II.4 The indicatrix of uniaxial crystals
The direction-dependent properties of light in minerals can best be imagined by a three-dimensional geometric surface corresponding to the numeric values of a particular physical property in all directions (in mathematic terms we call these vectors). An example of this we have already seen in the ray velocity surface. Inversely related to the velocity is the refractive index and the corresponding three-dimensional surface is known as the indicatrix. This indicatrix has the shape of an ellipsoid. Every mineral has its own characteristic indicatrix that qua shape and orientation is determined by the symmetry of the crystal class to which the mineral belongs.
In general we can distinguish three main groups
1) the indicatrix is a sphere. The refractive index is the same in all vibrational directions. Such a crystal is called optically isotropic and is only found in minerals belonging to the isotropic or cubic crystal class.
2) The indicatrix is a biaxial ellipsoid.
3) The indicatrix is a triaxial ellipsoid
Uniaxial crystals belongs to the second group, where the indicatrix has the shape of an ellipsoid characterised by two perpendicular axes. In the equatorial plane the refractive indices are the same for all vibrational directions in that plane. This plane has the shape of a circle and is called the circular section. The normal to the circular section is known as the optical axis. Because the ellipsoid has only one such a circular section, minerals in this group are known as optically uniaxial minerals. The refractive index corresponding to the circular section is called ω (corresponding to the ordinary ray), the refractive index parallel to the optical axis is called ε (corresponding to the extraordinary ray, see calcite example in II.3).
When ε > ω the mineral is called optically positive. The indicatrix has the shape of a citrus (prolate) (Fig. 2.6). An example of this is the mineral quartz
When ω > ε the mineral is called optically negative. The indicatrix has the shape of a mandarin (oblate) (Fig. 2.6). An example of this is the mineral calcite that we have seen before (II.3)
Fig. 2.6 The indicatrix for optically positive and optically negative minerals
Sometime other symbols are used for the refractive index ε; for example nε, Nε, E, ne and Ne. Similarly for ω; nω, Nω, O, no, No
We can prove that the indicatrix can be determined from and has the same shape as the ray velocity surface. Let’s have a look at the situation in Fig. 2.7, which shows an section through the extraordinary ray wave surface XRY with the optical axis given as OX.
Fig. 2.7 Section through a biaxial indicatrix, giving the ellipse X’R’Y’
Here we can see that:
OY = vo =c/ω (1) ↔ ω = c/vo
OX = ve = c/ε (2) ↔ ε = c/ve
Where vo ≥ ve i.e. ω ≤ ε (uniaxial, positive)
The refractive index in the ellipse is represented by OX’ = 1/OY and OY’ = 1/OX. This ellipse has the same shape as the corresponding section through the ray velocity surface since, based on (1) and (2) we can say:
OY/ε = c/ω x 1/ε and OX/ω = c/ε x 1/ω
So: OY/ε = OX/ω = c/εω = constant !!!!!! (3)
To the ray OR belongs a front normal ON. Now we need to prove that ε' =c/ON, where ON = vε'.
For an ellipse we can state that the product ON' x OD = constant = ε x w
OD ┴ ON; OD = ε'; ON' x ε' = ε x w ® ε' = εw/ON' ® ON' = εw/ε' (4)
cos RON = ON/OR and
cos R'ON' = ON'/OR' so ON/OR= ON'/OR' ® ON/ON' = OR/OR' (5)
substitute (3) in (5): ON/ON' = c/εw (6)
so ON' = εw/c x ON (7)
and ε' = εw/ON' = εw x c/(ON x εw) = c/ON