The choice of housing materials for athermalizing optics can drastically change the length and weight of the optic. This video uses a previously derived Figure of Merit (FOM) to estimate the length of a passively athermal housing based on housing material selection. A 40% reduction in length of the athermal optic is realized by replacing Aluminum with ALLVAR Alloy 30 in the commonly used Aluminum-Delrin material pair. Interestingly, a reduction to the length of a re-entrant designed is found by reducing the Delrin in the same pair.

Transcript Below

**Introduction**

We’ve got the figure of merit, that is L_{2} [length of inner tube or housing] by f [focal length]. And that’s equal to the coefficient of thermal defocus [δ] minus the coefficient of thermal expansion of the outer housing [α_{1}] divided by the coefficient of thermal expansion of the outer housing [α_{1}] minus the coefficient of thermal expansion of the inner housing [α_{2}] now we can take this equation and rearrange it in such a way where we distribute this denominator α1 minus α2 times delta [δ] minus α1 divided by α1 minus α2. This turns into the equation for a line, where this is the slope and this is a constant that gives you an offset. For a particular material pair 1 and 1 you can fit a particular coefficient of thermal defocus [δ].

**Aluminum + Delrin Athermal Optic Example**

What you wind up seeing is the equation for a line. Now this [graph contains] actually real information from real data, and this would be the aluminum and Delrin system. So α2 or the inner housing is Delrin, and α1 is aluminum, and the slope of this line is equal to 1 divided by α1 minus α2. This term, everything becomes 0, right here – so L2 [Length of outer Delrin housing] over f [focal length] becomes 0 at the point where the coefficient of thermal defocus [δ] is equal to the coefficient of thermal expansion [α] of aluminum.

That’s that condition where the focal length is equal to the length of the outer housing. If you want a coefficient of thermal defocuses [δs] that are lower than the coefficient of thermal expansion [α] for aluminum. Actually, I forgot to do this this is 0, 100, and the units here are ppm per degree Celsius – that would be negative two hundred [ppm°C] negative three hundred [ppm/°C] …zero …one …two …three. What this tells us is if you need a coefficient of thermal defocus [δ] at -100 ppm/°C, you will need your L_{2} [length of standoff or inner housing] to be approximately 1.5x the length of your focal length [f]. Now we’ve got a way to look at a δ [coefficient of thermal defocus] and understand what the length of a system would look like.

**Reducing the size of the athermal optic by using ALLVAR Alloy 30 in place of Aluminum**

So now finally after all that work, we’re able to start to compare the curves or the lines of different material pairs. If you do something very simple like replace the aluminum with ALLVAR, so ALLVAR [Alloy 30] has a coefficient of thermal expansion of -30 ppm/°C. So δ [coefficient of thermal defocus] here is equal to alpha of ALLVAR alloy 30.

The curve starts at that -30 ppm/°C, moves up kind of like that. Again, where this slope is equal to 1 over the coefficient of thermal expansion of the ALLVAR [α_{allvar}] minus the coefficient of thermal expansion of the Delrin [α_{Delrin}]. You’ll also notice, in addition to the decreased slope, you also have a shifting, so you don’t have to use that re-entrant design for values above negative 30 ppm/°C. And if you look at the savings or the reduction, overall, it’s approximately 40%. So, there’s about a 40% reduction in the overall length [L_{overall}] of that total athermal housing when you compare aluminum to ALLVAR for the outer housing while keeping Delrin the same for the inner housing.

### Replacing Delrin with ALLVAR Alloy 30 to create a smaller re-entrant design for athermal optics.

If we take a step back and look at the aluminum + Delrin system, again it’s bounded by the coefficient of thermal expansion of aluminum [α_{aluminum}], and instead of replacing the aluminum with ALLVAR alloy 30 we replace the Delrin with the ALLVAR alloy 30, we see something interesting. First, the coefficient of thermal expansion of ALLVAR [α_{ALLVAR}] at -30 ppm/°C is where this line starts. The slope is a lot steeper than the aluminum + Delrin system, but you do get a region where it is still shorter than the aluminum Delrin system. This is ALLVAR alloy 30 plus aluminum. So, all the way down to about -120 ppm/°C, you still get a length savings of the ALLVAR Alloy 30 + aluminum system compared to the traditional aluminum and Delrin system.

This presents some interesting and strange things, for example, you can machine ALLVAR and aluminum to much tighter tolerances than Delrin because they are alloys and the magnitude of the thermal expansion coefficients [α] are smaller. You can also potentially have some weight and linearity savings.

Now we’ve got something really exciting because of this figure of merit that we developed we can now start to plug in different coefficients of thermal expansion and of different materials and start to really directly compare the results, so we can have a really good understanding of not just if it’s going to save in the length, but how much [total length] it’s going to save [in the optic.]