In anistropic media, the velocity of light, and
thus the refractive index, depends on the
direction of vibration of light. The variation
of refractive index with vibration direction is
represented by a three dimensional ellipsoid
called an indicatrix.
An ellipsoid is a 3-D version of an ellipse,
such that any point on the surface of the
ellipsoid obeys the relationship:
x2/a2 + y2/b2 + z2/c2
The optical indicatrix is an imaginary 3-D
surface constructed such that the distance from
the center of the indicatrix to any point on its
surface is equal to the refractive index of light
vibrating in that direction. The shape and
orientation of the optical indicatrix is constant
for a mineral of fixed composition, but its
actual position in the mineral is not important.
The 2-D ellipse is a special case, representing the
locus of all points P(x,y), the sum of whose
distance from two points is a constant, or:
x2/a2 + y2/b2 = 1
where a and b are the major and minor axes of the
Types of ellipsoids:
Triaxial Ellipsoid: the general case: a ≠ b ≠ c ; α = β = γ = 90o
The optical indicatrices of minerals in the triclinic, monoclinic, and orthorhombic crystal systems are triaxial
ellipsoids. Three refractive indices must be determined to completely specify their indicatrix (ηα, ηβ, ηγ). The 2
fold rotational axes coincide with the major, intermediate, and minor axes, while mirror planes coincide with
each of the pairs of axes. Minerals with triaxial indicatrices are, in fact, termed biaxial because they contain two
circular sections, each with a perpendicular “optic axis”.
Ellipsoid of Revolution: a = b ≠ c
(x2 + y2)/a2 + z2/c2 = 1
The optical indicatrices of minerals in the trigonal, hexagonal, and tetragonal crystal systems are
ellipsoids of revolution, with the axis of revolution corresponding to the axis of highest rotational
symmetry. Two refractive indices are required to completely specifiy their indicatrix (ηε, ηω).
Minerals with indicatrices that are ellipsoids of revolution are termed uniaxial because they
contain only one circular section, with one perpendicular optic axis.
a = b = c
x2 + y2 + z2 = a2
The optical indicatrices of minerals in the isometric crystal systems are spheres. These minerals
are termed isotropic, and only one refractory index must be specified.
Principle of Double Refraction:
In an anisotropic medium, there are always two rays associated with any wave front
normal. Each of the two rays is plane polarized, but the planes of vibration are at
right angles to each other. The planes of polarization contain the major and minor
axes of the ellipse of section through the indicatrix that is perpendicular to the wave
front normal. The refractive indices of the 2 rays are proportional to the lengths of the
major and minor axes of the ellipse of section.
All sections passing through the center
of the indicatrix are ellipses:
Principle sections are those whose major
and minor axes both correspond to two of the
3 axes of the indicatrix.
Semi-principle sections are those that
whose major or minor axis corresponds to
one of the 3 axes of the indicatrix.
General sections are those whose major and
minor axis do not correspond to any of the 3
axes of the indicatrix.
Unixaial minerals are characterized by optical indicatices that are ellipsoids of revolution, in
which the axis of revolution of the indicatrix corresponds to the higher order symmetry axis in
the tetragonal, hexagonal, or trigonal systems. There are two types of uniaxial indicatrix:
Prolate (+): c > a ηε > ηω
c < a ηε < ηω
According to the law of double refraction, light entering a uniaxial mineral is split into two rays,
each plane polarized vibrating at right angles to each other and parallel to the major and minor
axis of the ellipse of section through the indicatrix perpendicular to the wave front normal. The
two rays will travel at different speeds within the mineral because in general they have different
refractive indices, given by the relative lengths of the major and minor axis of the ellipse of
happens to these two rays
within the crystal we
employ Huygen’s method,
constructing two different
ray velocity surfaces, one
for the ray and one for
Upon rotation of the microscope stage, the dot corresponding to the ηε ray will revolve about the
dot corresponding to the ηω ray. The ηε ray vibrates along the line joining the ηε and ηω rays,
while the ηω ray vibrates perpendicular to this line.
Interference and Birefringence:
The really interesting thing about the law of double refraction, however, is what happens
when the two plane polarized rays are forced to interfere with each other in the polarizing
analyzer filter of the microscope. Retardation or path difference Δ = VC × (time difference)
= VC × (d/V1 – d /V2)
= d × (VC/V1 –VC/V2)
= d × (2-1)
If the two plane polarized rays were vibrating in the same plane, then:
If the path difference between
the two rays is an even multiple of the
= d (2-1) = n
= Δ / air
If the path difference between
the two rays is an odd multiple of the
wavelength/2, the two rays will
Retardation Δ = d (2-1)
If, however, the two plane polarized
rays are vibrating in planes at right
angles to each other, the situation is
more complex because of the rotation
of the recombined light wave with
respect to the microscope’s analyzer
filter. Destructive interference occurs
when the path difference is a multiple
of the wavelength, and constructive
interference occurs when the path
difference is an odd multiple of half the
If the stage is rotated such that the mineral’s
vibration directions parallel those of the
complete extinction will occur. Any given
anisotropic grain will thus go extinct 4 times
(every 90o) in one complete rotation of the
The white light of your microscope is composed of many
wavelengths. Thus for some given retardation (Δ)
produced by a mineral grain, wavelengths such that Δ/λ is
an integer will not be transmitted, while those such that
Δ/λ is (2n+1)/2 will be transmitted at full intensity. All
other wavelengths will be partially transmitted, and the
light that reaches you eye will be the summation of the
intensities of all the wavelengths that make it through the
The net result is a spectrum of interference colours or
increasing value of retardation. The interference colours
repeat approximately every 550 mμ (λ of yellow light)
with reddish interference colours that divide the spectrum
into octaves or “orders”. Low order colours are bright,
while high order colours become pastel, eventually taking
on a “flesh” colour called high order white. Visual
estimates of the retardation produced by a grain are given
by indicating the interference colour and its order, eg.
second order yellow.
The retardation or path difference (Δ) produced by a
crystal is a function of the thickness of the crystal (d), the
birefringence of the mineral (Β = ηmax - ηmin), and the
orientation of the crystal (ηbig – ηsmall).
Δ = d × (ηbig – ηsmall)
Because the thickness (d) of grains is not
constant, individual grains will exhibit
increasing retardation towards their centers,
giving rise to concentric interference colour
bands within each mineral grain.
Thin sections are cut to a constant thickness, thus any given mineral grain will exhibit
only one interference colour. Different grains of the same mineral in one thin section,
however, will exhibit different interference colours because of their different
orientations, which gives rise to difference ellipses of section of their indicatrix and
thus different values for (ηbig – ηsmall). Only the maximum interference colour is a true
measure of the mineral’s birefringence. For uniaxial minerals, these would be grains
whose c axis (coincident with ηE) lie in the plane of the microscope stage.
Rotation of the stage
from an extinction position will
cause the isogyre to break into
two shadows that leave the field
of view in the quadrants into
which the c axis moves. This
phenomena can be used to
identify the vibration
direction and determine the
optic sign of a uniaxial mineral,
if optic axis figures are difficult
It is easy to determine the orientation of the 2 vibration directions in any grain, simply by turning the
microscope stage until the grain goes extinct. In this position, the two vibration directions of the grain
parallel those of the microscopes analyzer and polarized. It is also easy to tell which of the vibration
directions is ηbig and which is ηsmall. This is done by introducing an accessory plate of known
vibration directions and retardation into the light path when the grain is rotated to the 45o position.
If ηbig of the grain parallels ηbig of the accessory plate, then the retardation of the mineral will be
added to that of the accessory plate, and the resultant interference colour will go up.
If ηsmall of the grain parallels ηbig of the accessory plate, then the retardation of the mineral will be
subtracted from that of the accessory plate, and the resultant interference colour will go down.
Measurement of Refractive Indices in Uniaxial Minerals
The measurement of the refractive indices of a uniaxial mineral grain involves the separate
determination of and . Every grain of a uniaxial mineral will exhibit a vibration direction.
In order to measure it, the grain is simply rotated to the extinction position in which the vibration
direction of the grain parallels the vibration direction of the microscope polarizer and then doing a
comparative refractive index test with the analyzer out. The vibration direction can be
distinguished from the vibration direction in any grain by having determined the minerals optic
sign, and using an accessory plate to distinguished big from small.
The refractive index of light vibrating in the extraordinary direction (.’), however, varies with
orientation from to . . The true value of . is seen only in grains whose optic axes lie in the
plane to the microscope stage. Such grains can be recognized under crossed-polars because they
exhibit maximum interference colours ( = most number of colour rings) and, if you are lucky, a
uniaxial flash figure. The true value of can be measured in such grains by rotating them to the
extinction position in which the vibration direction of the grain parallels the vibration direction of
the microscope polarizer, and then doing a comparative refractive index test with the analyzer out.
Measurement of Birefrigence in Thin Sections:
The refractive indices of a mineral can not be directly measured in thin sections, although limits can be
placed by doing comparative refractive index tests with respect to the mounting glue (1.54) or adjacent
A good estimate of a mineral’s birefringence can, however, be made by identifying the grain(s) of the
mineral that exhibit the maximum interference colours () in the thin section, and thus have both ηε and
ηώ in the plane of the microscope stage. Such grains should give a centered uniaxial flash figure when
viewed with the Bertrand lens. The exact birefringence () of the mineral can be obtained if the thickness
(d) of the thin section is known:
Conversely, the exact thickness of a thin section can be determined by identifying the maximum
interference colour () of a known mineral.
Other Optical Properties of Uniaxial Minerals
Many minerals are coloured in the microscopes plane polarized light because of the selective
absorption of certain wavelengths of light passing through them. In anisotropic minerals, the different
vibration directions may exhibit differential absorption, such that the colour of a mineral grain
changes upon rotation of the microscope stage in plane polarized light (with the analyzer out). This
property is called pleochroism and is described by determining the true colours of and
vibration directions by successively rotating to the extinction the position in which each is parallel to
the microsope’s polarizer and then identifying the grain’s colour with the analyzer out (ie. proceeding
as if you were actually measuring the refractive indices as on previous page). In some cases,
pleochroism is a reflection of different degrees of total absorption and is not associated with any
Most people determine the vibration direction of their microscope’s polarizer by taking advantage
of the marked pleochroism exhibited by biotite. Although biotite is biaxial, it is almost uniaxial
(remember the phlogopite sheet in lab 3). The ηα vibration direction of biotite is light yellow
brown in colour, while ηγ, and ηβ have a dark reddish brown colour. The ηα vibration direction is
approximately perpendicular to biotite’s perfect cleavage, thus when a biotite crystal is rotated to
the extinction position in which the trace of the cleavage plane parallels the microscope’s polarizer,
the grain will have a dark brown colour in plane polarized light.
Other Optical Properties of Uniaxial Minerals.
Anomalous Interference Colours:
Strongly pleochroic minerals often exhibit anomalous interference colours because of the selective
absorption of different wavelengths of light. These anomalous colours are typically blues and greens
where none should be in the Michel-Levy interference colour chart. Grains that would normally
exhibit low first-order interference colours commonly exhibit their pleochroic colour instead, which
can make the determination of addition and subtraction with the accessory plate difficult. See your lab
notes for hints on how to proceed in such cases.
Sign of Elongation:
Many uniaxial minerals are elongated parallel to their c-axes, which coincide with the
vibration direction and the optic axis. In such cases the sign of elongation can be determine with
the accessory plate:
Positive uniaxial minerals ( >
, quartz) have positive signs of
elongation (+) and are termed
Negative uniaxial minerals ( <
, tourmaline) have negative
signs of elongation (-) and are
termed length fast.
Note: This property can provide a convenient way of determining the optic sign of an
elongated uniaxial mineral if one is having trouble obtaining grains that give an
optic axis interference figure.
An extinction angle is the angle between a linear crystallographic feature
of the mineral and one of its vibration directions, thus the angle between
the linear feature and a cross hair of the microscope at the extinction
position under crossed-polars.
Possible linear features include:
direction of elongation of a crystal
trace of a cleavage plane
trace of a crystal face
trace of a twin plane
There are three types of extinction:
parallel extinction - the linear feature is parallel to one the
microscope’s cross hairs at all extinction positions.
symmetric extinction – the microscope’s crosshairs bisect
symmetrically equivalent linear features at extinction, such as the
intersecting traces of symmetrically equivalent cleavage planes.
inclined extinction – a linear feature is inclined to the
microscope’s cross hairs at the extinction position.
Uniaxial minerals may exhibit parallel or symmetric extinction,
but not inclined extinction.