Direct Energy, 2018a

58 3.3 Electro-Optics
The rst two terms can be combined, and ɛ 0 | −→ E | can be distributed out.
∣ −→ P
[
∣ ∣ = (χ e +1)+χ (2) ∣∣ −→
∣ ∣∣ E + χ
(3)
∣ −→ E
]
∣ 2 ∣ ∣∣ −→
∣ ∣ ∣∣ ∣∣ −→
∣ ∣∣
+ ... ɛ 0 E − ɛ0 E (3.7)
The rst term is the displacement uxdensity.
−→ −→
∣
D = ɛr eoE =
[(χ e +1)+χ (2) ∣∣ −→
∣ ∣∣ E + χ
(3)
∣ −→ ]
E ∣ 2 ∣ ∣∣ −→
∣ ∣∣
+ ... ɛ 0 E (3.8)
The quantity in brackets in Eq. 3.8 is the relative permittivity, ɛ r eo . Since
we are considering electro-optic materials, it depends nonlinearly on the
applied external eld. Assuming the material is a perfect dielectric with
μ = μ 0 , the indexof refraction is the square root of this quantity. It
represents the ratio of the speed of light in free space to the speed of light
in this material, and it also depends nonlinearly on the applied external
eld.
n eo = √ ɛ r eo (3.9)
The indexof refraction must be larger than one because electromagnetic
waves in materials cannot go faster than the speed of light, so the quantity
1
ɛ r eo
must be less than one.
Some authors expand the term 1
ɛ r eo
in a Taylor expansion instead of the
material polarization, and electro-optic coecients are dened with respect
to this expansion [42].
1
= 1
+ γ ∣ −→ E ∣ + s ∣ −→ E ∣ 2 + .... (3.10)
ɛ r eo ɛ r x
The coecient γ is called the Pockels coecient, and it has units m V . The
coecient s is called the Kerr coecient, and it has units m2 . In the
V 2
absence of nonlinear electro-optic contributions, we can denote the relative
permittivity as ɛ r x and the indexof refraction as n x where
ɛ r x = n 2 x = χ e +1. (3.11)
The expansion of Eq. 3.10 is guaranteed to converge because
1
ɛ r eo
< 1.
Example values of the Pockels electro-optic coecient are listed in Table
3.1.
With some algebra, the overall indexof refraction n eo can be written in
terms of the Pockels and Kerr coecients. Equations 3.9 and 3.10 can be
combined.