How refractive index changes with temperature.

How do illumination and sensing optics perform under temperature variation? Simulation software can tell us, but only if we tell the software about two key properties of the optical materials: (i) thermal expansion and (ii) refractive index change with temperature. Expansion is the easier part: We take CTE_l, the linear coefficient of thermal expansion, from the data sheet and scale our optical parts accordingly. Refractive index change is more difficult: For many optical plastics and molding glasses, the datasheets may give us the index at a specified wavelength, and if we’re lucky, the Abbe number. Information about index change with temperature is rare.

The Lorentz-Lorenz equation

At least for optical plastics, Lorentz and Lorenz come to the rescue! I’m talking about Hendrik Lorentz and Ludvig Lorenz, the 19^th century Dutch/Danish physicists. They are famous for many discoveries, but here, we need the “Lorentz-Lorenz equation” which they independently found around 1870. They applied classical electromagnetic theory to a polarizable dielectric material, assuming that it’s the individual molecules which become polarized in an external electric field, and that there is only induced (no permanent) polarization. Then, thermal expansion means fewer molecules per volume, reducing the polarizability of the material. Light travels slower in polarizable materials (how and why exactly is fascinating, but beyond our scope here), and it is the refractive index which tells us how much slower. Putting it all together, Lorentz and Lorenz were able to relate refractive index n to mass density ρ:

where the constant depends on the material, but, crucially, not on wavelength or temperature.

Deriving the refractive index as function of wavelength and temperature

Now, let’s assume we know n for a certain wavelength λ and a certain temperature T₀. The datasheets are mostly silent about T₀, but we may assume 22 °C, a typical room temperature. It’s also the temperature used by Schott [1] for measuring their glass properties. At any rate, the precise value for T₀ is not so important for plastics, because the index change over a few Kelvin is typically less than the precision of the index value in the datasheet.

Then, we can apply the Lorentz-Lorenz equation to derive n at other temperatures.

First, we compute the density ρ as function of temperature from the linear CTE:

Why is there a factor of three? Because we need the volume coefficient of thermal expansion, and that is three times the linear CTE (you can see that by computing the volume of a cube and neglecting higher orders).

Next, we define the relative density,

Our final ingredient is the material constant b:

Some straightforward algebra allows us to express n(λ,T):

This relation for n(T) at λ doesn’t need the absolute density, only n at λ and T₀, as well as the linear CTE, both commonly available from the datasheet.

Application to Zeonex 480R

We take Zeonex 480R data from digitizing Figure 2 in "Liquid Crystals and Polymer-Based Photonic Crystal Fibers" by T.R. Wolinski et al., https://doi.org/10.1080/15421406.2014.917471 . There, index values are reported from 0°C to 80°C and 404.7 nm to 780 nm.

The linear CTE for Zeonex 480R is somewhat difficult to find. ZEON Japan reports CTE = 5.8e-5 for Zeonex K22R, a similar compound, https://www.zeon.co.jp/en/business/enterprise/resin/pdf/200323391.pdf 

but I found no data specific for 480R.

So let's use the K22R value and apply this Lorentz-Lorenz model. Comparing to the measured data, we obtain this plot:

The plot shows the comparison of measured values with the Lorentz-Lorenz model. We choose 25°C as T₀. Model and measurement must coincide at this termperature. For all other values, the CTE (taken from the data sheet for the related K22R compound) is the only degree of freedom in the model.

We see that Lorentz-Lorenz works very well for Zeonex across the whole temperature and wavelength range. The deviations are within the manual plot digitization error.

The Matlab® Live Script to create this plot and its pdf version are available here:

LL_example.zip

Download ZIP • 506KB

.

Other optical plastics

For PMMA and polycarbonate, Cariou et al. (https://doi.org/10.1364/AO.25.000334) have shown excellent validity of Lorentz-Lorenz across a wide temperature range. -100°C to +150°C. In this temperature range, there are certain phase changes (e.g. glass transition), and at those temperatures, the value of a changes suddenly, only to be again constant in the temperature range where there is no phase change.

Beaucage et al. (doi.org/10.1002/polb.1993.090310310) analyzed very thin (300 nm) polystyrene films. It is difficult to measure thickness and thus thermal expansion and possible phase changes of such thin films. Beaucage et al. reversed the logic: They measured the refractive index of the film with an ellipsometer, and deduced transition temperatures from applying Lorentz-Lorenz.

Glasses

For glasses, the situation is different: Lorentz-Lorenz is not readily applicable.

Schott has published a technical note (TIE 19, https://mss-p-009-delivery.stylelabs.cloud/api/public/content/63b62a0d73a640efa8c6fde92f2ad5c2 ) about the "Temperature Coefficient of the Refractive Index". There, it is stated that while the CTE is positive for all optical glasses (with the possible exception of Zerodur, which is more used for large mirrors), the refractive index variation with temperature may be positive or negative, clearly not following Lorentz-Lorenz.

I would speculate that for plastics, the assumption of "polarization happens within the molecule" is valid: Polymer chain molecules are long, with substantial longitudial polarizability, but with comparatively small interaction between electrons in neighboring molecules. On the other hand, for glasses, there are no molecules, but only an amorphous arrangement of single atoms, where the interaction between electrons of adjacent atoms may play a substantial role that varies with composition, explaining the variety of index-vs-temperature relations for glasses.

Unfortunately, Schott publishes data only for their "optical glasses" like N-BK7, but not for their "molding glasses" like B270.

Fortunately, however, both thermal expansion and index change with temperature are much smaller for glasses than for plastics.