This tutorial emphasizes the use of transmitted polarized light microscopy. The reasons for this are practical (it's a relatively cheap and widely available method), historical (it was the earliest method after stereomicroscopy and we've learned most of what we know about petrology this way); and scientific (it's a darn good method). Learning mineral optics, ideally, occupies several weeks of an undergraduates career, at some time early in their program of study. It would not be unusual for such study to be combined with crystallography in a semester-long course. For students who encounter this tutorial prior to this sort of training (and some undoubtedly will), it is recommended that they independently seek out a resource such as Nesse (2004). If you must carry on with the tutorial before studying such a resource, here is a bare-bones tutorial:
Observations is plane-polarized transmitted Light
Okay, so what's the speed of light? Any science student (even some random guy on the street) is likely to spout: 186,000 miles/second! That's right, of course... in a vacuum! But what if light isn't in a vacuum? What if it's in air, or better yet, in a calcite crystal? Well, then a completely different answer applies, and that, leads us to the useful and very beautiful world of transmitted polarized light microscopy. Not only is the speed of light slower inside of solids, it varies, depending on the physical properties of the material. The ratio of the speed of light in a vacuum to the speed of light in matter is called the index of refraction (n=Vv/V)(and indeed, this number can be used to make predictions about the bending of light rays by different materials). Now WHY does the light slow down? Check out any good physics text. From a practical standpoint, you should know that minerals that have a relatively high index of refraction (i.e., slow the light down a lot) will appear to have relatively high "relief" compared to the epoxy and other minerals that surround them. They will appear to your eye to actually stand out, to have more starkly defined edges, etc. Why does this happen? Again, please go read Nesse (2004) or some other resource on optics.
Observations in cross-polarized light
Now in some materials, the speed of light is the same in every direction. The light doesn't notice whether it is going along parallel to, say, the a particular direction in the crystal or some other one. We call such materials isotropic. But in other materials direction does make a difference. Such materials are called anisotropic. If you understand just a bit about crystallography, it's easy to imagine that, to a light ray, the various atoms in the crystal might be arrayed in different patterns, with different proximities, as you traverse the crystal along different paths, creating different opportunities for the interactions that slow the light. In anisotropic materials the light is, in essence, split into two rays, each traveling at different velocities (a slow one and a fast one) and vibrating in specific directions allowed by the crystal (i.e., the crystal itself is a polarizer). The difference in speed between these two rays is called the retardation. Now, if you recall, light has the properties of a wave, with a particular wavelength. Now these two waves, traveling at different velocities as they do, get out of sync-- allowing them to interfere either constructively or destructively.
Now in order to use this phenomenon in a useful way, the polarizing microscope is rigged to supply light waves that vibrate only in a single direction (plane-polarized light). Within the anisotropic crystal this single wave gets twisted around (polarized again) and becomes two (slow and fast, with a certain separation between them). On exiting the crystal, these interfering light waves now have to make it through a second polarizer (called the analyzer) that is set exactly perpendicular to the lower one. In the case of no splitting into separate waves (as in isotropic crystals), well, nothing has changed, so guess what: exactly nothing comes through and the crystal appears BLACK (and by the way-- the same thing will happen if there is no crystal structure at all, as in some opals and glasses). But in an anisotropic crystal, the light is no longer completely parallel to the lower polar, and so, will no longer be cut out completely by the upper one. And the difference (the retardation) between the two waves yields a color, knows as birefringence. The birefringent colors vary in a systematic way with the retardation and the thickness of the crystal traversed. "Interference color charts" show this color variation. Such charts are typically displayed on the walls in petrography labs and you can also find them in optical mineralogy books. Of course, it is possible to intentionally align a crystal (by rotating the stage) so that its transmitted light is cut out by the upper polar-- a position known as extinction.
Now to go a bit further, anisotropic minerals come in two varieties that are related to crystal form: uniaxial (tetragonal and hexagonal crystals) and biaxial (orthorhombic, monoclinic, and triclinic crystals). In uniaxial crystals, there are indeed only two rays (at most, and no splitting up at all when exactly parallel to the c-axis), but in biaxial crystals, things get a bit more complicated with slow, intermediate, and fast rays (though across a single direction of traverse, only two apply at a time). Now imagine your thin section. Mother Nature doesn't usually provide you with crystals in anything but random orientations. Even when she does, will we think to notice and cut a thin section in a way that will reveal this? What this means is that crystals of a single mineral will likely display quite a range of birefringent colors across the thin section, depending on how they are sliced. In one slice, slow and fast rays may come through at their maximum separation and you will see a color representative of the highest possible birefringence for that mineral. A crystal of the same mineral in another orientation may show a color representative of a lesser degree of separation. Suppose you are looking straight down the c-axis of a uniaxial mineral: then there is no separation between slow and fast rays and you see something that, if you didn't know better, might be something isotropic!
On a practical basis, these various optical properties (index of refraction, birefringence, extinction) are used collectively and in combination with other clues (color, crystal form and appearance, general geological knowledge) to identify minerals in thin sections. For carbonate rocks as emphasized in this tutorial, a general knowledge of the range of index (high or low?), birefringence, extinction behaviors (parallel to the cross hairs or not?) for just a few minerals may suffice as you learn to make basic observations.The extent to which you ultimately learn crystallography and crystal optics will serve to enhance your abilities to appreciate the details that can be extracted from the rocks and the ease with which you handle observations of the unusual and the unexpected (those things every scientist hopes to encounter!).