 =
will come back to this in Lecture 8
z-axisor c-axis
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Anisotropic linear median1=n2 n3 n1 n2 n3
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3
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Nonlinear Susceptibility Tensor
Nonlinear susceptibility is a 3-rd rank tensor 𝜒./0(2)
(3 x 3 x 3 tensor)
the nonlinear susceptibility tensor 𝜒./0(2)
When all of the optical frequencies are detuned from the resonance frequencies of the optical medium
has full permutation symmetry:
𝜒./0 = 𝜒.0/= 𝜒/.0= 𝜒/0.= 𝜒0./= 𝜒0/.
Full permutation symmetry can be deduced from a consideration of the field energy density within a nonlinear medium(see Boyd or Stegeman books).
And also from quantum mechanics !
... and does not actually depend of frequencies that participate, so instead of writingwe write simply 𝜒./0
(2)
Kleinman’s symmetry condition
𝜒./0(2)(𝜔5 = 𝜔6+𝜔2, 𝜔6, 𝜔2)
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Nonlinear Susceptibility TensorConsider mutual interaction of three waves at 𝜔1, 𝜔2, and 𝜔3 = 𝜔1 + 𝜔2
Assume E is the total field vector 𝑬 = 𝑬=6 + 𝑬=2 + 𝑬=5
Or in vector components 𝐸. = 𝐸.,=6 + 𝐸.,=2 + 𝐸.,=5𝑖 = 1,2,3 (= 𝑥, 𝑦, 𝑧)
For example, 𝑥 −component of the nonlinear polarization P is:
𝑃6 = 𝜖EF/,0
𝜒6/0 𝐸/𝐸0 = 𝜖E { 𝜒666𝐸6𝐸6 + 𝜒662𝐸6𝐸2 + 𝜒665𝐸6𝐸5 +
+ 𝜒626𝐸2𝐸6 + 𝜒622𝐸2𝐸2 + 𝜒625𝐸2𝐸5 +
+ 𝜒656𝐸5𝐸6 + 𝜒652𝐸5𝐸2 + 𝜒655𝐸5𝐸5 }
𝑃2 = 𝜖EF/,0
𝜒2/0 𝐸/𝐸0 = …
𝑃5 = 𝜖EF/,0
𝜒5/0 𝐸/𝐸0 = …
x
y
z
𝑃. = 𝜖EF/,0
𝜒./0 𝐸/𝐸0 𝑖, 𝑗, 𝑘 = 1,2,3 (= 𝑥, 𝑦, 𝑧)then
E are real fields~ 𝐸 cos(𝜔𝑡)
note that the terms underlined by the same color are equal
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4
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Nonlinear Susceptibility Tensor
Now inroduce the tensor: 𝑑./0 =12𝜒./0
(2)(historical convention !)
and write:
𝑃6 = 2𝜖E ( 𝑑666𝐸6𝐸6 + 𝑑622𝐸2𝐸2 + 𝑑655𝐸5𝐸5 + 2𝑑625𝐸2𝐸5 + 2𝑑665𝐸6𝐸5 + 2𝑑662𝐸6𝐸2)
𝑃2 = 2𝜖E ( 𝑑266𝐸6𝐸6 + 𝑑222𝐸2𝐸2 + 𝑑255𝐸5𝐸5 + 2𝑑225𝐸2𝐸5 + 2𝑑265𝐸6𝐸5 + 2𝑑262𝐸6𝐸2)
𝑃5 = 2𝜖E ( 𝑑566𝐸6𝐸6 + 𝑑522𝐸2𝐸2 + 𝑑555𝐸5𝐸5 + 2𝑑525𝐸2𝐸5 + 2𝑑565𝐸6𝐸5 + 2𝑑562𝐸6𝐸2)
now reduced to only 18 componnents
Reduce 3D tensor to 2D matrix
𝑗𝑘 → 𝑙
𝑗𝑘 = 11 22 33 23,32 31,13 12,21𝑙 = 1 2 3 4 5 6
Second-order susceptibility described using contracted notation
Matrix with 6x3 components
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Nonlinear Susceptibility Tensor
𝑃%𝑃'𝑃(
=
𝐸%2
𝐸'2
𝐸(22𝐸'𝐸(2𝐸%𝐸(2𝐸%𝐸'
the way to find nonlinear polarizations in 3D case using contracted notation
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5
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Nonlinear Susceptibility Tensor
By further applying Kleinman symmetry, we find that 𝑑.R matrix has only 10 independent elements
for example: d14 is d123
but d25 is d213 =d123
and d36 is d312 =d123
d14=d25=d36à etc.
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Nonlinear Susceptibility Tensor
Only 10 independent elements
à
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Nonlinear Susceptibility Tensor
Spatial symmetries of crystals further reduce the amound of independednt tensor elements
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Nonlinear Susceptibility Tensor
‘Cubic’ crystalse.g. GaAs, GaP, InAs,ZnS, ZnSe– are optically isotropicbut nonlinear !
KTP
LiNbO3
GaSe
KDP
GaAszero nonlinearity
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Nonlinear Susceptibility Tensor
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Nonlinear Susceptibility Tensor
andKTP (KTiO2PO4) crystal
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Nonlinear Susceptibility Tensor
KTP (KTiO2PO4) crystal
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Nonlinear Susceptibility Tensor
Crystal of class 3m (e.g. Lithium Niobate, LiNbO3)
only d22 , d31 and d33
Class: 3m àother crystals of this class
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Physical origin of off-diagonal elements in dijk tensorafter Stegeman NLO book
x
yz
oxygenoxygen
oxygen
oxygen
phosphorus
active electron
bondsbonds
E-field applied in xy
electron motion
KDP crystal
e-
motion in in xy
motion in z
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Physical origin of off-diagonal elements in dijk tensor
x
yz
As
Ga
active electron
bondsbonds
E-field applied in xy
electron motion
GaAs crystal
e-
motion in in xy
motion in z
As
As
As
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10
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Nonlinear Susceptibility Tensor
Crystal of class 43𝑚 (e.g. Gallium Arsenide, GaAs)
only d14
GaAs
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Nonlinear Susceptibility Tensor
GaAs
=2𝜖E
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Nonlinear Susceptibility Tensor
𝑃%𝑃'𝑃(
=
𝐸%2
𝐸'2
𝐸(22𝐸'𝐸(2𝐸%𝐸(2𝐸%𝐸'
How to calculate effective nonlinearity ?
1) 𝐿𝑖𝑁𝑏𝑂3 𝑑55 ≠0
Simple case: SFG 𝜔3 = 𝜔1 + 𝜔2
all waves (𝜔1𝜔2𝜔3) are polarized along z-axis𝐸( 𝑡 = 𝐸6,(𝑒.=Z[ + 𝐸2,(𝑒.=\[input fields
Need to find z-component of 𝑃(,=5
(]^) = 𝑃(𝜔5)𝑒.=_[
𝑃(]^ (𝑡) = 2𝜖E𝑑55𝐸(2 = 2𝜖E𝑑55
6` (𝐸6,(𝑒.=Z[ + 𝐸2,(𝑒.=\[ + c.c.)2= ...
2𝜖E𝑑bcc𝐸(2 𝑑bcc=𝑑55
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Nonlinear Susceptibility Tensor
2𝜖E𝑑556`
(𝐸6,(𝑒.=Z[ + 𝐸2,(𝑒.=\[ + c.c.)2= ...
= 2𝜖E𝑑556`(𝐸6,(2 𝑒2.=Z[ + 𝐸2,(2 𝑒2.=\[ + 2𝐸6,(𝐸2,(𝑒.(=Zd=\)[ + 2𝐸6,(𝐸2,(∗ 𝑒.(=Zf=\)[ + c. c. ) + 2𝜖E𝑑55
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(𝐸6,(𝐸6,(∗ +𝐸2,( 𝐸2,(∗ ) =...
𝑃(]^ 𝑡 =
for SFG term: 𝑃 𝜔5 =𝑃 𝜔6 + 𝜔2 – pick only components with ± 𝜔6 + 𝜔2
𝑃(]^ 𝑡 = 2𝜖E𝑑55
6`(2𝐸6,(𝐸2,(𝑒.(=Zd=\)[ +𝑐. 𝑐. ) = 𝜖E𝑑55(𝐸6,(𝐸2,(𝑒.(=Zd=\)[ +𝑐. 𝑐. )
12(𝑃( 𝜔5 + 𝑐. 𝑐)
at 𝜔5
𝑃( 𝜔5 =𝜖E2𝑑55(𝐸6,(𝐸2,()
= 𝜖E𝑑55(𝐸6,(𝐸2,(𝑒.(=Zd=\)[ +𝑐. 𝑐)
real Fourier componnent 𝑃 = 𝑃( 𝜔5 cos(𝜔5t)
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Nonlinear Susceptibility Tensor
same as (5.2)–(5.3) with 𝑑 = 62𝜒(2)
𝑃 𝜔6 + 𝜔2 = 2𝜖E𝑑55𝐸6𝐸2
𝑃 2𝜔6 = 𝜖E𝑑55𝐸62
𝑃 2𝜔2 = 𝜖E𝑑55𝐸22
once you know 𝑑bcc, you can treat fields as scalars
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Nonlinear Susceptibility Tensor
2) Effective nonlinearity, GaAs
focus on this component:
SFG 𝜔3 = 𝜔1 + 𝜔2
𝑬 𝑡 = 𝑬𝟏𝑒.=Z[ + 𝑬𝟐𝑒.=\[for input fields
Need to find 𝑃(𝜔5)
𝐸6 𝐸2xy
𝑃=5
(]^) 𝐸5z
y
x
GaAs
beam k-vector: goes into the page (so-called 110 direction)
𝐸6 →12(𝐸6%+𝐸6')𝑒.=Z[
𝐸2 →12(𝐸2%+𝐸2')𝑒.=\[
𝑃𝑁𝐿(𝑡) = 𝑃((𝑡) = 𝜖E2𝑑6` 2𝐸%𝐸' = 4𝜖E𝑑6`626`(𝐸6𝑒.=Z[ + 𝐸2𝑒.=\[ + 𝑐. 𝑐)2
𝐸% =12(𝐸6𝑒.=Z[ + 𝐸2𝑒.=\[)
𝐸' = 𝐸%
𝑃𝑁𝐿 𝑡, 𝜔5 = 𝜖E𝑑6`62(2𝐸6𝐸2𝑒.(=Zd=\)[ + 𝑐. 𝑐. ) = 𝜖E𝑑6`(𝐸6𝐸2𝑒.(=Zd=\)[ + 𝑐. 𝑐. ) = 𝜖E2𝑑6`[
62
(𝐸6𝐸2𝑒.(=Zd=\)[ + 𝑐. 𝑐. )]
𝑑bcc=𝑑6`𝑃 𝜔6 + 𝜔2 = 2𝜖E𝑑6`𝐸6𝐸2 Can treat fields as scalars keeping in mind that 𝐸6 𝐸2 are in xy and 𝐸5 is in z -direction.
for SFG à
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Nonlinear Susceptibility Tensor
Because of the off-diagonal tensor elements, we can generate SFG with the output polarization perpendicular to the input polarizations
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First-order electrooptic effect in anisotropic crystals
The electrooptic effect (Pockels effect) is the change in refractive index of a material inducedby the presence of a static (or low-frequency) electric field.
Linear anisotropic medium:
Dielectric tensor is represented as a diagonal matrix (by a propoer choice of the coordinate system).
the dielectric constant is a second-rank tensor
in the principal dielectric axes
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First-order electrooptic effect in anisotropic crystals
beam k-vector
z
x,y
the index ellipsoid
two allowed directions of polarization with two distinct ncoefficients
𝑥2
𝑛62+𝑦2
𝑛22+𝑧2
𝑛52= 1
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First-order electrooptic effect in anisotropic crystals
direction of propagationbeam k-vector
z
x,y
Uniaxial crystal: n1=n2=no; n3=ne
two allowed directions of polarization with two distinct n coefficients
𝑥2
𝑛m2+𝑦2
𝑛m2+𝑧2
𝑛b2= 1
θ
perpendicular to the plane of the figure
in the plane of the figure
Uniaxial crystals: The indicatrix is an ellipsoid of revolution.
For the direction of polarization perpendicular to theoptic axis, known as the ordinary direction, the index is independent of the direction of propagation.
For the other direction of polarization, known asthe extraordinary direction, the index changes between the value of the ordinary index n0, when the wave normal is parallel to the optic axis (z) and the extraordinary index ne, when the wave normal is perpendicular to the optic axis.
The two beams of light so produced are often referred to as o-rays and e-rays, respectively.
When the wave normal is in a direction θ tothe optic axis, the index is given by:
n=𝑛m
1𝑛(𝜃)2 =
𝑐𝑜𝑠(𝜃)2
𝑛m2+sin(𝜃)2
𝑛b2
à
𝑛b>𝑛m‘positive’ crystal
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First-order electrooptic effect in anisotropic crystals
direction of propagationbeam k-vector
z
x,y
Uniaxial crystal: n1=n2=no; n3=ne
two allowed directions of polarization with two distinct n coefficients
𝑥2
𝑛m2+𝑦2
𝑛m2+𝑧2
𝑛b2= 1
θ
perpendicular to the plane of the figure
in the plane of the figure
Uniaxial crystals: The indicatrix is an ellipsoid of revolution.
For the direction of polarization perpendicular to theoptic axis, known as the ordinary direction, the index is independent of the direction of propagation.
For the other direction of polarization, known asthe extraordinary direction, the index changes between the value of the ordinary index n0, when the wave normal is parallel to the optic axis (z) and the extraordinary index ne, when the wave normal is perpendicular to the optic axis.
The two beams of light so produced are often referred to as o-rays and e-rays, respectively.
When the wave normal is in a direction θ tothe optic axis, the index is given by:
n=𝑛m
1𝑛(𝜃)2 =
𝑐𝑜𝑠(𝜃)2
𝑛m2+sin(𝜃)2
𝑛b2
à
𝑛b < 𝑛m‘negative’ crystal
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First-order electrooptic effect in anisotropic crystals
the index ellipsoid
𝑥2
𝑛62+𝑦2
𝑛22+𝑧2
𝑛52= 1
General case:
(only 6 independent terms)
General case for the the index ellipsoid
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First-order electrooptic effect in anisotropic crystals
or
The essense of linear electrooptic effect
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First-order electrooptic effect in anisotropic crystals
KDP
Electrooptic modulator, KDP
Electrooptic effect and NLO effects are present in the same classes of crystals
Apply field along z-axis
This causes (via r63) induced index change in xy plane
y
zx
𝑛%t=
𝑛't=
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First-order electrooptic effect in anisotropic crystals
Evolution of the vertical polarization originally sent to the modulator
Half-wave voltage ~ kV range
Electrooptic modulator, KDP
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First-order electrooptic effect in anisotropic crystals
Electrooptic modulator, lithium niobate LiNbO3
Apply field along x-axis (transverse effect)
This causes (via r22) induced index change in xy plane
y
zx
𝑛%t 𝑛'tV
Half-wave voltages are much lower than in KDP
NLO tensor
light
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Electrooptic effect vs NLO effect
Let us now have another look at this phenomenon.
Before we saw that we can generate SFG with the output polarization perpendicular to the input polarizations.
SFG 𝜔3 = 𝜔1 + 𝜔2
EO 𝜔 = 0 + 𝜔
orthogonal polarizations
The electrooptic effect can be seen as a frequency-mixing interaction (SFG or DFG) between the incident radiation and an externally applied DC voltage.
𝜔6 → 0, 𝜔5 → 𝜔2
𝑑./ = −𝑛`
4 𝑟/.General relation between Pockels and NLO coefficients
Electrooptic effect : - see Yariv ‘Opt Waves in Cryst’ (p 561) - for rijk vs dijk relations
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