Efficiency as function of spectrograph aperture

Figure 1 shows the encircled energy within the beam at the spectrograph entrance pupil computed for PSF d .
The diameter of the aperture is in units of the aperture size without oversizing (speed spectrograph = speed telescope). The diameter with oversizing of 20% is marked with a long-dashed line.
The energy is normalized to the total energy which leaves the slit, i.e. only the additional loss of transmission at the spectrograph aperture is shown. This additional loss is mainly caused by the diffraction due to the gaps between the facets of the MEMS slit.
The red lines show the energy which passes the spectrograph aperture as a function of aperture size. The four curves differ in the position of the PSF within the slit. The chosen positions are the 4 corners of the 50 x 100mas dither rectangular.
The blue line shows the average of the encircled energies for ALL dither positions, including but not limited to the dither position represented by the red curves. The yellow shaded region around the blue curves shows the mean +/- one sigma of the energy averaged over those positions.
The green lines show the behavior of an ideal slit, i.e. a fully transparent slit of size 200mas x 600mas. For this perfect slit, the PSF is well contained at each of the dither positions. Therefore, all these curves look similar. This contrasts the large spread seen for for the MEMS generated slit.

The same calculation as the one shown above was carried out using the four previously considered PSFs. In addition, the same PSFs were used with the dead area around each facet reduced from 8 to 4 mas. The results are shown in Figure 2. The upper panel is the same plot as shown in figure 1, but only the average of all dither positions is shown for each PSF. The blue line is therefore identical to the blue line in figure 1, only the limits of both axis have changed. The solid lines are the results from the four different PSFs. The dashed lines are for two different PSFs and reduced dead area around each facet. The lower panels shows for each model the quantity n_dither= (3*rms / mean /0.1)^2. n_dither corresponds to the number of dither position needed to average out the uncertainty added by diffraction. Note that this is different from the previously used n_dither which included the geometrical effect of light which does not pass the slit because of the gaps between facets.

Conclusions

• Although oversizing more than 20% obviously will increase the throughput, the gain is a painfully slow function of the oversizing. In order to gain say 5% in throughput on top of the 20% oversizing, we would need to oversize by 70%! This makes the mirror surface about (1.7/1.2)^2 ~ 2 times larger, the mass presumably even by a larger factor.
• 20% oversizing seems indeed to be a sensible choice: The steep part of the curve throughput vs oversizing is already fully exploited. To go from 0% to 20% oversizing, we gain about 10% in throughput. This result depends only weakly on what PSF is used.
• Based on the models I used, we could actually compromise to only 15% oversizing if necessary. Most of the gains are in this range.
• If the gaps between the facets are smaller than what we have assumed, the gain from further oversizing is even smaller. (dashed curves in figure 2. )
• If the dead area can be reduced to half the "conservative estimate" of 8mas, the total slit throughput (geometrical + diffraction effects) looks much more acceptable.