Optical Retroreflectors: How Are the Faces So Accurately Righted? This question is about Optical Retroreflectors (corner cubes) and how the extreme precision in their manufacturing is achieved. I suspect there is interesting basic physics involved, which is why the question should fit here. In any case, a discussion of how this accuracy is achieved would be interesting background for many experimenters.
Manufacturers of corner cubes routinely claim deviations from mutual face orthogonality of the order of one arc SECOND. To put this into perspective: if one bounced a laser off a corner cube with a one arc second face deviation attached to the International Space Station, the return beam would land within three meters of one's feet. Or, another perspective: one arc second is less than fifty ${\rm Si 0_2}$ lattice periods over a one centimeter wide face. 
So how would one go about creating faces on glass which are mutually orthogonal to within such an astounding accuracy? A brute force method I can think of which would work would be polishing under closed loop control of an interferometer, but this would be slow and clumsy and I am not sure of the details (i.e. how to set up the correct reference for the interferometer). I suspect this isn't how corner cubes are manufactured, though (simply from the low price of such amazing objects) and so I suspect there is a faster, more natural method grounded on natural lattice cleaving. 

Background Corner cubes work by reflecting an incoming beam from three mutually orthogonal planes. Recall that if $\hat{N}$ is a unit normal vector to a surface, the operator describing reflexion from a plane surface with this normal, and acting on the direction vector of a ray / plane wave is the $3\times 3$ matrix:
$$\mathrm{id} - 2\,\hat{N}\otimes\hat{N} = \mathrm{id} - 2\,\hat{N}\,\hat{N}^T$$
(the expression on the right holding when $\hat{N}$ is written as a column vector, so that a corner cube with faces defined by $\hat{N}_j;\;j=1,\,2,\,3$ is:
$$\prod\limits_{j=1}^3 \left(\mathrm{id} - 2\,\hat{N}_j\otimes\hat{N}_j\right)$$
which works out to $-\mathrm{id}$ if the $\hat{N}_j$ are mutually orthogonal. One can also derive expressions for the total deviation from the above: a deviation from mutual orthogonality of three planes is defined by two latitudinal deviations $\delta_1,\,\delta_2$ and one longitude angle $\phi$, which angles completely define the relative orientations of the three $\hat{N}_j$; the magnitude of the deviation of a reflected ray from an exact half turn (spatial inversion) can be shown from the above expression to be $2\,\sqrt{\delta_1^2+\delta_2^2}$, to first order.
 A:  Comparison of fabrication techniques for hollow retroreflectors describes, in great detail, the two obvious methods:


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*The use of a precision solid prism corner cube as a mandrel to hold the glass plates prior to gluing of the edges. 

*An adjustable set of precision mirror mounts, designed to hold the three plates
In both cases interferometric techniques are used to verify alignment. The art is in the bonding process. The second method is more flexible.
I've used hollow retroreflectors in many projects; they are a crucial element in delay lines, and require careful aligment ... the path needs to be flat, straight, and parallel to the incomig beam. The return beam is usually picked off by the displacement.  Front surface mirrors are the only way to go with ultrafast laser pulses. 
With solid retroreflectors,  the light passes through the block, resulting in chromatic dispersion, and thermal limitations under high power. The manufacturing process is obvious: parallel surfaces are standard in the optical business, and have been since before Newton. The final step is the interferometric test.
Next-generation hollow retroreflectors for lunar laser ranging, by the same author, also goes over the manufacturing process in detail.
