# 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.

• Given the symmetry of the element it seems that one needs a grinding machine that can turn the prism by exactly 120 degrees during the manufacturing process while grinding under a constant angle. That only leaves two variables to control, both of which can be done either by fabricating precise work holders (i.e. the precision alignment is done on the holder, rather than the individual workpiece) or with a CNC machine that has exactly the kind of interferometric control that you are thinking about. – CuriousOne May 22 '16 at 8:55
• The idea of cleaving a crystal to make a precision optical normal sounds rather cute, by the way. :-) – CuriousOne May 22 '16 at 9:05
• @CuriousOne I like that: that's a key insight you notice there. If I understand right, you're saying that we're really dealing with the accuracy of the angles between the projections of the intersections between the faces onto the plane of rotation of a CNC machine, so that our problem is essentially a 2D one rather than the full 3D alignment problem I have been thinking about. I hadn't thought about that before, even though I've wracked my brains over this for a couple of days. – WetSavannaAnimal May 22 '16 at 9:12
• The way I understand optical machining, at the core is a three step process with cutting, grinding and polishing. I believe the grinding step defines the geometry of the element. If the machine can rotate the part precisely and the angle under which it grinds is correct, the depth to which each face is ground only changes the center of the cube, which usually doesn't matter much (?). The grinding angle and the rotation can then probably be fine-tuned in an "outer" loop by comparing the optical performance of the final workpiece to the ideal. That's just my guess. – CuriousOne May 22 '16 at 9:30
• If you really want to know the details, by the way (and you don't want to read a book on optical manufacturing techniques), you can always call an optical workshop which makes custom prisms. Part of the business of these folks is to educate the customer about their manufacturing process. If you ask nicely, chances are that one of the design or production engineers will give you "the full download" (and then some :-)) on how they would manufacture this part and what machine they would use. – CuriousOne May 22 '16 at 9:40

Comparison of fabrication techniques for hollow retroreflectors describes, in great detail, the two obvious methods:

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

2. 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.

• Hi Peter,. I too have done all these things. I tend to achieve minutes of arc, typically 2 minutes is already quite a feat for most optical mounting problems (I may differ from you in that I work in very small optics). Talking with manufacturers I believe that they find the same thing - only they're of course a bit more practised than I am at it. I'd be interested in hearing more about what accuracy you can achieve with the above techniques. Certainly one can do a bit better than these figures hand tweaking with an interferometer, but I'm skeptical that you could turn out pieces for ... – WetSavannaAnimal May 22 '16 at 13:54
• ... a few hundred dollars at the most with these techniques. There's something that makes this problem quite different from other alignment problems: I think that CuriousOne's insight into the problem is highly relevant, i.e. this isn't the whole thorny 3D alignment of two independent surfaces relative to a third but rather a much simpler 2D alignment problem where the projections of the plane intersections onto the rotation plane of a CNC machine need to be accurately set to $120ˆ\circ$ - if you can achieve $120ˆ\circ$, you're there. – WetSavannaAnimal May 22 '16 at 13:58
• The information given is a summary of the papers; the papers describe the actual techniques currently in use for the construction of hollow retroreflectors. – Peter Diehr May 23 '16 at 2:10
• Hi Peter: thanks for the references, most helpful now that I've had time to get to a place where I can download and read them. – WetSavannaAnimal May 23 '16 at 3:06