#DevOptical Part 30: Athermal Doublets

In our [»] previous post, we have seen how to design achromatic thin-doublets from any two-glasses combinations using their Abbe number. If the glasses Abbe numbers are noted V₁ and V₂ respectively, we obtain achromaticity by selecting a [»] power partition factor equals to

This works for any glass combination but obviously crashes when V₁ equals V₂. More specifically, as V₁ approaches V₂, the magnitude of the partition factor gets larger and larger, requiring extremely powerful elements (meaning higher curvature and more higher-order aberrations). Practicality therefore imposes that we select two glasses with very different Abbe numbers.

That being said, this leaves a very large amount of combinations possible. If we restrict ourselves to the Schott catalog, more than 10,000 glasses combinations have |X_p|≤5 !

To reduce the number of candidates, we saw that some glass combinations yield better achromatic performances than others. These are obtained by using at least one abnormal dispersion glass. The downside is that these glasses are usually expensive and don’t perform well in terms of environmental resistance. For instance, N-FK58 ED glass performs very poorly in contact with chemicals such as acids, bases and cleaning agents.

But achromatic performance and chemical resistance are not the only discriminating factor in glass selection. Some combinations also have better temperature stability than others.

Indeed, glasses are not immune to temperature effects: glass blocks grow as temperature increases and their refractive index changes as well. For lenses, this has two effects:

- The overall lens will grow proportional to the temperature through its coefficient of thermal expansion (CTE), α. As the lens grows larger, the curvature of the lens becomes smaller (or, inversely, its radii of curvature becomes larger) resulting in a loss of power proportional to the temperature. The expansion of glass is linear with temperature for small temperature changes, typically through -30°C to +70°C. At larger temperatures, the behavior changes but becomes linear again on a second temperature range. Glass manufacturers therefore define two coefficients of thermal expansion, one for low temperature ranges, and one for higher temperature ranges. We will usually focus only on the former as most optical systems are designed to be used within that range.

- Because the refractive index changes with temperature, the light bending capacity of the lens for a given shape will change as well. This effect can be either positive or negative and is relatively complex to compute. I will detail the mathematical developments necessary to compute the change of refractive index with temperature at the end of this post.

The two effects combine to change the overall power of the lens according to its thermal power, G,

The power of a lens increases proportional to its refractive index change with temperature, and decreases proportional to its coefficient of thermal expansion.

An athermal lens is a lens that does not change power with temperature, or by very little amounts. Indeed, it is not necessary to have perfect athermal behavior as long as the focus change is within the depth of field of the instrument.

Very few lenses have zero athermal power on their own but we do not necessarily need that all lenses in our system are athermal when isolated. What we need is that the whole system be athermal.

For a thin-doublet, the total power of the system is given by

such that its thermal behavior is given by

using the pre-computed thermal powers of each lens, G₁ and G₂.

We could therefore compute a partition ratio for any two glass combinations such that it gives an athermal doublet:

To my knowledge, this is however not common because such lens will have unpredictable chromatic properties. It is more common to combine both athermalism with achromatism, even for monochromatic systems (to decrease their sensitivity to wavelength changes).

For a [»] thin achromatic doublet, we know that

such that

Here, we can only affect the thermal behavior of the system by a careful selection of glasses since it is not possible to solve the partition ratio of a thin-doublet system for both athermalism and achromatism.

For perfect athermalism we need to satisfy the relation

We might however not necessarily want our system to have zero thermal power. As mentioned previously, athermalism needs to be considered as a whole and our lenses are usually mounted in a barrel made of some material M having thermal coefficient of expansion α_M. As temperature increases, our lens will therefore move relative to the image plane leading to an out-of-focus condition.

There are two ways to handle this problem. The first one is to select an athermal doublet and to make the barrel itself athermal by a proper choice of material or material combination. The second is to select a doublet with a thermal power that compensate as best as possible the change in focus due to the barrel.

I will leave the former option aside for this post and focus on the latter since this post is on athermal doublets and not on barrel design.

Imagine our barrel grows with temperature change ∆T by an quantity ∆x. For a telescope system, this means that we need the focal length of our telescope to change its back focal length by the same amount. In a thin-doublet, the back-focal length is equal to the effective focal length such that ∆f=∆x.

The change of power of the thin doublet is equal to

and the change in focal length is therefore

but we also have

such that

To create an athermal telescope, we need to select a glasses combination such that its total thermal power is equal to the inverse of the barrel coefficient of expansion.

For instance, for a barrel made of brass with α_M=18 ppm/°C, we need to select two glasses that meets the condition

The glasses N-FK5 and N-LASF9 have respective Abbe numbers of 70.406 and 32.170 and thermal powers -11.50 and -2.00 leading to an equivalent coefficient of expansion of -19.49 ppm/°C. For a 100 mm telescope with a 10°C temperature shift, the total defocus will be

This defocus will be inside the depth of field, i.e. the total system will be athermal, provided the f-number is at least f/1.2 (computed at 550 nm).

Historically, to select glass combinations that would yield a given thermal power, optical designers would use plots of glasses thermal power versus 1/V. The total thermal power of an achromatic thin doublet being given by the intersection with the origin axis of a line passing by the two glasses. This is illustrated in Figure 1 with a line passing by the glasses N-FK5 and N-LASF9 and interesting the origin at -19.49 ppm/°C.

The method of Figure 1 is still in use today and most softwares like ANSYS OpticStudio provide thermal power charts to assist the optical designer.

That being said, I estimate we can do better in today’s computer age. Indeed, since all glasses combinations yielding a specific total thermal power must be joined by lines intersecting the axis at that specific coordinate, we end up with radial lines centered on the point (0, G_T) where G_T is the targeted thermal power. Any glasses combination that are radial to this point are potential candidates. Also, we already know that the best candidates will be those with a large difference between their Abbe numbers, V.

I therefore propose the updated (V, θ) chart of Figure 2 with θ computed as

This time, we need to identify glasses that line up on a vertical (same angle relative to the (0, G_T) point) and favor those that are the most separated as to keep a large Abbe number difference. For instance, in Figure 2, we see that the couple N-FK5 and N-LASF9 meet our conditions. We also see that N-KF9 and F2 would yield a closer thermal target match but that they are closer along the Abbe number axis so they would yield a doublet with probably more higher-order aberrations. The same conclusions can be drawn for other couples like N-K5 or N-LAK7 with N-KZFS11. An exhaustive search will yield a family a solutions that we could depart based on our requirements.

Note that the method illustrated in Figure 1 and Figure 2 are specific to thin achromatic doublets. For instance, in the case of split doublets the equation becomes

and therefore

or, in terms of thermal powers,

That being said, just like for standard achromatic doublets, we can combine multiple athermal doublets to give an overall athermal system. For instance, if P₁ and P₂ are athermal doublets in the equation above, we get

This, however, works only for purely athermal combinations and would therefore require an athermal barrel too. Tricks can be used on a per-case basis, for instance, in the formula above, if P₁ and P₂ are set, we can adjust the total thermal power by using an athermal doublet for P₁ and a crafted doublet P₂ such that

with α_M the thermal coefficient of expansion of the telescope barrel.

Let’s now come back to the definition of the thermal power of a singlet lens that we introduced at the top of this post:

As we saw, there are mainly two terms driving the value of the material thermal power: refractive index changes with temperature and the glass thermal coefficient of expansion, α.

An infinite number of combinations of refractive index changes with temperature and glass coefficient of thermal expansion can therefore lead to the same thermal power. We might however want to restrict our glass choice based on the absolute value of those terms because two glasses with the same overall thermal power can behave completely differently depending on how those two elements combines.

While there is not much to say about refractive index changes with temperature, having a material with large coefficient of expansion can be problematic. Indeed, as temperature increases the glass will expand. This is not without consequences because a glass that expands too much may crack under the applied thermal load.

Glasses have very similar heat capacity as metals but much worse heat transfer properties. This means that strong thermal gradients may appear in lenses when heated abruptly, resulting in catastrophic failure of the glass block element. This impacts both the usage and the manufacturing process of the glass. Problems can still occur with slower heating but are usually less problematic than abrupt changes. Typically, glass under stress will have their refractive index change and the change not equal depending on the axis relative to the applied stress. This results in rays that behave differently in terms of their initial direction and yield a blurred spot (you can think of this as a form of thermal astigmatism). This goes beyond the context of this post so I will not discuss these aspects further but you need to be aware of their existence.

More specifically, it is not much the coefficient of expansion itself that will characterize glass sensitivity to thermal load but the thermal stress factor, φ_w, which depends on the thermal and mechanical properties of the glass:

with α the coefficient of expansion, E the Young modulus, and µ the Poisson ratio of the glass.

Actual stress on the glass will be proportional to the thermal stress factor, φ_w, the thermal load, ∆T, and an empirical stress factor, f, that depends on the geometry of the glass piece, how the glass piece is held and how fast the temperature changes. Schott recommends using f=1 for temperature shocks, f=0.5-0.7 for moderate temperature changes and f<0.5 for slow temperature changes. The notion of high/low temperature changes is inherently bound to the thermal conductivity of the glass and the dimensions of the glass piece as much as the temperature change itself. Again, this goes beyond this post, and I would recommend to use φ_w×∆T as an upper bound for the thermal stress (that is, assuming the worst case scenario with f=1).

Refractive index will change proportional to this stress and depending on the direction of the ray relative to the stress orientation. As an order of magnitude, N-BK7 glass has a thermal stress factor of 0.481 MPa/°C and refractive index changes of -0.84 ppm/MPa and -3.54 ppm/MPa depending on the ray orientation relative to the stress. This yields a maximum (f=1) difference of refractive index of 1.3 ppm/°C between the two rays. This might not look much but corresponds to a change in optical path of 130 nm for a 10°C change over a 1 cm path. This is a the limit of what will start degrading optical performances in a system designed in the visible range. Note that this is the worst case scenario with f=1 and applications with slower heating will have less optical path change. That being said, this accounts only for the thermal stress and neglects mechanical pre-load that is already applied to secure the lens in terms of shocks and vibrations. Again, this goes beyond the scope of this post so I will not develop it further. More information on thermal stress and stress effects can be obtained [∞] here and [∞] here.

Last but not least, we are now left with the mathematical derivations required to compute the refractive index change with temperature that I discussed earlier. In practice, you don’t need to compute the value yourself but can rely on your optical design software to provide the values for you. I made a [∞] glass selection application that can compute the different thermal charts we have seen here. The derivations can be found in Schott’s [∞] TIE-19 document.

For each glass, manufacturers will measure how the absolute refractive index of the glass changes with temperature and express it as polynomial coefficients of the formula below:

with T₀ the reference temperature (usually 20°C), ∆T the temperature increases relative to T₀, λ the wavelength of the light, D_0-2, E_0-1 and λ_TK constants given by the glass manufacturer.

The refractive index change from T₀ to T₀+∆T is obtained through integration of the previous formula:

and

Until now, the maths are relatively straightforward. The problem comes from the fact that optical software always defines refractive indices relative to air such that it is possible to assume n_air=1 for any temperature, pressure and wavelength.

This is expressed as

with n_rel the relative index of refraction of the glass at wavelength λ, temperature T and pressure P, n_abs the absolute refractive index of the glass at those wavelength, temperature and pressure and n_air the absolute refractive index of air at those values.

The variation of relative refractive index with temperature is therefore

where the subscripts λ, T and P have been omitted for clarity.

The refractive index of air can be obtained with satisfactory accuracy using

and

and

with P₀=0.101325×10⁶ Pa, λ given in µm and T given in °C.

As mentioned, you don’t need to implement those equations yourself as your optical design software will provide these values for you. But as I don’t like blackboxes, it is important for me to provide all the details :)

That is all for today! You should be now experts in designing achromatic athermal doublets!

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