I did a small analysis to estimate what the effect of FRD variations would be, given our current understanding of the instrument. In the Figure "FRD_effect.png" I am showing my results.

I have simulated 4 images in focus, close to the centre of detector and I have used FRD of 0, 6, 12 and 18 mrad. This numbers were chosen given the analysis by Belland et al. which claimed smallest FRD of roughly 6 mrad. 4 rows of the image show these 4 simulations.

Columns are as follows: 1. image of the exit pupil - you can see hot the FRD is increasing towards to bottom, changing the illumination of the edge of the exit pupil; 2. final image in the focus; 3. difference of the final result and of the result with frd=0 mrad. 

As we can see, at the moment the biggest effect is in the core of the line. I have to study literature and talk with Jim to understand how much sense do these results make.

The updated figures show much more realistic and useful analysis of the FRD effect. Improvement comes from primarily two factors: 1. insights about FRD effects given by Brent 2. input from discussion at PU on October 30, 2018.

What this simulation assumes: 1. It has wrong details of FRD modelling, i.e., it assumes that effective FRD is given by pure Gaussian convolution and that change due to misalignment at the entrance to the fiber is just change in the width of the convolving Gaussian.

2. we assume perfect alignment of the camera and the fiber

3. we assume that illumination of the entrance pupil==illumination on the exit pupil

There are 10 figures, 5 figures for a spot at the center of the detector and 5 figures for a spot on the edge of the detector. Figure 5 is the most important. 

They show:

Figure 1. Changes of flux as function of FRD. Columns are as follows: 1. image of the exit pupil - you can see how the FRD is increasing towards to bottom, changing the illumination of the edge of the exit pupil; 2. final image in the focus; 3. difference of the PSF with this FRD and FRD smaller by 3 mrad.

Figure 2. Changes of flux as function of FRD. Same as plot 1 expect for the last column.  So the columns are as follows: 1. image of the exit pupil - you can see how the FRD is increasing towards to bottom, changing the illumination of the edge of the exit pupil; 2. final image in the focus; 3. difference of the PSF with this FRD and FRD smaller by 3 mrad divided by the current PSF - this shows fractional change of the PSF.  This figures makes is a bit more obvious what the main effect is -> sending more flux in the diffraction wings of the PSF.

Figure 3. Image of the PSF, 6 pixels wide along which I make "extraction" in figure 5.

Figure 4. 1d projections of the illumination of the entrance pupil, showing more explicitly effect of FRD on the illumination of the entrance pupil. It also shows how much flux in going to be lost by changing FRD by 3 mrad (roughly of the order of 1.5%).

Figure 5. Change of the extracted flux as a function of pixels in the wavelength direction, by purest box-car extraction. Difference of extracted flux is given by different colors. Dashed line shows total flux as a function of wavelength. Black number are showing total flux at each wavelength (so they correspond to the black curve). Brown numbers are values of the brown curve, showing difference between changing "effective FRD" from 9 to 12 mrad. This is roughly the order of the change that we would expect in the real systems (given the work at Caltech and presentation by Brent).  

We see that effect of expected changes if of the order 0.5-2.5% across the whole 20 pixels elements. Note that in our two cases depicted here, the strength of wings far away from the center (more than 5 pixels from center) is roughly the same as expected continuum (cca 100 count), but this of course depend on the strength of the line. 

Even though we looked at this before, soon we will get new data with different stops, i.e., different illuminations of the pupil and we will we able to compare our predictions with the actual data.

I am reporting here on the result of the first analysis, comparing the results coming from the differently illuminated pupils.

1. Figure ComparisonDifferentIllumination_spot56.png shows:
i) top left: ``old data'', taken with F/3.3 stop
ii) top, center: full model describing the ``old data''
iii) top, right: difference between the model and the data
iv) bottom left: ``new data'', taken with F/2.8 stop
ii) bottom, center: model describing the ``new data'', derived from the ``old'' model and only modifying the parameters describing the illumination of the pupil
iii) bottom, right: difference between the model and the data

We obviously see systematic residuals (bottom right), which is at least partly the consequence of the fact that the fit to the defocused data with poor illumination was not sensitive to the areas that are illuminated now.

I am then interested to see if there a substantial difference between two images in focus, i.e., to test if my model is at least roughly correct. Of course, I do not expect the results to be perfect given how we can from image above how only changing the parameters that describe the illumination of the pupil is not sufficient to describe the ``new'' defocused images very well.

2. ComparisonDifferentIllumination_Out_Focus_spot96.png shows comparison of two defocused images, taken with different stops, that I am using for this analysis (I could not use the same spot as above as there are some weird gaps in the new focused data for that particular spot).

3. ComparisonDifferentIllumination_In_Focus_spot96.png shows comparison of the focused images taken with different stops, that are corresponding to the defocused images just mentioned.

4. FWHM_comparisons_spot96.png shows the cuts along the wavelength direction, comparing these two images and the two models (full model to the ``old data'' and the ``predicted'' model, constructed from the old model and only changing the illumination parameters). Given the simplicity of my modeling the agreement seems good. This gives me confidence that the previous analysis, conducted in October 2018 makes at least some sense.

I propose the close this ticket now, and capture the possible future investigations of this type in new and more focused tickets.