Optic axis of a crystal is the direction in which a ray of transmitted light suffers no birefringence (double refraction). Light propagates along that axis with a speed independent of its polarization
However, if the light beam is not parallel to the optical axis, then, when passing through the crystal the beam is split into two rays: the ordinary and extraordinary, to be mutually perpendicular polarized.
A crystal which has only one optic axis is called uniaxial crystal. An uniaxial crystal is isotropic within the plane orthogonal to the optical axis of the crystal.
A crystal which has only two optic axis is called biaxial crystal.

The refractive index of the ordinary ray is constant for any direction in the crystal, and of the extraordinary ray is variable and depends on the direction.
In a uniaxial crystal for the direction parallel to the optical axis the refractive indices are equal.
Positive birefringence
Negative birefringence
BBO is negative uniaxial crystal
Fast Axis
Direction having a low refractive index is the fast axis; at right angles to it is the slow axis, with a high index of refraction.

Type I cut k p
= k s
+ k i n p w p
=2n s w s cos q s for w s
= w p
/2 n p
=n s cos q s
For collinear down-conversion the index of refraction at wavelengths differing by a factor of two must be equal. For isotropic material it is impossible, but by using birefringent material we can achieve this condition via the different indices of refraction polarizations.
of orthogonal linear
Index of refraction of BBO (negative uniaxial crystal): blue solid curve: ordinary polarization; red/dash-dot curve: full extraordinary polarization; green/dash curve: extraordinary polarization at the phase matching angle (29 o )

Calculating Characteristics of Noncollinear Phase Matching in Uniaxial and Biaxial Crystals
N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, A. Migdall
Originally published in: Opt. Eng. 39 (4), 1016-1024 (April 2000). http://units.nist.gov/Divisions/Div844/facilities/cprad/index.html

Malus' law , which is named after Etienne-Louis Malus (1775 –
1812) , says that when a perfect polarizer is placed in a polarized beam of light, the intensity, I , of the light that passes through is given by where
I
0 is the initial intensity, and θ i is the angle between the light's initial polarization direction and the axis of the polarizer.
His discovery of the polarisation of light by reflection was published in 1809 and his theory of double refraction of light in crystals , in 1810.

A retarder, or waveplate, is an optical device that resolves a light wave into two orthogonal linear polarization components and produces a phase shift between them. The resulting light wave generally is of a different polarization form.
Ideally, retarders do not polarize, nor do they induce an intensity change in the light beam. They simply change its polarization form.
Retarders are used in applications where control or analysis of polarization states is required.

A retarder that produces a λ/2 phase shift is known as a half wave retarder. Half wave retarders can rotate the polarization of linearly polarized light to twice the angle between the retarder fast axis and the plane of polarization.
Placing the fast axis of a half wave retarder at 45 ° to the polarization plane results in a polarization rotation of 90 °. Passing circularly polarized light through a half wave plate changes the "handedness" of the polarization.

If the orthogonal electric field components are equivalent, a phase shift λ/4 in one component will result in circularly polarized light.
Retarders that cause this shift are known as quarter wave retarders.
They have the unique property of turning elliptically polarized light into linearly polarized light or of transforming linearly polarized light into circularly polarized light when the fast axis of the quarter wave plate at
45 ° to the incoming polarization plane. (Light polarized along the direction with the smaller index travels faster and thus this axis is termed the fast axis. The other axis is the slow axis).