## Introduction to Athermalizing Optics with ALLVAR Alloy 30

Hello everyone and welcome back to another ALLVAR University video. In this video will be talking about athermalizing optics with ALLVAR Alloy 30. You have some aluminum, some ALLVAR, some more aluminum and a sensor and you want to make sure that the lens is going to focus light on that sensor regardless of temperature. And you can athermalize the system by tuning the housing’s coefficient of thermal expansion by playing with the length of ALLVAR relative to aluminum. But before you get to something like this, you can use some of our newly available lens tubes. This is an SM1 tube, and it mates perfectly with normal commercially off the shelf, aluminum SM1 lens tubes. So, you can breadboard with our ALLVAR alloy 30 and create an athermal system that you can test in the lab.

## Doublet and Housing Characterization

So, to get started, we can use an off the shelf lens that is already well understood and well characterized so that we can athermalize the housing to match what the lens does. And this example we’re going to be talking about a doublet, but you can do this with much more complex systems. So if you have a doublet that has two lens elements, Element 1 and Element 2, you guys already know this, the index of a fraction will change with temperature and the radius of curvature is going to change with temperature due to thermal expansion. So these two contributions to the change in focus, which will here in this case, will push this lens out of focus if the housing doesn’t match the change in focal length. So if you have a focal length from here to here, that’s the focal length of the doublet, and the temperature changes, you’re going to get a change in the focal length of the doublet due to a change in temperature. So we can, all we have to do is match this change in focal length with the same change in length of the housing. So if for example you have Element 3 right here is Housing 3 and Element 4 the Housing. You can choose two different materials, two different materials for SM1 lens tubes for example or for an assembly and if this is the length of Element 3 and this is the length of Element 4. Element 4 will also change, due to change in temperature and Element 3 will also change in length due to a change in temperature and we can actually reference this as the total length of the housing is equal to the focal length.

And we’ll also notice that the length of the housing is going to change also. And we can kind of combine these, we know that there’s some relationships here: the length of the housing is equal to the focal length of the doublet, it’s also equal to L3 plus L4. We also have a relationship between, the change in length of the housing is equal to the change in length of Element 3 plus the change in length of Element 4. So how do we athermalize? Well it’s quite simple. You really just want the change in length of the housing to be the same as the change in the focal length of that doublet.

## Separating out the mechanical and optic contributions

So to solve this, we can break it up into the mechanical portion and the optic portion. If we look at the optic portion, there’s a term called the thermo-optic coefficient Beta, and this is defined as one over the focal length times the instantaneous or the 1st derivative of focal length with temperature. This is approximately equal to one over the focal length times Delta f over Delta T. This is a linear assumption that says, you know, for small differences in temperature or for linear changes in and the focal length with temperature you will observe, or you can calculate the thermo-optic coefficient.

On the mechanical side you have thermal expansion. Thermal expansion is defined as the 1st derivative of strain with temperature. Now this equals one over the original length times the 1st derivative of length with temperature. Notice that this is very very similar, the thermo-optic coefficient and thermal expansion coefficient have very similar forms, the same form actually, and this is approximately equal to one over L Delta L over Delta T. Again, this being the linear assumption. So we can rearrange these to kind of make it a little easier to work with and be able to plug into this equation. So for thermal expansion, the change in length is equal to the thermal expansion coefficient times the original length times Delta T of the lens tube. Thermo-optic coefficient, Delta f, or the change in focal length is equal to the thermo-optic coefficient times the focal length times Delta T.

## Reducing terms needed for athermalizing optics

So now we can plug-in the two elements here. So in this case, if we look at the thermal expansion coefficient and the length of the housing, here we have the thermal expansion coefficient of the housing times the length of the housing times Delta T is equal to the thermo-optic coefficient of the doublet times the focal length of the doublet times Delta T. Now Delta T will cancel because it’s on both sides and remember that the focal length and the lens housing are the same length, at least when everything is in focus, and so at the reference temperature, so we can eliminate these two as well. So as long as the thermal expansion coefficient of the housing matches the thermo-optic coefficient of the doublet, the system will stay in focus, regardless of the temperature.

## Understanding the relationship between the thermal expansion coefficient, the housing lengths, and changes in length of each element

Now, I told you that we have an off shelf optic or an off the shelf lens and we want to athermalize it and we can do that by choosing the length of L3 relative to L4 to tune the thermal expansion coefficient of this air gap right here. Now to do that we need to get this, the housing Delta L, in terms of the change in length of the, of Element 4 and of Element 3. So to do that we take this equation and this relationship right here: the change in length of the housing is equal to the change in length of Element 3 plus the change and length of Element 4. We can expand these using this

thermal expansion relationship. And so the thermal expansion coefficient of the housing times the length of the housing times Delta T is equal to the thermal expansion coefficient of Element 3 times the length of Element 3 times Delta T plus the thermal expansion coefficient Element 4 times L4 times Delta T. Just like in this equation, changes in temperatures cancel, and we’re left with a relationship between the thermal expansion coefficient, the lengths, and the relative lengths of the two Elements 3 and 4. We can rearrange this so that

## Rearranging terms to determine element 4’s length (ALLVAR Alloy 30) needed to athermalize the system

We can rearrange this so that we get this equation in terms of the thermal expansion coefficient of the housing alone. So thermal expansion coefficient of the housing is equal to thermal expansion coefficient of Element 3 times L3 over LH plus the thermal expansion coefficient of Element 4 times L4 over LH. Now, this isn’t exactly useful for us because we want to be able to eliminate either L3 or L4 and get this term in the expression of only one of them because we know that there’s a relationship between LH, L3 and L4. To do that we rearrange this and say L3 is equal to LH minus L4, and we plug that into this equation. Alpha H is equal to thermal expansion coefficient Element 3 times LH minus L4 divided by LH plus thermal expansion coefficient Element 4 times L4 over LH. We can expand this and distribute the thermal expansion coefficient and what we have is the housing thermal expansion is equal to the thermal expansion coefficient Element 3 minus the thermal expansion coefficient of Element 4 times L4 over LH plus the thermal expansion coefficient Element 4 times L4 over LH. Now we can take this and plug it in to this half of the equation. So, in this case, Beta d, the thermo-optic coefficient of the doublet is equal to everything on this side of the equation. L4 over LH times Alpha 4 and then L4 over LH minus Alpha 3 and then Alpha 3. So now we can take this equation, and we can rearrange it because what we really want to know is how long does Element 4 need to be athermalize the system if Element 4 and Element 3 have different thermal expansion coefficients? So, this is known, the length of the housing is known, the thermal expansion coefficients of the elements are known Elements 3 and 4 and so we’re solving for the length of Element 4.

So in this equation, we can bring together the LH over L4, so we have thermal expansion of Element 3 minus L4 over LH, make sure it’s negative (PAUSE) we’ll do that, and times the thermal expansion coefficient of Element 4 minus the thermal expansion coefficient of Element 3 is equal to the thermo-optic coefficient of the doublet. Then we can rearrange this, we want to get everything in terms of Element 4. The length of Element 4 is equal to the thermo-optic coefficient of the doublet minus the thermal expansion coefficient of Element 3 divided by the thermal expansion coefficient of Element 4 minus the thermal expansion coefficient Element 3, times the length of the housing. And remember the length of the housing is equal to the focal length of the doublet so that can be substituted directly.

## What if I need to athermalize something other than a doublet?

Now, the beauty of this equation is this thermo-optic coefficient doesn’t necessarily need to be the thermo-optic coefficient of a doublet. It can be the thermo-optic coefficient of any complex lens

system, and if you have a single air gap that you’re trying athermalize, for example between lens and a sensor, you can use this equation to determine what the length of let’s say the ALLVAR piece relative to aluminum piece should be, so you can get the thermo-optic coefficient from a software program and plug it into this equation and you’ll get a very good approximation for how to athermalize this system.