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5 Discussion and conclusions

We have analyzed the effects of pupil apodisation in the TESOS filtergraph on the angular resolution and accuracy of Doppler velocity measurements of solar small scale features. We confirm Beckers' observation that a telecentric mount FPI filtergraph suffers from significant pupil apodisation; even more so when also the deformation of optical phase is included in the analysis. The latter gives rise to an enhanced halo in the point spread function in the blue wing of a line when the numerical aperture is large, in comparison to calculations where the phase effects are not taken into account.

The dependence of the phase between intensity and velocity error on structure size is easy to understand. The PSF in the blue wing of a spectral line has a sharp core and broad wings compared to the PSF in the red wing where the core is broader but wings are essentially absent. A spectral line in the presence of large scale intensity fluctuations will be blurred in the blue wing because of the extended wings of the PSF and show more contrast in the red wing. The net result is an apparent blueshift in the brighter parts of the structure and an apparent redshift in the darker parts, resulting in an anti-correlation of intensity and lineshift.

In the presence of small scale intensity fluctuations, the opposite is true, because now the sharp core of the blue wing PSF wins over the broader core in the red wing, and intensity and lineshift will be correlated. Practically, there will be a mixture of scales in the observed solar structure, and the simultaneous presence of intermediate and small scales may cause correlated superposition effects which enhance the error through the phase relation. A conservative estimate of the velocity error will therefore be the sum of the peaks at intermediate and small scales. One can expect in the worst case a peak-to-valley error of 45 m/s for the low resolution mode and of 9 m/s for the high resolution mode and for a contrast of 15% rms. The error will scale linearly with the intensity contrast.

With respect to the performance of TESOS, we make the following observations:

The pupil apodisation effects of TESOS are comparable to those of a single Fabry-Pérot etalon filtergraph with the same spectral bandwith. The addition of the other two interferometers only serve to increase the free spectral range of the filtergraph;
The low resolution mode just performs as such. The spatial resolution is slightly better than one arcsec and is quite well adapted to the pixel scale. Although the modulation transfer function drops rapidly at scales of 1 arcsec, there is remaining signal at higher frequencies which could be recovered by deconvolution if the quality of the data warranted it;
The spatial resolution of the high resolution mode does not suffer significantly from pupil apodisation degradation;
Velocity errors in the low resolution mode are comparable to or smaller than other sources of velocity error in TESOS (see below);
The high resolution mode does not suffer from significant velocity errors at all scales and contrasts which are relevant to solar observations.

\par\includegraphics[width=8cm,clip]{ms9986f10a.eps}\par\includegraphics[width=8cm,clip]{ms9986f10b.eps}\end{figure} Figure 10: Velocity error as a function of intensity modulation period for the low resolution (F = 1/128, top) and high resolution (F = 1/256, bottom) modes as a function of modulation period and structure contrast

\par\includegraphics[width=8cm,clip]{ms9986f11.eps}\end{figure} Figure 11: Dependency of Strehl ratio on numerical aperture in the telecentric beam of TESOS

The random velocity error in the low resolution filtergrams of TESOS is estimated to be of the order of 100 m/s (Schlichenmaier & Schmidt 1999). The most significant source is the wavelength sampling of typically 2.5 pm, which corresponds to a Doppler velocity of 1.5 km s-1. Other sources are incomplete calibration of the filtergrams, straylight and detector noise. A systematic variation of seeing during a line scan gives also rise to small scale contrast-dependent spurious velocity signals when one line wing is better resolved than the other. Taken all together, these effects are worse than the errors caused by pupil apodisation.

It is conceivable that very fine structure such as small magnetic flux concentrations and umbral dots have contrasts which exceed the range that we have investigated. But even with contrasts of up to 100% the velocity error in the high resolution mode should not amount to more than 30 m/s which is still comparable to other error sources. We therefore conclude that TESOS is very well suited for high precision and high resolution investigations of the sun.

This may not be the case for other combinations of etalon characteristics and numeric aperture in a telecetric mount filtergraph. We have therefore calculated the Strehl ratio for the TESOS filtergraph for magnifications corresponding to F/512 and F/64 to give an indication of the variation with numeric aperture. The results are shown in Fig. 11 together with the Strehl ratio of the current magnifications. It is evident that for a filtergraph with the spectral resolution of TESOS, the spatial resolution is significantly affected for numerical apertures larger than 0.002, corresponding to F/250, and decays rapidly with $S \approx (NA)^{-4.4}$. Pupil apodisation effects therefore require careful consideration in the design of a filtergraph.

The alternative would be to use the etalons in a collimated configuration close to a pupil. A simple calculation shows that such a mount is not at all without problems, either. Each surface of an etalon cavity shows deviations from a perfectly flat plane at the scale of typically $\lambda / 100$ which origin from the polishing process. Microroughness at the level of a few nm rms adds to these deviations. Let $\Delta(\vec{r})$ be the thickness variation of the etalon's cavity as function of position $\vec{r}$ perpendicular to the direction of propagation of light with $\Delta(\vec{r}) << \lambda_{{\rm c}}$. It is then easy to show from Eqs. (1) and (2) that the optical phase $\phi (\vec{r})$ of an originally plane wavefront transmitted by the etalon is given with

 \begin{displaymath}\phi(\vec{r}) \ \ = \ \ \arctan \frac{2 R^{2}}{1 - R^{2}} \,
\frac{2 \pi}{\lambda} \Delta(\vec{r}) .
\end{displaymath} (10)

For TESOS, $R^{2} \, = \, 0.97$ and the resulting wavefront error would amount to about 160 times the wavefront error of a single etalon plate if all three etalons were placed in the collimated beam, or to more than a wave of aberration for optics with a quality of $\lambda / 100$. Such wavefront errors would make high spatial resolution observations impossible.

The authors wish to express their gratitude to J.M. Beckers, National Solar Observatory, for discussion and many helpful comments.

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