Download Optical Indicatrix

The Optical Indicatrix
IN THIS LECTURE
– Optical Indicatrix
– Isotropic Indicatrix
– Uniaxial Indicatrix
• Ordinary and Extraordinary Rays
• Optic sign
• Use of the Indicatrix
– Biaxial Indicatrix
• Optic Sign
• Crystallographic Orientation of Indicatrix Axes
• Use of the Indicatrix
The Optical Indicatrix
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A geometric figure that shows the index of
refraction and vibration direction for light
passing in any direction through a material is
called an optical indicatrix.
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The indicatrix is constructed by plotting
indices of refraction as radii parallel to the
vibration direction of the light.
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Ray p, propagating along Y, vibrates parallel
to the Z-axis so its index of refraction (np) is
plotted as radii along Z.
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Ray q, propagating along X, vibrates parallel
to Y so its index of refraction (nq) is plotted as
radii along Y.
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If the indices of refraction for all possible light
rays are plotted in a similar way, the surface
of the indicatrix is defined. The shape of the
indicatrix depends on mineral symmetry.
Constructing an Optical Indicatrix
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The primary use of the indicatrix is to determine the indices of
refraction and vibration directions of the slow and fast rays given the
wave normal direction followed by the light through the mineral.
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The basic steps are:
1. Construct a section through the indicatrix at right angles to the
wave normal. This section is parallel to the wave front. In the
general case, the section through the indicatrix is an ellipse.
2. The axes of the elliptical section are parallel to the vibration
directions of the slow and fast rays and the lengths of radii parallel
to those axes are equal to the indices of refraction.
Optical Indicatrix with Wave Normal
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Shown is an indicatrix showing a wave
normal direction (WN) along which
the light propagates.
An elliptical section through the
indicatrix perpendicular to the wave
normal is parallel to the wave front.
The long axis of this elliptical section
is parallel to the slow ray vibration
direction and the radius parallel to
this direction is equal to the slow ray
index of refraction (nslow).
The short axis of the elliptical
section is parallel to the fast ray
vibration direction and the radius
parallel to this direction is equal to
the fast ray index of refraction
(nfast).
Ray Paths for Optical Indicatrix
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To find the ray paths, which are
paths followed by an image through
the mineral such as those seen with
the calcite rhomb, tangents to the
indicatrix are constructed parallel
to the vibration directions of the
slow and the fast rays.
In the general case in which the
indicatrix is a triaxial ellipsoid, both
rays diverge from their associated
wave normals.
Isotropic Indicatrix
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Optically isotropic minerals all
crystallise in the isometric crystal
system.
One unit cell dimension (a) is required
to describe the unit cell and one index
of refraction (n) is required to describe
the optical properties because light
velocity is uniform in all directions for a
particular wavelength of light.
The indicatrix is therefore a sphere.
All sections through the indicatrix are
circles and the light is not split into two
rays.
Birefringence may be considered to be
zero.
Uniaxial Indicatrix
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Minerals that crystallize in the tetragonal and
hexagonal crystal systems have two different
unit cell dimensions (a and c) and a high degree
of symmetry about the c axis.
Two indices of refraction are required to define
the dimensions of the indicatrix, which is an
ellipsoid of revolution whose axis is the c
crystal axis.
The semiaxis of the indicatrix measured parallel
to the c axis is called ne, and the radius at right
angles is called nw. The maximum birefringence
of uniaxial minerals is always [ne - nw].
All vertical sections through the indicatrix that
include the c axis are identical ellipses called
principal sections whose axes are nw and ne.
Random sections are ellipses whose dimensions
are nw and ne’ where ne’ is between nw and ne.
The section at right angles to the c axis is a
circular section whose radius is nw. Because this
section is a circle, light propagating along the c
axis is not doubly refracted as it is following an
optic axis.
Because hexagonal and tetragonal minerals have
a single optic axis, they are called optically
unixial.
Ordinary and Extraordinary Rays
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Recall that if a cleavage rhomb of calcite is placed on a dot or other
image on a piece of paper, two images appear, each composed of planepolarised light vibrating at right angles to the other.
The light passing up through the calcite can be considered to be
incident at right angles.
Based on Snell’s Law, the wave normal for this light is not bent, it
remains perpendicular to the bottom surface of the rhomb.
When light moves in uniaxial crystals in any direction other than parallel
to the c axis, it is broken into two rays travelling with different
velocities, what we have previously called the fast ray and the slow ray.
One of these rays vibrates in the basal plane whilst the other vibrates
at right angles to it and thus in a plane that includes the c axis, ie a
principal section.
The ray, whose waves vibrate in the basal plane is called the ordinary
(e) ray whilst the ray whose waves vibrate in a principal section is called
the extraordinary (w) ray.
To determine whether the extraordinary ray is the fast or the slow ray
we need to know the optic sign of the mineral.
Ordinary Rays
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Ordinary rays or e rays have waves vibrating
in the basal plane, which is represented by a
sphere and thus all the waves travel the
same distance in the same time. They thus
behave in an ordinary manner.
Put another way all ordinary rays have the
same velocity and thus the same index of
refraction.
The ordinary ray vibration vector is always
parallel to the (001) plane in uniaxial minerals
which is the only plane in which electron
density is uniform.
Regardless of propagation direction one of
the two rays produces as a consequence of
double refraction in unaxial minerals is
always an ordinary ray.
If the calcite rhomb is rotated about a
vertical axis, the position of the ordinary ray
image remains fixed.
Extraordinary Rays
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Extraordinary rays or w rays have waves
vibrating in a principal section, which is
represented by an ellipsoid and thus the waves
travel different distances in the same time
depending on the orientation of the incident
beam.
They are called extraordinary rays because the
ray path and the wave normal do not coincide.
The wave normal for an extraordinary ray is
parallel to the normally incident light striking
the bottom of the mineral, in this case calcite,
and it therefore coincident with the ordinary
ray and wave normal.
However, the extraordinary ray path diverges
from the wave normal
The index of refraction of extraordinary rays
varies with direction between nw and ne where ne
may be either higher or lower than nw.
For any propagation direction except
perpendicular to the c axis, the index of
refraction of the extraordinary ray is
designated ne’ and is between nw and ne.
Optic Sign – Uniaxial Minerals
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The dimensions of the indicatrix along the c axis may be either greater
or less than the dimensions at right angles.
We can use this to define the optic sign in uniaxial minerals
1. In optically positive minerals, ne is greater than nw and thus the
extraordinary rays are slow rays.
2. In optically negative minerals, ne is less than nw and thus
extraordinary rays are fast rays.
Uniaxial Indicatrix - Positive
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Thus if the uniaxial mineral is
optically positive it will be
shaped like a prolate spheroid
Uniaxial Indicatrix - Negative
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If the mineral is optically
negative the indicatrix will be
shaped like an oblate spheriod
Use of the Indicatrix
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We can now examine the behaviour of light passing through grains of a
uniaxial minerals in different orientations either in a thin section or a
grain mount.
Orthoscopic illumination is used (ie auxilliary condenser lens removed)
so that the light strikes the bottom surface of the sample more or less
normal to the surface.
This means that the wave normal of the light entering the mineral is not
bent and the wave front is parallel to the bottom surface of the
mineral.
Use of the Indicatrix
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In this example the mineral is oriented
so that its optic axis is horizontal.
We will assume that the mineral is
uniaxial positive
The wave normal is through the centre
of the indicatrix and light is incident
normal to the bottom surface of the
grain.
Because the optic axis is horizontal, this
section is a principal section, which is an
ellipse whose axes are nw and ne. The
ordinary ray therefore has index of
refraction nw and the extraordinary ray
ne, which is its maximum because the
mineral is optically positive.
The extraordinary ray vibrates parallel
to the trace of the optic axis (c axis)
and the ordinary ray vibrates at right
angles.
Therefore, birefringence and hence
interference colors are maximum values.
Use of the Indicatrix
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In this case the mineral sample is
oriented so that the optic axis is
vertical.
The section through the indicatrix
perpendicular to the wave normal is the
circular section whose radius is nw.
Light coming from below is not doubly
refracted, birefringence is zero and
the light preserves whatever vibration
direction it initially had.
Between crossed polars this mineral
should behave like an isotropic mineral
and remain dark as the stage is
rotated.
However, because the light from the
substage condenser is moderately
converging, some light may pass
through the mineral.
The mineral may display interference
colors but they will be the lowest order
found in that mineral.
Use of the Indicatrix
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In this sample the mineral sample is
oriented in a random orientation so
that the light path is at an angle q to
the optic axis.
The section through the indicatrix
parallel to the bottom surface of
the mineral is an ellipse whose axes
are nw and e
The extraordinary ray vibrates
parallel to the trace of the optic
axis as seen from above, while the
ordinary ray vibrates at right angles
Both birefringence and interference
colors are intermediate because ne’
is intermediate between nw and ne.
Biaxial Indicatrix
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Minerals that crystallise in the orthorhombic, monoclinic and triclinic crystal
systems require three dimensions (a, b and c) to describe their unit cells and
three indices of refraction to define the shape of their indicatrix.
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The three principal indices of refraction are na, nb and ng where na < nb < ng
The maxmium birefringence of a biaxial mineral is always ng - na
Construction of a biaxial indicatrix requires that three indices of refraction are
plotted
However, while three indices of refraction are required to describe biaxial
optics, light that enters biaxial minerals is still split into two rays.
As we shall see, both of these rays behave as extraordinary rays for most
propagation paths through the mineral.
The wave normal and ray diverge like the extraordinary ray in uniaxial minerals
and their indices of refraction vary with direction, the same of the fast and slow
rays that we are familiar with.
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The index of refraction of the fast ray is identified as na’ where na < na’ < nb
and the index of refraction of the slow ray is ng’ where nb < ng’ < ng
Biaxial Indicatrix – (1)
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The biaxial indicatrix contains three
principal sections, the YZ, XY and XZ
planes.
The XY section is an ellipse with axes na
and nb, the XZ section is an ellipse with
axes na and ng and the YZ section is an
ellipse with axes nb and ng.
Random sections through the indicatrix
are ellipses whose axes are na’ and ng’.
The indicatrix has two circular sections
with radius nb that intersect the Y axis.
The XZ plane is an ellipse whose radii vary
between na and ng. Therefore radii of nb
must be present.
Radii shorter than nb are na’ and those
that are longer are ng’.
The radius of the indicatrix along the Y
axis is also nb
Biaxial Indicatrix – (2)
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Therefore the Y axis and the nb radii in the
XZ plane define the two circular sections.
Like uniaxial minerals, the circular sections in
biaxial minerals are perpendicular to the
optic axes, hence the term biaxial.
Because both optic axes lie in the XZ plane of
the indicatrix, that plane is called the optic
plane.
The angle between the optic axes bisected by
the X axis is also called the 2Vx angle, while
the angle between the optic axes bisected by
the Z axis is called the 2Vz angle where 2Vx
+ 2Vz = 180°.
The Y axis, which is perpendicular to the
optic plane is called the optic normal.
Optic Sign – Biaxial Minerals
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The acute angle between the optic axes is called the optic angle or 2V
angle.
The axis (either X and Z)that bisects the optic angle is the acute
bisectrix or Bxa.
The axis (either Z or X) that bisects the obtuse angle between the
optic axes is the obtuse bisectrix or Bxo.
The optic sign of biaxial minerals depends on whether the Z or Z
indicatrix axis bisects the acute angle between the optic axes.
1. If the acute bisectrix is the X axis, the mineral is optically
negative and 2Vx is less than 90°
2. If the acute bisectrix is the Z axis, the mineral is optically positive
and 2Vz is less than 90°
3. If 2V is exactly 90° so neither X nor Z is the acute bisectrix, the
mineral is optically neutral.
Optic Sign and Biaxial Indicatrix
Crystallographic Orientation of
Indicatrix Axes
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The term optic orientation refers to the relationship between
indicatrix axes and crystal axes.
Because the optical properties of minerals are directly controlled by
the symmetry of the crystal structure, optic orientation must be
consistent with mineral symmetry.
Indicatrix Axes and Orthorhombic Crystals
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Orthorhombic crystals have three
mutually perpendicular crystallographic
aces of unequal length. These crystal
axes must coincide with the three
indicatrix axes and the symmetry
planes in the mineral must coincide
with principal sections in the
indicatrix. Any crystal axis may
coincide with any indicatrix axis
however.
The optic orientation is defined by
indicating which indicatrix axis is
parallel to which mineral axis.
1. Aragonite X = c, Y = a, Z = b
2. Anthophyllite X = a, Y = b, Z = c
Indicatrix Axes and Monoclinic Crystals
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In monoclinic minerals, the b
crystallographic axis coincides with the
single 2-fold rotation axis and/or is
perpendicular to the single mirror plane.
The a axes and c axes are perpendicular
to b and intersect in an obtuse angle.
One indicatrix axis, which could either be
X, Y or Z, is always parallel to the b
crystallographic axis, and the other two
lie in the {010} plane and are not parallel
to either a or c except by chance.
The optic orientation is defined by
specifying which indicatrix axis coincides
with the b axis and the angles between
the other indicatrix axes and the a and c
crystal axes.
Optical Indicatrix and Triclinic Crystals
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Triclinic minerals have three
crystallographic axes of different
lengths, none of which is at right
angles.
Because the only possible symmetry
is centre, the indicatrix axes are not
constrained to be parallel to any
crystal axis.
In most cases the optic orientation
is specified by indicating the
approximate angle between
indicatrix and crystal axes.
Use of the Biaxial Indicatrix
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The biaxial indicatrix is used in the same way as the uniaxial indicatrix.
It provides information about the indices of refraction and vibration
direction given the wave normal direction that light is following through
a mineral.
Birefringence depends on how the sample is cut.
Birefringence is:
– A maximum if the optic normal is vertical
– A minimum if an optic axis is vertical
– Intermediate for random orientations