1. Uniaxial Optics

2. Anisotropic crystals Calcite experiment and double refraction
O E
O-ray (Ordinary)
Obeys Snell's Law and goes straight
Vibrates perpendicular to the plane containing ray and c-axis
E-ray (Extraordinary)
deflected and therefore does not obey Snell's Law. It vibrates in plane containing ray and c-axis
Fig 6-7 Bloss, Optical Crystallography, MSA

3. Anisotropic crystals
IMPORTANT: A given ray of incoming light is restricted to only 2 (mutually perpendicular) vibration directions once it enters an anisotropic crystal
Called privileged directions
Each ray has a different n
w = no
e = nE
w < e (in the case of calcite)
O E

4. n > 1 for all anisotropic substances
Indicatrix no longer a sphere
Indicatrix = ellipsoid of revolution
Hexagonal and tetragonal crystals have one unique crystal axis (c axis) perpendicular to two axes of equal length.
The optical properties reflect this as well: optical properties parallel to c differ from those parallel to the ab-plane

5. For light travelling parallel to c, all vibration directions
Fig 6-10 Bloss, Optical Crystallography, MSA
For light travelling parallel to c, all vibration directions
perpendicular to c are the same:
circular section of indicatrix (perpendicular to the optic axis = c -axis); light parallel the optic axis vibrates in the circular section = behaves as isotropic, optic axis = axis of isotropy
only one ray (O-ray) with n = w (doesn’t split into two rays)
extinct with analyzer in and stays that way as rotate stage
(unpolarized light will remain so)

6. For light travelling perpendi-cular to c: principal section of indicatrix:
get 2 rays
O-ray with n = w
E-ray with n = e
this e (parallel c) is the maximum possible deviation in n from w (true e)
Fig. 6-12
For random vibration direction  same situation as above
Except that E-ray has some n between e and w
All intermediate values are called e’ (a variable value between e and w)

7. The types of intersection between
the faces of a crystal and its indicatrix:
The heavy radii of the ellipse indicate the crystal’s privileged directions for light entering by normal incidence on this face.
Minerals with one circular section
have only one optic axis = c-axis
= uniaxial minerals

8. ellipsoid and conventions:
(+) crystal = prolate e > w = uniaxial positive
(-) crystal = oblate e < w = uniaxial negative
(-) crystal:
w > e
 oblate
(+) crystal:
e > w
 prolate
Fig 6-11 Bloss, Optical Crystallography, MSA

9. Essentially 2 possibilities
O E
1. Circular section
look down OA
like isotropic
only one ray
n = w
extinct in XPL
stays so as rotate
Fig 6-7 Bloss, Optical Crystallography, MSA
2. Elliptical section
any orientation other than circular
2 rays
only 2 vibration directions
O ray with n = w
E ray with n = e’ or e
does not stay same as rotate

10. Consider rotating the crystal as you watch:
B = polarizer vibration direction
parallel to e  only E-ray
Analyzer in  extinct
C = polarizer vibration direction
parallel to w  only O-ray
also  extinct with analyzer e w
polarizer w e

11. Consider rotating the crystal as you watch: D
Polarized light has a component of each
Splits  two rays
one is O with n = w
other is E with n = e
When the rays exit the crystal they recombine
No extinction with crossed polarizers w e
polarizer

12. Transmission by the Analyzer
A function of:
a) Angle between analyzer and polarizer (fixed at 90o)
b) Angle between polarizer and closest privileged direction of crystals
When polarizer parallel to either one of the privileged vibration direction ® extinct, since only one ray
Every crystal goes extinct 4 times in 360o rotation (unless )
c) D = path difference = t (N-n),
where N-n is the difference between the greater
refractive index N and the smaller refractive
index n and t is the thickness of the crystal.

13. Interference Colour Chart (properly call a Michel-Levy Chart)

14. Example: Quartz w = 1.544 e = 1.553 w e 1.544 1.553
Data from Deer et al
Rock Forming Minerals John Wiley & Sons

15. Example: Quartz w = 1.544 e = 1.553 Sign?? (+) because e > w
e - w = and is called the birefringence (d) = maximum interference color
What color is this?? Use your chart.
1) Follow line in toward origin
2) Where it crosses 30 micron thickness (the standard for thin sections) we get a yellowish tan (see when quartz oriented with OA in plane of stage)
For other orientations get e' - w ® progressively lower color
Rotation of the stage changes the intensity, but not the hue
Extinct when privileged direction N-S (every 90o) and maximum brightness at 45o; 360o rotation ® 4 extinction positions exactly 90o apart

16. Estimating birefringence
1) Find the crystal of interest showing the highest order colors (D depends on orientation)
2) Count in isochromes from edge, if necessary, to determine maximum color and order - accessory plate may help too
3) Go to color chart
thickness = 30 microns
use 30 micron line + color, follow radial line through intersection to margin & read birefringence
Suppose you have a mineral with second-order green
What about third order yellow?

17. So far, all of this has been orthoscopic (the normal way)
All light rays are ~ parallel and vertical as they pass through the crystal
xtl has particular interference color = f (birefringence, thickness, orientation)
Points of equal thickness will have the same color
isochromes = lines connecting points of equal interference color
At thinner spots and toward edges will show a lower color
Count isochromes (inward from thin edge) to determine order
Orthoscopic viewing
Fig 7-11 Bloss, Optical Crystallography, MSA

18. By changing the light path, we can get conoscopic illumination
By changing the light path, we can get conoscopic illumination. Using conoscopic illumination will allow us to obtain interference figures.
Rotate the sub-stage condensor lens into position.
Use the highest magnification objective lens
Observe the interference figure by rotating the Bertrand lens into position or removing the ocular (eye piece). The interference figure is focused on the objective lens in the microscope tube.

19. In order to obtain an interference figure from a uniaxial minerals, one needs to observe a grain parallel to its optic axis. In this orientation, the grain will be extinct under crossed polars on rotation of the stage.
Before changing to conoscopic illlumination, it is necessary to focus on the grain using the highest magnification objective.

20. Uniaxial Optic Axis Figure
Circles of isochromes
Note vibration directions:
w tangential
e' radial & variable magnitude
Black cross (isogyres) results from locus of extinction directions
Center of cross (melatope) represents optic axis
Approx 30o inclination of OA will put it at margin of field of view
Fig. 7-14

21. Uniaxial Figure Centered axis when rotate stage cross does not rotate
Off center: cross still E-W and N-S, but melatope rotates around center
Melatope outside field: bars sweep through, but always N-S or E-W at center
Flash Figure: OA in plane of stage Diffuse black fills field brief time as rotate
Fig. 7-14

22. Accessory Plates We use an insertable 1st-order red (gypsum) plate
Fig 8-1 Bloss, Optical Crystallography, MSA
We use an insertable 1st-order red (gypsum) plate
Slow direction is marked N on plate
Fast direction (n) || axis of plate
The gypsum crystal is oriented and cut so that D = (N-n) ® 550nm retardation
 thus it has the effect of retarding the N ray 550 nm behind the n ray
If insert with no crystal on the stage ® 1st-order red in whole field of view

23. Accessory Plates N
Suppose we view an anisotropic crystal with D = 100 nm (1-order gray) at 45o from extinction n
If Ngyp || Nxl ® Addition
Addition since ray in xl || Ngyp
already behind by 100nm & it gets further retarded by 550nm in the gypsum plate
® 650nm
On your color chart what will result?
Original 1o grey ® 2o blue N

24. Accessory Plates N n
Now rotate the microscope stage and crystal 90o ® Ngyp || nxl (D still = 100 nm)
Ngyp || nxl ® Subtraction
Now the ray in the crystal that is parallel to Ngyp is ahead by 100mm
550mm retardation in gypsum plate ® 450nm behind
On your color chart what will result?
1o orange N

25. Optic Sign Determination
For all xls remember e' vibrates in plane of ray and OA, w vibr normal to plane of ray and OA
O E e ' w
1) Find a crystal in which the optic axis (OA) is vertical (normal to the stage) 2) Go to high power, insert condensing and Bertrand lenses to ® optic axis interference figure
(+) crystals:
e’ > w
so w faster

26. Optic Sign Determination
Inserting plate for a (+) crystal:
® subtraction in NW & SE where n||N
® addition in NE & SW where N||N
Whole NE (& SW) quads add 550nm
isochromes shift up 1 order
Isogyre adds ® red
In NW & SE where subtract
Each isochrome loses an order
Near isogyre (~100nm)
get yellow in NW & SE
and blue in NE & SW e ' w
sub
add
add
sub
(+) crystals:
e’ > w
so w faster N

27. (+) OA Figure with plate
(+) OA Figure without plate
(+) OA Figure with plate
Yellow in NW is (+)

28. Optic Sign Determination
Inserting plate for a (-) crystal:
® subtraction in NE & SW where n||N
® addition in NW & SE where N||N
Whole NW (& SE) quads add 550nm
isochromes shift up 1 order
Isogyre still adds ® red
In NE & SW where subtract
Each isochrome loses an order
Near isogyre (~100nm)
get 650 blue in NW & SE
and 450 yellow in NE & SW e ' w
add
sub
sub
add
(-) crystals:
e’ < w
so w slower N

29. (-) OA Figure without plate (same as (+) figure)
(-) OA Figure with plate
Blue in NW is (-)

30. Sign of Elongation 45 degree position of crystals!!!!
If N || elongation will always add ® length slow
If n || elongation will always subtract ® length fast N n
ω/ε
ω/ε
ω/ε
ε /ω
ε /ω
ε /ω N
Independent if uniaxial(+) or (-),
Elongation is positive
Independent if uniaxial(+) or (-),
Elongation is negative