Optical Indicatrix

The optical indicatrix is a geometric figure that encodes both the index of refraction and the vibration direction for light traveling in any direction through a material. It is constructed by plotting the refractive index for each light ray as a radius drawn parallel to that ray’s vibration direction. When this is done for all possible propagation directions, the resulting surface is an ellipsoid - the indicatrix - whose shape is entirely determined by the mineral’s crystal symmetry. ^[1]

Quick Revision

• Isotropic minerals → indicatrix is a sphere; no double refraction; birefringence = 0. ^[1]
• Uniaxial minerals (tetragonal, hexagonal) → indicatrix is an ellipsoid of revolution about the c axis; two principal indices nω and nε; one optic axis along c. ^[1]
• Biaxial minerals (orthorhombic, monoclinic, triclinic) → indicatrix is a triaxial ellipsoid; three principal indices nα < nβ < nγ; two optic axes perpendicular to the two circular sections. ^[1]
• Optic sign (uniaxial): positive if nε > nω; negative if nε < nω. ^[1]
• Optic sign (biaxial): positive if Z bisects the acute angle (2V⊂z < 90°); negative if X bisects the acute angle (2V⊂x < 90°). ^[1]
• Optic orientation is the relationship between indicatrix axes and crystallographic axes; it is symmetry-constrained. ^[1]

How to Use the Indicatrix

The primary practical use of the indicatrix is to find the indices of refraction and vibration directions of the slow and fast rays for a given wave normal direction. The procedure: ^[1]

1. Cut the indicatrix with a plane perpendicular to the wave normal. In the general case this section is an ellipse. ^[1]
2. Read the vibration directions from the ellipse axes - the long axis is parallel to the slow ray vibration direction and its radius equals the slow ray index of refraction; the short axis is parallel to the fast ray vibration direction and its radius equals the fast ray index of refraction. ^[1]
3. Find ray paths by constructing tangents to the indicatrix parallel to the vibration directions. In anisotropic minerals, both ray paths diverge from the wave normal. ^[1]

Isotropic Indicatrix

Isotropic minerals all crystallize in the isometric crystal system. Because light velocity is the same in all directions, only one index of refraction (n) is needed, and the indicatrix is a sphere. Every section through a sphere is a circle, so there is no preferred vibration direction and light is never split into two rays. Birefringence is zero. ^[1]

Uniaxial Indicatrix

Minerals in the tetragonal and hexagonal crystal systems have two distinct unit cell dimensions (a and c) and high rotational symmetry about the c axis. Two indices of refraction are needed to define the indicatrix, which is an ellipsoid of revolution - its axis of revolution is the c crystal axis. The radius measured parallel to the c axis is called nε (the extraordinary index) and the radius at right angles is called nω (the ordinary index). The maximum birefringence is always |nε − nω|. ^[1]

The indicatrix contains three types of sections: ^[1]

• Principal sections: all vertical sections that include the c axis. These are identical ellipses with axes nω and nε. They carry both the maximum and minimum indices. ^[1]
• Random sections: ellipses with axes nω and n′ε, where n′ε is intermediate between nω and nε. ^[1]
• Circular section: the section perpendicular to the c axis. Its radius is nω throughout. Light propagating along the c axis encounters only this circular section, so it is never doubly refracted - the c axis is the single optic axis, explaining why these minerals are called optically uniaxial. ^[1]

Ordinary and Extraordinary Rays

Double refraction in uniaxial minerals always produces one ordinary ray (ω ray) and one extraordinary ray (ε ray). Regardless of propagation direction, one of the two rays produced is always ordinary. ^[1]

The ordinary ray behaves as though it were traveling through an isotropic material: its wave normal and ray path coincide, and its index of refraction nω is the same in all propagation directions. In calcite, nω = 1.658. The ordinary ray vibration vector is always parallel to the (001) plane. ^[1]

The extraordinary ray behaves very differently: its ray path diverges from its wave normal (the image it carries is displaced sideways), and its index of refraction varies with propagation direction between nω and nε. For any direction except perpendicular to the c axis, the extraordinary index is designated n′ε and lies between nω and nε. In calcite, nε = 1.486. For the specific case where the wave normal is perpendicular to a calcite cleavage surface, the extraordinary ray diverges 6.2° from the wave normal and has n′ε = 1.566. Extraordinary rays are labeled ε′ when their index is n′ε and ε when their index equals the true nε. ^[1]

Uniaxial Optic Sign

The dimension of the indicatrix along c may be greater or less than the dimension at right angles, which defines optic sign: ^[1]

• Optically positive (uniaxial+): nε > nω. The indicatrix is a prolate spheroid (elongate, like a rugby ball). Extraordinary rays are slow rays. ^[1]
• Optically negative (uniaxial-): nε < nω. The indicatrix is an oblate spheroid (flattened, like a discus). Extraordinary rays are fast rays. ^[1]

Behavior of Uniaxial Mineral Grains in Three Orientations

The indicatrix predicts the birefringence and interference color for any grain orientation in orthoscopic illumination (the auxiliary condensing lens removed, so light enters nearly parallel): ^[1]

• Optic axis horizontal: The indicatrix section is a principal ellipse with axes nω and nε. Birefringence is maximum = |nε − nω|. Highest interference color for the mineral. ^[1]
• Optic axis vertical: The section through the indicatrix is the circular section (radius nω). No double refraction; birefringence zero. The grain behaves like an isotropic mineral and stays dark between crossed polarizers. In practice, converging light from the condenser means a small amount of color may appear in high-birefringence minerals. ^[1]
• Optic axis at random angle θ: The section is an ellipse with axes nω and n′ε. Birefringence is intermediate. This is the most common grain orientation in a thin section. ^[1]

Biaxial Indicatrix

Minerals in the orthorhombic, monoclinic, and triclinic crystal systems require three unit cell dimensions (a, b, c) and three principal indices of refraction to define the indicatrix. The three principal indices are nα, nβ, and nγ, where nα < nβ < nγ. By convention, nα is plotted along the X axis, nβ along Y, and nγ along Z. The resulting shape is a triaxial ellipsoid, elongate along Z and shortened along X. The maximum birefringence is always nγ − nα. ^[1]

Despite needing three indices to describe the shape, light entering biaxial minerals is still split into only two rays. Both behave as extraordinary rays - wave normal and ray diverge, and their indices vary with direction. They are called the fast ray (n′α, where nα < n′α < nβ) and the slow ray (n′γ, where nβ < n′γ < nγ).

The biaxial indicatrix has three principal sections: XY (ellipse, axes nα and nβ), XZ (ellipse, axes nα and nγ), and YZ (ellipse, axes nβ and nγ). Random sections are ellipses with axes n′α and n′γ. ^[1]

Circular Sections and Optic Axes

The biaxial indicatrix has two circular sections, both with radius nβ. They arise because in the XZ plane, the radii vary between nα and nγ, so there must be radii equal to nβ. These nβ radii in XZ, combined with the nβ radius along the Y axis, define the two circular sections, which intersect along the Y axis. Each circular section is perpendicular to one optic axis - so there are two optic axes, which is why these minerals are called biaxial. Both optic axes lie in the XZ plane. ^[1]

Because both optic axes lie in the XZ plane, that plane is called the optic plane. The angle between the optic axes bisected by X is called 2V⊂x; the angle bisected by Z is 2V⊂z; and 2V⊂x + 2V⊂z = 180°. The Y axis, which is perpendicular to the optic plane, is called the optic normal (ON). ^[1]

Biaxial Optic Sign

The acute angle between the two optic axes is the optic angle (2V). The indicatrix axis that bisects this acute angle is the acute bisectrix (Bxa); the axis that bisects the obtuse angle is the obtuse bisectrix (Bxo). Optic sign depends on which indicatrix axis is the Bxa: ^[1]

• Biaxial positive: Z is the Bxa; 2V⊂z < 90°. ^[1]
• Biaxial negative: X is the Bxa; 2V⊂x < 90°. ^[1]
• Biaxial neutral: 2V = exactly 90°; neither X nor Z is the Bxa. ^[1]

The values nα, nβ, nγ, and 2V⊂z are mathematically interrelated by the equation: cos²2V⊂z = (nβ² − nα²)(nγ² − nβ²) ÷ [(nγ² − nα²)(nβ² − nα²)].

Uniaxial as a Special Case of Biaxial

The uniaxial indicatrix is a limiting case of the biaxial. If nβ is decreased toward nα, the two circular sections converge into one (in the XY plane), the two optic axes converge on the Z axis, and the result is a uniaxial positive indicatrix where nα = nω and nγ = nε. Conversely, if nβ is increased toward nγ, the circular sections converge in the YZ plane, the optic axes converge on the X axis, and the result is a uniaxial negative indicatrix where nα = nε and nγ = nω. ^[1]

Crystallographic Optic Orientation

Optic orientation is the term for the relationship between the indicatrix axes (X, Y, Z) and the crystallographic axes. Because optical properties are controlled by the symmetry of the crystal structure, optic orientation must always be consistent with mineral symmetry. ^[1]

Orthorhombic minerals have three mutually perpendicular crystallographic axes. All three crystal axes must coincide with the three indicatrix axes, and the symmetry planes coincide with principal indicatrix sections. Any crystal axis may coincide with any indicatrix axis, however - the assignment is mineral-specific. In aragonite, X = c, Y = a, Z = b; in anthophyllite, X = a, Y = b, Z = c. ^[1]

Monoclinic minerals have a single 2-fold symmetry axis (or a perpendicular mirror plane), which is always the b crystallographic axis. One indicatrix axis - which may be X, Y, or Z - is always parallel to b. The other two indicatrix axes lie in the {010} plane but are not parallel to either the a or c crystal axis except by coincidence. Optic orientation in monoclinic minerals is reported by stating which indicatrix axis coincides with b, plus the angles from the remaining indicatrix axes to a and c. By convention, an angle is positive if the indicatrix axis lies in the obtuse angle between a and c, negative if it lies in the acute angle. For example, an optic orientation of X^a = -5°, Y = b, Z^c = +15° means the monoclinic β angle = 90° - 5° + 15° = 100°. ^[1]

Triclinic minerals have three crystallographic axes of unequal length, none perpendicular to the others. No symmetry constraint forces any indicatrix axis to coincide with any crystal axis. Optic orientation must therefore be defined by specifying the full angular relationship between each indicatrix axis and the crystal axes, and because triclinic minerals can be described in multiple axis settings, the reported optic orientation may vary between sources depending on which convention for crystal axes is used.

Using the Biaxial Indicatrix in Practice

The biaxial indicatrix is used in exactly the same way as the uniaxial indicatrix: given the wave normal direction of light passing through a mineral, the indicatrix reveals which indices of refraction apply and how the vibration directions are oriented for that path. ^[1]

The key variable that determines what a petrographer sees under the microscope is how the mineral is cut relative to the indicatrix. The three reference cases illustrate the range of behaviour. When the optic normal (the Y axis) is vertical, the section through the indicatrix has axes nα and nγ, so birefringence reaches its maximum value of nγ − nα and the mineral displays its highest interference color. When an optic axis is vertical, the indicatrix section is the circular section with radius nβ, birefringence is zero, and the mineral remains dark between crossed polarizers. All other grain orientations are intermediate and yield birefringence between these extremes. ^[1]

This cut-dependence is why no single grain of a biaxial mineral tells you the mineral’s true maximum birefringence - the grain must happen to be oriented with the optic normal vertical. In practice, a petrographer scans many grains and uses the highest interference color observed as the best estimate of maximum birefringence. Grains oriented with an optic axis vertical are the diagnostic opposite: they show no color at all and identify the optic axes position.

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