lecture 07 g278 2007 optical indicatrix

8/2/2019 Lecture 07 G278 2007 Optical Indicatrix

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The Optical IndicatrixThe 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


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The Optical IndicatrixThe Optical Indicatrix

A geometric figure that shows the index ofrefraction and vibration direction for light

passing in any direction through a material is

called an optical indicatrix.

The indicatrix is constructed by plotting

indices of refraction as radii parallel to the

vibration direction of the light.

Ray p, propagating along Y, vibrates parallel

to the Z-axis so its index of refraction (np) is

plotted as radii along Z.

Ray q, propagating along X, vibrates parallel

to Y so its index of refraction (nq) is plotted asradii along Y.

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.


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Constructing an Optical IndicatrixConstructing an Optical Indicatrix

The primary use of the indicatrix is to determine the indices ofrefraction and vibration directions of the slow and fast rays given thewave normal direction followed by the light through the mineral.

The basic steps are:1. Construct a section through the indicatrix at right angles to thewave normal. This section is parallel to the wave front. In thegeneral case, the section through the indicatrix is an ellipse.

2. The axes of the elliptical section are parallel to the vibrationdirections of the slow and fast rays and the lengths of radii parallel

to those axes are equal to the indices of refraction.


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Optical Indicatrix with Wave NormalOptical Indicatrix with Wave Normal

Shown is an indicatrix showing a wavenormal direction (WN) along whichthe light propagates.

An elliptical section through theindicatrix perpendicular to the wavenormal is parallel to the wave front.

The long axis of this elliptical sectionis parallel to the slow ray vibrationdirection and the radius parallel tothis direction is equal to the slow rayindex of refraction (nslow).

The short axis of the ellipticalsection is parallel to the fast rayvibration direction and the radiusparallel to this direction is equal tothe fast ray index of refraction(nfast).


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Ray Paths for Optical IndicatrixRay Paths for Optical Indicatrix

To find the ray paths, which arepaths followed by an image throughthe mineral such as those seen withthe calcite rhomb, tangents to theindicatrix are constructed parallel

to the vibration directions of theslow and the fast rays.

In the general case in which theindicatrix is a triaxial ellipsoid, bothrays diverge from their associatedwave normals.


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Isotropic IndicatrixIsotropic Indicatrix

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 describethe 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.


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Uniaxial IndicatrixUniaxial Indicatrix

Minerals that crystallize in the tetragonal andhexagonal crystal systems have two differentunit cell dimensions (a and c) and a high degreeof symmetry about the c axis.

Two indices of refraction are required to definethe dimensions of the indicatrix, which is anellipsoid of revolution whose axis is the ccrystal axis.

The semiaxis of the indicatrix measured parallel

to the c axis is called nI, and the radius at rightangles is called n[. The maximum birefringence

of uniaxial minerals is always [nI

- n[].

All vertical sections through the indicatrix thatinclude the c axis are identical ellipses calledprincipal sections whose axes are n

[and n

I.

Random sections are ellipses whose dimensionsare n

[and n

I where n

I is between n

[and n

I.

The section at right angles to the c axis is acircular section whose radius is n[. Because thissection is a circle, light propagating along the caxis is not doubly refracted as it is following anoptic axis.

Because hexagonal and tetragonal minerals havea single optic axis, they are called opticallyunixial.


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Ordinary and Extraordinary RaysOrdinary and Extraordinary Rays

Recall that if a cleavage rhomb of calcite is placed on a dot or otherimage on a piece of paper, two images appear, each composed of plane-polarised light vibrating at right angles to the other.

The light passing up through the calcite can be considered to beincident at right angles.

Based on Snells Law, the wave normal for this light is not bent, itremains perpendicular to the bottom surface of the rhomb.

When light moves in uniaxial crystals in any direction other than parallelto the c axis, it is broken into two rays travelling with differentvelocities, what we have previously called the fast ray and the slow ray.

One of these rays vibrates in the basal plane whilst the other vibratesat right angles to it and thus in a plane that includes the c axis, ie aprincipal section.

The ray, whose waves vibrate in the basal plane is called the ordinary(I) ray whilst the ray whose waves vibrate in a principal section is calledthe extraordinary ([) ray.

To determine whether the extraordinary ray is the fast or the slow raywe need to know the optic sign of the mineral.


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Ordinary RaysOrdinary Rays

Ordinary rays or I rays have waves vibratingin the basal plane, which is represented by asphere and thus all the waves travel thesame distance in the same time. They thusbehave in an ordinary manner.

Put another way all ordinary rays have thesame velocity and thus the same index ofrefraction.

The ordinary ray vibration vector is alwaysparallel to the (001) plane in uniaxial mineralswhich is the only plane in which electrondensity is uniform.

Regardless of propagation direction one of

the two rays produces as a consequence ofdouble refraction in unaxial minerals isalways an ordinary ray.

If the calcite rhomb is rotated about avertical axis, the position of the ordinary rayimage remains fixed.


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Extraordinary RaysExtraordinary Rays

Extraordinary rays or[

rays have wavesvibrating in a principal section, which isrepresented by an ellipsoid and thus the wavestravel different distances in the same timedepending on the orientation of the incidentbeam.

They are called extraordinary rays because theray path and the wave normal do not coincide.

The wave normal for an extraordinary ray isparallel to the normally incident light strikingthe bottom of the mineral, in this case calcite,and it therefore coincident with the ordinaryray and wave normal.

However, the extraordinary ray path divergesfrom the wave normal

The index of refraction of extraordinary raysvaries with direction between n[ and nI where nImay be either higher or lower than n[.

For any propagation direction exceptperpendicular to the c axis, the index ofrefraction of the extraordinary ray isdesignated nI and is between n[ and nI.


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Optic SignOptic Sign Uniaxial MineralsUniaxial Minerals

The dimensions of the indicatrix along the c axis may be either greateror less than the dimensions at right angles.

We can use this to define the optic sign in uniaxial minerals

1. In optically positive minerals, nI is greater than n[ and thus theextraordinary rays are slow rays.

2. In optically negative minerals, nI is less than n[ and thusextraordinary rays are fast rays.


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Uniaxial IndicatrixUniaxial Indicatrix -- PositivePositive

Thus if the uniaxial mineral isoptically positive it will beshaped like a prolate spheroid


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Uniaxial IndicatrixUniaxial Indicatrix -- NegativeNegative

If the mineral is opticallynegative the indicatrix will beshaped like an oblate spheriod


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Use of the IndicatrixUse of the Indicatrix

We can now examine the behaviour of light passing through grains of auniaxial minerals in different orientations either in a thin section or agrain mount.

Orthoscopic illumination is used (ie auxilliary condenser lens removed)so that the light strikes the bottom surface of the sample more or lessnormal to the surface.

This means that the wave normal of the light entering the mineral is notbent and the wave front is parallel to the bottom surface of themineral.


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In this example the mineral is orientedso 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 incidentnormal to the bottom surface of thegrain.

Because the optic axis is horizontal, thissection is a principal section, which is anellipse whose axes are n[ and nI. Theordinary ray therefore has index ofrefraction n[ and the extraordinary raynI, which is its maximum because themineral is optically positive.

The extraordinary ray vibrates parallelto the trace of the optic axis (c axis)and the ordinary ray vibrates at rightangles.

Therefore, birefringence and henceinterference colors are maximum values.


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In this case the mineral sample isoriented so that the optic axis isvertical.

The section through the indicatrixperpendicular to the wave normal is thecircular section whose radius is n[.

Light coming from below is not doublyrefracted, birefringence is zero andthe light preserves whatever vibrationdirection it initially had.

Between crossed polars this mineralshould behave like an isotropic mineraland remain dark as the stage isrotated.

However, because the light from the

substage condenser is moderatelyconverging, some light may passthrough the mineral.

The mineral may display interferencecolors but they will be the lowest orderfound in that mineral.


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In this sample the mineral sample isoriented in a random orientation sothat the light path is at an angle Utothe optic axis.

The section through the indicatrix

parallel to the bottom surface ofthe mineral is an ellipse whose axesare n[ and RI

The extraordinary ray vibratesparallel to the trace of the opticaxis as seen from above, while theordinary ray vibrates at right angles

Both birefringence and interferencecolors are intermediate because nIis intermediate between n[ and nI.


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Biaxial IndicatrixBiaxial Indicatrix

Minerals that crystallise in the orthorhombic, monoclinic and triclinic crystalsystems require three dimensions (a, b and c) to describe their unit cells andthree indices of refraction to define the shape of their indicatrix.

The three principal indices of refraction are nE, nF and nK where nE < nF < nK The maxmium birefringence of a biaxial mineral is always nKnE 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 mostpropagation 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 slowrays that we are familiar with.

The index of refraction of the fast ray is identified as nE where nE < nE < nFand the index of refraction of the slow ray is nK where nF < nK < nK


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Biaxial IndicatrixBiaxial Indicatrix (1)(1)

The biaxial indicatrix contains threeprincipal sections, the YZ, XY and XZplanes.

The XY section is an ellipse with axes nEand nF, the XZ section is an ellipse withaxes nE and nK and the YZ section is an

ellipse with axes nF and nK. Random sections through the indicatrixare ellipses whose axes are nE and nK.

The indicatrix has two circular sectionswith radius nF that intersect the Y axis.

The XZ plane is an ellipse whose radii vary

between nE and nK. Therefore radii of nFmust be present. Radii shorter than nF are nE and those

that are longer are nK. The radius of the indicatrix along the Y

axis is also nF


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Biaxial IndicatrixBiaxial Indicatrix (2)(2)

Therefore the Y axis and the nF radii in theXZ plane define the two circular sections. Like uniaxial minerals, the circular sections in

biaxial minerals are perpendicular to theoptic axes, hence the term biaxial.

Because both optic axes lie in the XZ plane ofthe indicatrix, that plane is called the opticplane.

The angle between the optic axes bisected bythe X axis is also called the 2Vx angle, whilethe angle between the optic axes bisected bythe Z axis is called the 2Vz angle where 2Vx+ 2Vz = 180.

The Y axis, which is perpendicular to theoptic plane is called the optic normal.


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Optic SignOptic Sign Biaxial MineralsBiaxial Minerals

The acute angle between the optic axes is called the optic angle or 2Vangle.

The axis (either X and Z)that bisects the optic angle is the acutebisectrix 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 opticallynegative and 2Vx is less than 90

2. If the acute bisectrix is the Z axis, the mineral is optically positiveand 2Vz is less than 903. If 2V is exactly 90 so neither X nor Z is the acute bisectrix, the

mineral is optically neutral.


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Optic Sign and Biaxial IndicatrixOptic Sign and Biaxial Indicatrix


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Crystallographic Orientation ofCrystallographic Orientation ofIndicatrix AxesIndicatrix Axes

The term optic orientation refers to the relationship betweenindicatrix axes and crystal axes.

Because the optical properties of minerals are directly controlled bythe symmetry of the crystal structure, optic orientation must beconsistent with mineral symmetry.


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Indicatrix Axes and Orthorhombic CrystalsIndicatrix Axes and Orthorhombic Crystals

Orthorhombic crystals have threemutually perpendicular crystallographicaces of unequal length. These crystalaxes must coincide with the threeindicatrix axes and the symmetryplanes in the mineral must coincide

with principal sections in theindicatrix. Any crystal axis maycoincide with any indicatrix axishowever.

The optic orientation is defined byindicating which indicatrix axis isparallel to which mineral axis.

1. Aragonite X = c, Y = a, Z = b

2. Anthophyllite X = a, Y = b, Z = c


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Indicatrix Axes and Monoclinic CrystalsIndicatrix Axes and Monoclinic Crystals

In monoclinic minerals, the bcrystallographic axis coincides with thesingle 2-fold rotation axis and/or isperpendicular 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 bcrystallographic axis, and the other twolie in the {010} plane and are not parallelto either a or c except by chance.

The optic orientation is defined byspecifying which indicatrix axis coincideswith the b axis and the angles betweenthe other indicatrix axes and the a and ccrystal axes.


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Optical Indicatrix and Triclinic CrystalsOptical Indicatrix and Triclinic Crystals

Triclinic minerals have threecrystallographic axes of differentlengths, none of which is at rightangles.

Because the only possible symmetry

is centre, the indicatrix axes are notconstrained to be parallel to anycrystal axis.

In most cases the optic orientationis specified by indicating theapproximate angle betweenindicatrix and crystal axes.


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Use of the Biaxial IndicatrixUse of the Biaxial Indicatrix

The biaxial indicatrix is used in the same way as the uniaxial indicatrix.

It provides information about the indices of refraction and vibrationdirection given the wave normal direction that light is following througha 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

lecture 07 g278 2007 optical indicatrix

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