Direct Energy, 2018a

3 PYROELECTRICS AND ELECTRO-OPTICS 57
We can write the magnitude of the material polarization as a function
of powers of the applied external eld.
∣ −→ ∣ ∣∣ −→
∣ ∣ ∣∣
P ∣ = ɛ 0 χ e E + ɛ0 χ (2) ∣∣ −→
∣ ∣∣
2 ∣
E + ɛ0 χ (3) ∣∣ −→
∣ ∣∣
3
E + ... (3.4)
The quantity χ (2) is called the chi-two coecient, and it has units m V . The
quantity χ (3) is called the chi-three coecient, and it has units m2 [27] [42,
V 2
ch. 1].
If an innite number of terms are included on the right side of Eq. 3.4,
any arbitrary material can be described. In most materials, only the rst
term of Eq. 3.4 is needed while χ (2) ,χ (3) , and all higher order coecients
are negligible, and these materials are not electro-optic. Materials with
χ (2) , χ (3) or other coecients nonzero are called electro-optic. It is rare to
need more coecients than χ e ,χ
∣ (2) , and χ (3) todescribe a material.
∣∣
The eect due tothe ɛ 0 χ (2) −→
∣ ∣∣
2
E term is called the Pockels eect or
linear electro-optic eect. It was rst observed by Friedrich Pockels in
1893 [3, p. 382] [10]. In this case the material polarization
∣
depends on
∣∣
the square of the external eld. The eect due to the ɛ 0 χ (3) −→
∣ ∣∣
3
E term is
called the Kerr eect or the quadratic electro-optic eect. In this case,
the material polarization depends on the cube of the external electric eld.
John Kerr rst described this eect in 1875 [3, p. 382] [10].
While some authors use the coecients χ e , χ (2) and χ (3) , this eect is
most often studied by optics scientists who instead prefer index of refraction
n, a unitless measure introduced in Sec. 2.2.3. In electro-optic materials,
the index of refraction is a nonlinear function of the strength of the external
electric eld. Instead of expanding the material polarization in a power
series as a function of the external eld strength as in Eq. 3.4, the index of
refraction is expanded. Pockels and Kerr coecients are dened as terms
of this expansion.
As described by Eq. 2.3, material polarization is the dierence in m C 2
between an external electric eld in a material and the eld in the absence
of the material.
| −→ P | = | −→ D|−ɛ o | −→ E | (3.5)
With some algebra, we can identify the displacement ux density component
and the overall index of refraction. Add two terms which sum to zero
toEq. 3.4.
∣ −→ ∣ ∣∣ −→
∣ ∣∣
P ∣ = ɛ 0 χ e E + ɛ0 | −→ ∣
E | + ɛ 0 χ (2) ∣∣ −→
∣ ∣∣
2 ∣
E + ɛ0 χ (3) ∣∣ −→
∣ ∣∣
3 ∣ ∣∣ −→
∣ ∣∣ E − ɛ0 E (3.6)