# Quantification of the expected residual dispersion of the MICADO Near-IR imaging instrument

- Direct link to the publication: MNRAS (open access)
- The
`AstroAtmosphere`

PYTHON package, was developed during this research. It allows you to easily calculate the refractive index, atmospheric refraction and atmospheric dispersion using many models (more than discussed in the publication). The Git-Lab page also includes all scripts to reproduce the results in the publication. For more info: Git-Lab or PyPI.org.

During the development of the MICADO *Atmospheric Dispersion Corrector* the question arose as to which model, describing the atmospheric dispersion, we should use in the control algorithm of the ADC. Unfortunately, it will not be possible to measure the atmospheric dispersion directly during a science observation. The instrument will have to rely on models with inputs such as temperature, pressure levels and relative humidity to determine the amount of atmospheric dispersion. The requirements state that the MICADO ADC should be able to decrease the amount of atmospheric dispersion to a level lower than 2.5 milli arcseconds of chromatic elongation of the point spread function (PSF), or roughly one third of the J-band PSF size.

From a study done by Dr. Paolo Spanò in 2014^{[1]}P. Spanò (2014). *“Accurate astronomical atmospheric dispersion models in ZEMAX”*, Proc. SPIE 9151, Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation, 915157. DOI:10.1117/12.2057072, we knew that the atmospheric model used in Zemax OpticStudio might not be the most accurate model available. I set out to compare various models describing atmospheric dispersion and see whether or not these are sufficient for MICADO or that we would have to develop a new one. Fortunately, we think that will not be necessary.

## Atmospheric refraction

The principle of atmospheric refraction is very simple indeed. Just like light refracts at any interface where to media meet with difference indices of refraction, so does star light when it hits the Earths atmosphere. However, the atmosphere is not a medium with a constant index of refraction. For example, as the atmospheric pressure changes from (near) vacuum to about 1 bar at sea level, the refractive index changes from about 1.0000 to about 1.0003 and light bends accordingly.

With perfect knowledge about the refractive index as a function of wavelength, n(), and altitude, it is possible to correctly predict the atmospheric refraction from the *refraction integral*,

Here is the refractive index at the observer and is the zenith angle.

Trouble arises, however, when we do not know much about the conditions except for those at the observer. Many different models exist, with various assumptions, that try to approximate the refraction integral. Below a list of several, most of which are mentioned in the MNRAS publication.

**Plane-parallel atmosphere model** – Assuming the atmosphere is a flat homogeneous medium, we can derive a very simple model from Snell’s law. The atmospheric refraction is then described by

**Cassini’s homogeneous atmosphere model** – A natural improvement to the plane-parallel atmosphere model is found by taking the curvature of the Earth and the atmosphere into account. Using Snell’s law and the law of sines, one can quite easily derive that the atmospheric refraction can be described by

**Oriani’s model** – It is often possible to use a Taylor expansions on the model describing refraction, to arrive at a form typically known as Oriani’s model. Depending on the required accuracy, one could increase the included orders of the expansion.

**Error function model** – Found in A. Danjon’s book *Astronomie Generale*^{[2]}A. Danjon (1980). *“Astronomie Generale – Astronomie Spherique et elements de mecanique celeste”*, 2nd edition., this model offers a method to describe refraction that does not diverge near the horizon. This model assumes an exponential decrease in the air density as a function of altitude. Invoking the Gladstone-Dale relationship, it is then possible to derive the following equation for the atmospheric refraction:

where is the local air refractivity, is the ratio of the reduced height of the atmosphere and the radius the Earth. The function in the equation above contains the error function that gives this model its name and is defined as

**Hohenkerk & Sinclair model** – This is the model used by Zemax OpticStudio and the popular SLALIB package. C.Y. Hohenkerk and A.T. Sinclair^{[3]}C.Y. Hohenkerk, A.T. Sinclair (1985). *“The Computation of Angular Atmospheric Refraction at Large Zenith Angles”*, NAO Technical Note No. 63. http://astro.ukho.gov.uk/data/tn/naotn63.pdf assumed a linear decrease in temperature from sea level to 11 kilometers altitude. From there up to 80 kilometers, they assumed a constant temperature. By incorporating a description of the refractive index, they are able to find a semi-analytic description for the refraction.

**Barometric exponential model** – Just after the publication, I received a message from Dr. Richard Mathar pointing me to a recent manuscript of his^{[4]}R. J. Mathar (2020). *“A Barometric Exponential Model of the Atmosphere’s Refractive Index: Zenith Angles and Second Order Aberration in the Entrance Pupil”*, arXiv:2004.11808. In it he assumed an exponential decay of the refractive index as altitude increases. Using various series expansions he does, in the end, also arrive at a form similar to Oriani’s form. The calculation of the coefficients takes up a large part of the manuscript.

As it turns out, all these models behave more or less the same with respect to atmospheric dispersion, the differential in the refraction for two different wavelengths of light. Not incredibly surprising, but good knowledge nonetheless. For refraction values they might differ more, but that is not of our present concern.

The primary reason why the accuracy of these models can be doubted is in the choice of refractivity model. These equations, describing the refractive index of air as a function of wavelength, temperature, pressure, humidity and sometimes even CO_{2} density, have a large impact on the overall refraction and dispersion values. This is especially true now that the required accuracy has increased so significantly with the advent of _extremely large telescopes. At this point, likely the Ciddor equations^{[5]}P. E. Ciddor (1996). *“Refractive index of air: new equations for the visible and near infrared”*. Applied Optics, 35:1566 are best suited to describe the refractive index in the optical and near-infrared. But as Dr. Paolo Spanò pointed out in his publication, the equations by Barrell and Sears^{[6]}H. Barrell, J. E. Sears (1939). *“The Refraction and Dispersion of Air for the Visible Spectrum”*. Philosophical Transactions of the Royal Society of London Series A, 238:1–64. DOI:10.1098/rsta.1939.0004 from the first half of the twentieth century are still regularly used. In many cases these equations are sufficiently accurate, but in the case of MICADO they are outdated.

## Other sources of expected residual dispersion for MICADO

While understanding the various models describing atmospheric refraction was the main goal of this work, we also spend a significant amount of time trying to predict where other parts of the overall residual dispersion might come from. There are some obvious sources that we could point out right away, such as the limited angular resolution that we’ll have to place the ADC prisms in the right location. Another one is the mismatch between the dispersive properties of the atmosphere and the ADC glass. Other effects, such as position knowledge and the source spectral distribution were less clear, but turned out to be very small.

While we were able to find values for these effects using a combination of Zemax OpticStudio and Python (using the ZOS-API), it was also part of my job as an academic student to better understand what was happening on a more fundamental level. Therefore, I derived an analytical model to describe the ADC and its optimum position, given some amount of atmospheric dispersion. Perhaps the largest advantage of doing this was the resulting increase in control over the different variables and the speed at which the analyses could be performed.

Near the end of the paper we proposed that there might also be other relevant dispersive effects, that we did not consider in this particular work. Since then we have found that, indeed, other optics in MICADO (and before MICADO, such as adaptive optics dichroic) introduce some chromatic dispersion. Furthermore, the ADC is only able to reduce the dispersion to a theoretical zero for a single point in the field. For other parts of the field there will also be some amount of residual dispersion present. At the far edges of the field of view this might be as large as 1 milli arcsecond. Not very large, but something that might have to be taken into account to achieve the highest level of performance.

## What’s next?

At NOVA we are now working towards the MICADO Final Design Review. The design of the *atmospheric dispersion corrector* is taking shape. Now that this study of the uncertainties with respect to residual dispersion sources is done, we have started to think about how to calibrate the device, preferably without having to do an extensive on-sky campaign.

Likely, we or somebody else has to verify that these models are correct. I do have some ideas about that, but I’ll let you know when I’m ready to talk about those publicly.

Perhaps you’ll see a page on that in the future.

## Further reading

If you’d like to know more about the topics I’ve written about on this page, these are some recommendations that I’d like to give.

Obviously, the publication itself and the references therein, go into much more detail and specifics, so that would be a fantastic start. However, some works might warrant some extra attention.

Perhaps the best start for an overview of atmospheric refraction is “Understanding Atmospheric Refraction” by Andrew Young (2006). Then, the chapter on atmospheric refraction in Danjon (1980) was invaluable to me to understand the error function model and how you can use a Taylor expansion to get Oriani’s model from there. It’s a bit unfortunate that the text is only available in French, but Google translate did a wonderful job translating it.

I recommend that you also read the Spanò paper, that started this whole work, if only to understand that there are significant differences between various refractivity models and that a recent publication date might not mean that the data and equations used are up to date.

Finally, I wholeheartedly recommend you to use my `AstroAtmosphere`

Python package, if you need to calculate the refractive index of air, the atmospheric refraction or the atmospheric dispersion, using any of the models and references described here and in the publication. They’re all included and make it easy to compare the values. And also, it makes the time I’ve spend on polishing (i.e. procrastinating) it, worth it!

Footnotes

↑1 | P. Spanò (2014). “Accurate astronomical atmospheric dispersion models in ZEMAX”, Proc. SPIE 9151, Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation, 915157. DOI:10.1117/12.2057072 |

↑2 | A. Danjon (1980). “Astronomie Generale – Astronomie Spherique et elements de mecanique celeste”, 2nd edition. |

↑3 | C.Y. Hohenkerk, A.T. Sinclair (1985). “The Computation of Angular Atmospheric Refraction at Large Zenith Angles”, NAO Technical Note No. 63. http://astro.ukho.gov.uk/data/tn/naotn63.pdf |

↑4 | R. J. Mathar (2020). “A Barometric Exponential Model of the Atmosphere’s Refractive Index: Zenith Angles and Second Order Aberration in the Entrance Pupil”, arXiv:2004.11808 |

↑5 | P. E. Ciddor (1996). “Refractive index of air: new equations for the visible and near infrared”. Applied Optics, 35:1566 |

↑6 | H. Barrell, J. E. Sears (1939). “The Refraction and Dispersion of Air for the Visible Spectrum”. Philosophical Transactions of the Royal Society of London Series A, 238:1–64. DOI:10.1098/rsta.1939.0004 |