OpticStudio can model linear and quadratic temperature variations of glass and air by using the flexible Gradient 4 surface type. This article shows how to model a linear thermal gradient in a double-pass system. The method can be easily extended to more complex gradients.

**Authored By Michael Pate**

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## Introduction

A linear thermal temperature gradient of an interferometer cavity can be modeled in OpticStudio with a Gradient 4 surface. This surface enabled you to create a linear or quadratic type index variation along one, two, or three axes. This can be understood by looking at the refractive index equation for the Gradient 4 surface:

n(x, y, z) = n_{0} + n_{x1}x + n_{x2}x^{2} + n_{y1}y + n_{y2}y^{2} + n_{z1}z + n_{z2}z^{2}

By using only the terms in y the required gradient can be easily produced. This article will model a thermal gradient between a point source and a mirror in an interferometer cavity with this method.

## Modeling the interferometer cavity

The interferometer cavity above 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 OpticStudio. I use the System Explorer to set the aperture type to Object Cone Angle with 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 Gradient 4 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 2000 mm thick as shown above. The mirror semi-diameter (radius of the substrate) is 175 mm 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 -2000 mm 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 Gradient 4 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 633 nm to match the typical interferometer wavelength.

## Defining the Vertical Gradient

To define a vertical (y) gradient we use the Gradient 4 surface with only the y-dependent terms non-zero.

I wanted to enter the index variation in OpticStudio by computing the index as a function of the Y height in the interferometer cavity with an external equation. From Michael Koch's website^{1} we find the refractive index of air with temperature and pressure is:

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 n_{y1} and n_{y2} coefficients. The Excel spreadsheet is attached for your calculation pleasure.

I found by trial and error that a ΔT (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.

This results in about 40 microns of separation between the input and output spots. The final system is attached.

## References

1. Koch, Michael. 2018. *Astro Electronic.* Accessed 2008. www.astro-electronic.de/faq3.htm#9.

KA-01530

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