The biaxial minerals are members of the lower symmetry systems:
orthorhombic, monoclinic and triclinic. Their optical indicatrix has no
circular symmetry anymore but is a general ellipsoid with three principal axes (a,
b and g)
which are the principal indices of refraction of the considered mineral. a
is the smaller index, g is the
larger. Such an ellipsoid, has two circular sections with indices equal to b.
The lines perpendicular to these circular sections are called optic axes, so
biaxial crystals have two optic axes. The indices a
and g are in the same plane as
the optic axes and they bisect the optic axes angles. The index b
is perpendicular to the optic plane. Figure 1 shows a section of the biaxial
indicatrix, the index b
is perpendicular to the plane of the figure. An optical sign can also be
defined for those crystals: a positive crystal has its large index g
as acute bisectrix of the optic axes angle like in figure 1. For a negative
crystal a
is the acute bisectrix .

The same geometrical construction can be made to view the conoscopic image
structure of a biaxial crystal perpendicular to the acute bisectrix as we deed
for uniaxial crystals. In figure 1, main features of such an image are
presented. To get a three dimensional picture, the conoscopic plane should be
rotated by 90° around an horizontal axis. The poles of both optic axes appear
in the figure with a birefringence equal to zero because both indices are b
(cyclic section), they are thus always dark. At the center of the figure, the indices are a
and b . Between
the poles of the optic axes and the center of the figure, the indices parallel
to the optic planes have values between a
and b as can be
seen on the indicatrix figure because such points in the conoscopic figure
correspond to sections situated between the circular section and the section
containing a
(marked by the red line in the figure). For the points beyond the poles of the
optic axes, the indices in the optic plane are greater then b
and are called g'
(sections between cyclic section and g
). The indices on this line being parallel to polarizer and analyzer belong to
the isogyres and are thus situated on a dark line.

Deduction of the orientation of refraction indices in other places of the figure
is a little bit more complicated then explained for uniaxial crystals. The
approximate solution is coming from the Biot-Fresnel law: draw the lines
connecting the optic axes to the point of interest in the conoscopic figure. The
bisectrix of these two lines determines the direction of one index in that
position, the second index being perpendicular to the first one. Two examples of
such a construction are given in figure 1. At point Y
the index g'
is the bisector of lines c and d
joining Y to the poles of the optic axes.
The other index is perpendicular to g'
and equal to a
because the section in the indicatrix normal to point Y
contains main index a.
The same is true for all the points of the vertical line passing through Y.
These two indices are parallel to polarizer and analyzer and thus belong to the
isogyres. The second
example is given by point X, where the
indices are bisector of lines a and b
joining X to the poles of the optic axes. In
that case, the direction of indices are no more colinear to the crossed polars
so the interference colors could be seen.

Figure1. Conoscopy of a biaxial crystal cut perpendicularly
to the acute bisectrix of the optical axes. The optic plane is parallel to
polarizer priviledged direction.

For a crystal with its optic plane parallel to polarizer or analyzer, the
isogyres are an asymmetric cross with a wider dark line normal to the optic
plane. The orientation of indices changes slower on the vertical line between
optic axes than in the region around optic axes as can be seen by experimenting
with the Biot-Fresnel construction. This explain why the isogyres are
thinner close to the poles of the optic axes, particularly if the birefringence
and thickness of the crystal plate are high.

The isochromes curves have also a more complicated shape as compared to the
uniaxial conoscopy. For high birefringence and thickness, some of them resemble
circles and ellipses around the optic axis. With the Biot-Fresnel construction,
it is easy to see that indices directions rotate around the pole of the optic
axis in a region close to it. At some distances from the axes, the extinction
curves have the shape of Bernouilli's lemniscates.

The rotation of the optic plane by 45° gives the image in figure 2. This image
was obtained in monochromatic light at 467 nm. Two groups of dark curves can be
seen on the figure. Point X belongs to the
isogyres because the indices obtained as bisector of the aXb
angle are parallel to polarizer and analyzer. If we repeat this geometric
construction, we get the complete isogyres which are now hyperboles passing
through the poles of the optic axes. Other dark points like Y
have indices in any orientation. They are dark because their retardation is a
multiple of the wavelength of the light and belong thus to the isochromes.

Figure 2. Same crystal as in figure 1 but optic plane
rotated at 45° from polarizer and analyzer. Monochromatic light (467nm).