MSc project: Quantifying the wavefront error budget of the MICADO ADC using power spectral density analysis

Introduction

By picking the Master’s track of Instrumentation and Informatics I conciously decided that I wanted more of an engineering approach to astronomy. During my first year of this Master I was introduced to the MICADO project and I have been working on this ever since.

MICADO, or the Multi-AO (Adaptive Optics) Imaging CamerA for Deep Observations, is one of three first light instruments in development for the 39 meter Extremely Large Telescope (ELT). This instrument will be an imaging camera for light of near-infrared wavelengths (800 nm to 2400 nm), but it also offers some spectroscopic capabilities. It is currently in development by a consortium lead by the Max-Planck-Institut für extraterrestrische Physik (MPE), in collaboration with the European Southern Observatory (ESO). An important scientific capability of MICADO will be that of highly precise relative astrometry, i.e. the precise positional measurement of stars or other objects in the field of view of MICADO with respect to each other. The goal is to do this to a level of 50 micro arc seconds, which is a level of accuracy not achieved before on ground based telescopes. For more info on MICADO in general I refer the reader to the ESO page or to Davies et al., 2018.

For my first small project on MICADO I tried to fit geometric distortions using a 3D fifth order polynomial. I’m still not exactly sure why, but I had tremendous trouble trying to get this to work and few results were achieved. Though some time after I got the fitting routine to work using orthogonal distance regression in stead of the linear least squares minimization that I had tried. For the very curious reader, here is the original project report and here is the update describing the new fitting routine and results.

Then, in September 2016, I started an internship at the NOVA Optical-IR Group in Dwingeloo, where I did some thermal simulations on the MICADO filter wheels to get to a concept for the cooling and heating mechanisms that would need to be used to cool the instrument to or heat the instrument from its operating temperature of 80 Kelvin (-193°C). The expected outcome of using a combination of radiative heat transfer and conduction was found. Besides this, I also looked at the possible science filters that could be included by comparing use cases as well as usage statistics on available telescopes. Finally, I measured the throughput of science filters of the BlackGEM telescopes for some practical experience. The full report can be downloaded here.

Finally we get to the subject of my Master’s thesis. As mentioned before, MICADO is aiming for 50 micro arc second astrometric accuracy. At this level it could be possible that the small imperfections of the optics have a significant impact on the astrometric performance. It was then up to me to try to find out how large these effects would be, and what surface error would be permitable for the MICADO atmospheric dispersion corrector (ADC).

The Atmospheric Dispersion Corrector

Just as light disperses as it travels through a glass prism, light also disperses as it travels through the atmosphere. Blue light gets refracted more then red light, resulting in a smearing effect. This effect is dependent on several parameters, of which the most important are the wavelength of the light, the width of the bandpass and the zenith angle of the observed object.

Atmospheric dispersion is one of the reason astronomers don’t like to point their telescope close to the horizon, especially when they’re observing at optical wavelengths. However, with the increase in resolution that comes with the increase in telescope diameter of the ELT, atmospheric dispersion also becomes a significant effect in the near infrared. This is even the case at relatively modest zenith angles of, for example, 30°. The ELT diffraction limit at MICADO wavelength is approximately 16 milli arc seconds, corresponding to about 4 pixels on the detector. The figure below shows how quickly the dispersion effects becomes important.

The total atmospheric dispersion at Cerro Armazones for various observing bands of MICADO. Note that broader bands have higher dispersion and the dispersion is larger for higher zenith angles. Also note that shorter wavelengths experience higher dispersion than longer wavelengths.

It is therefore not surprising that something should be done about this effect, which is where the Atmospheric Dispersion Corrector (ADC) comes in. The principle of an ADC is very simple and is even used by amateur astronomers for the observing of planets near the horizon. By using two prisms, that can be rotated or moved linearly, the dispersive effect of the atmosphere can be reversed. The dispersive effect is illustrated in the figure below.

The two types of ADC typically considered, the linear (top) and rotating (bottom) ADC, in their minimum and maximum dispersion configuration.

Simulating real optical surfaces with Power Spectral Density Analysis

While the low spatial frequency wavefront errors, typically described using Zernike polynomials, and high spatial frequencies are quite well understood, the effects of mid-spatial frequencies are still uncertain. With a mathematical framework, called Power Spectral Density analysis, we generated our own surfaces with many random mid-spatial frequency features. The equation below describes the Power Spectral Density function as the amplitude of a two dimensional Fourier transform of the surface height profile.

 $\textnormal{PSD}(f_x,f_y) = \frac{1}{ab} \left| \int\limits_{-b/2}^{b/2} \int\limits_{-a/2}^{a/2} h(x,y)e^{-2\pi i (f_x x + f_y y)} dx dy\right|^2$

Now, we can evaluate the frequency content of a given surface. Inversely, we can define a PSD profile and generate a surface profile. We decided to use the definition described by Sidick (2009).

 $\textnormal{PSD}'(u_m,v_n) = \frac{\sigma^2 A}{h_0}\frac{1}{1+(\rho_{mn}/\rho_{c})^{p}}$

with the normalisation factor

 $ h_0 = \sum^{M}_{m=1}\sum^{N}_{n=1}\frac{1}{1+(\rho_{mn}/\rho_{c})^{p}}$

and

 $\rho_{mn} = \sqrt{u_m^2 + v_n^2}.$

The influence of the p and ⍴c parameters on the surface profile map are illustrated for nine examples. From left to right p increases from 2 to 4, while from top to bottom ⍴c takes on a value of 1, 5 or 10. More power is contained in the lower spatial frequencies for higher p-values. Low cut-off frequency PSD maps with a steep slope show a similarity to maps modelled using Zernike polynomials.

This definition allows the user to generate surfaces using only three parameters, namely the RMS surface roughness σ, the cut-off frequency ρc and the power slope p. Below some examples for various values of the power slope and the cut-off frequency.

Distortions caused by surface errors on the ADC surfaces

Next I wrote a framework linking the optical design software ZEMAX OpticStudio to Python. By adding the generated surface profiles to the glass surfaces of the ADC I could investigate the distortions caused by these surface profiles. By generating geometric distortion plots, similar to the ones ZEMAX itself is able to generate I was able to do this. After subtraction of the distortions from the nominal system, i.e. without any added distortions, I would be able to determine the distortions introduced by the ADC surface imperfections.

Of course, this is easier said than done. The main problem was that the added surface perturbations had to be sufficiently sampled by the rays used to generate the distortions plot. For a system without these complicated surface imperfections only a single ray is required per field position. With the ADC located close to the optical pupil, I would have to use 512×512 rays per field position. By taking the average detector position of all these rays I determined an effective centroid, which could then be used as a point in the distortion grid.

To get a distortion grid of 11×11 field points, I had to trace a total of almost 32 million rays per simulation run. As I hadn’t figured out how to do this efficiently, the simulations took quite some time.

Doing this, I found that the distortions seemed to be relatively large. An optimistic surface profile (σ = 5 nm, ρc = 1 and p = 2) would give a root mean square distortion of more than 300 micro arcseconds, where we hoped to find something about a factor 10 smaller.

Exploring the parameter space resulted in reasonable relations between the magnitude of the geometric distortions and the changing parameter. However, when adding a small amount of white noise to the PSD curve, the results of the previous analyses became inconsistent. This led us to believe that a systematic problem was present, namely the pupil sampling.

Further analyses, after I had finished my Master’s degree, confirmed these suspicions. Then a more analytic approach to this problem was conceived, using the average tip and tilt of a surface profile over the footprint of the beam. Unfortunately, this analysis was also inconsistent with previously obtained results. Reworking this problem using diffractive beam propagation, Fourier Optics, would take significant effort. This was to be one of the main topics of my upcoming PhD research project.


Still interested? Click here to read the full thesis.

Or if you’re interested in how this story continued: Click here to go to the page describing the Fourier optics approach to this problem.