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# How to Model Thermal Gradients in an Interferometer Cavity

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**Introduction**

*Problem: How can we model a linear thermal temperature gradient of an interferometer cavity in Zemax? *

This question was posed by an interferometer expert in one of the online optics fourms. He wanted an analytical or modeling solution to this problem. I thought that I could model this in Zemax and took a quick look at the gradient index surface types and incorrectly decided it could not be done. So I came up with another way using non-sequential ray-tracing.

I would make rectangular volume objects and lay them all on top of each other to create a linear thermal gradient in the interferometer cavity. I mentioned this method, and a fellow Zemax user (Dave Schaack) mentioned that I could use a Gradient4 type surface in sequential Zemax to model this problem. I looked up Gradient4 in the manual and sure enough it had the capability, I just missed it when I read the manual too fast before – message: read the manual carefully before you make assumptions!

The key to putting in a vertical thermal gradient into the airspace between the point source and the mirror is to use the Gradient4 type surface. This type of gradient index enables one to create a linear or quadratic type index variation along one, two, or three axis. This can be understood by looking at the refractive index equation for the Gradient4 surface:

n(x, y, z) = n0 + nx1*x + nx2*x2 + ny1*y + ny2*y2 + nz1*z + nz2*z2

By using only the terms in y the required gradient can be easily produced.

Modeling the Interferometer Cavity

The interferometer cavity can be easily modeled like so:

One can see from the lens data editor the system setup consists of a point source at the focus of the transmission sphere of the interferometer, this point in the system is the OBJect surface in Zemax. I set the aperture on the General tab using the object space cone angle choice and gave it a 5 degree half angle cone of light from the point source to fill the mirror under test:

You can of course change this half angle to match your transmission sphere f/# or your mirror diameter. Next I have the Gradient4 type surface and this is the 2 meter or 2000 mm airspace between the point source and the mirror under test, so the airspace is 2000mm thick as shown above. The mirror semi-diameter (radius of the substrate) is 175mm so I set the airspace to be that same diameter as well.

Then I put in a standard surface spherical mirror with a radius of curvature of -2000mm or concave with a glass type of mirror of course so it will reflect the rays back to the point source. Finally I have a second Gradient4 material -2000 mm thick which is the return path back to the point source of the interferometer transmission sphere focal point. It is locked to the first pass using pickup solves. Finally we have the image plane surface which should be coincident back to the point source. This is the typical optical cavity setup for a Fizeau type phase measuring interferometer with a transmission sphere, as shown on the previous page.

Note I set the wavelength to 633nm to match the typical interferometer wavelength.

Defining the Verical Gradient

To define a vertical (y) gradient we use the Gradient4 surface with only the y-dependent terms non-zero

n(y) = n0 + ny1*y + ny2*y2

I wanted to enter the index variation in Zemax by computing the index as a function of the Y height in the interferometer cavity with an external equation. The refractive index of air with temperature and pressure is:

n = 1 + 0.000293246 (273.15K / T) (p / 1013hPa)

from Michael Koch’s website at: http://www.astro-electronic.de/faq3.htm#9

Using this equation, and knowing a temperature gradient that was measured in the interferometer cavity of 0.4 deg K/meter using several thermocouples at different heights in the cavity, it is possible to model and understand the effect of a linear temperature gradient in the cavity.

I set up this equation in an Excel spreadsheet so that I could choose the ny1 and ny2 coefficients. The Excel spreadsheet is attached for your calculation pleasure.

I found by trial and error that a delta_T (gradient index step size) of 10 mm gave good results. Smaller step sizes take longer to trace (obviously) but do not affect the results significantly. The spot size and ray fan clearly show astigmatism:

So the mirror cavity has just under 0.1 nm of astigmatism because of the 0.4° K/meter thermal gradient in y.

resulting an about 40 microns of separation between the input and output spots.

**Summary**

Zemax can model linear and quadratic temperature variations of glass and air by using the flexible Gradient 4 surface type. We have shown that one can model the thermal gradient that causes the refractive index of the air to vary linearly across the vertical or Y direction of a Fizeau interferometer cavity while testing a front surface mirror. We have also shown that this thermal gradient will cause a small amount of astigmatism in the return beam and a Y axis displacement of the spot going back into the Fizeau interferometer transmission sphere.