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4. Results

Figure 1 (click here) shows the width of the PSF as a function of wavelength displacement for the on-axis ray on the Fabry Perot centered on a wavelength displacement of 0 pm for the 75, 50 and 25 percentile peak values, the 50 percentile value corresponding to the full width at half maximum (FWHM). All values are normalized to the FWHM of an evenly illuminated pupil (the Airy disk) which corresponds to tex2html_wrap_inline780 where D is the telescope diameter. At displacements towards the blue the PSF narrows because of the increase in illumination towards the edges of the pupil. The opposite is the case for red wavelength displacements. The effects of the uneven illumination of the pupil caused by the telecentric configuration of the rays through the Fabry-Perot is considerable. Figure 2 (click here) gives a different evaluation of the PSF. It shows the amount of energy E contained within the first dark Airy ring of the same telescope but with uniform aperture illumination (radius tex2html_wrap_inline786). For an Airy disk the amount equals 82%. In the telecentric Fabry-Perot it ranges between tex2html_wrap_inline788 in the blue wing of the filter profile to tex2html_wrap_inline790 in the red wing. Table 1 (click here) lists the properties of the telecentric Fabry-Perot profile and PSF behavior with different f-ratios of the telecentric beam. Obviously the slower the f-ratio, the smaller the PSF effects.

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Figure 1: Point-spread-function shape as a function of wavelength. Shown are the PSF widths for three fractions of the peak intensity in units of the FWHM of the Airy function

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Figure 2: Percentage of energy of the PSF contained in the image area within the first dark Airy ring of a uniform illuminated aperture. tex2html_wrap_inline796 is the wavelength of peak the total filter transmission

 

 
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Table 1: Properties of Fabry-Perot interferometer in telecentric configuration for different f-ratios

The effect on the astronomical observations is, of course, dependent on the actual observational configuration used. Solar spectral lines have a typical width of 10 pm so that Doppler shift observations require two observations with the filter transmission shifted by approximately 5 pm to either sides of the line center. The diffraction limited PSF in the red wing will be broader than that in the blue wing which will give rise to artificial velocity signals.

To assess the magnitude of the effect, I calculated the velocity image which would be seen for a velocity point on the solar surface if it were obtained by subtraction of two intensity images taken in the blue and red wing of a Gaussian shaped absorption line at 550 nm wavelength, with a central intensity of 33.33% of the continuum intensity and a FWHM of 10 pm. Such a line is similar in properties to the often used 617.3 and 630.3 nm Fraunhofer lines. The images were taken at tex2html_wrap_inline838 pm from line center, where line center was chosen such that the spatially and spectrally integrated intensities were equal in the two wings. Figure 3 (click here) shows cross-sections through the resulting velocity image PSFs for Doppler velocities of 0 and tex2html_wrap_inline840 m/s. Also shown, on the same relative intensity scale, are the cross-sections for the case of a uniform pupil illumination as seen from behind the etalon (but assuming the same telecentric spectral transmission profile). One notices: (i) the fully artificial velocity signal in case of zero Doppler shift. Its amplitude equals approximately that of a real 30 m/s feature observed in the collimated configuration. For a uniform solar image without velocity structure this artificial signal, of course, averages out. But for an image with surface structure it doesn't, and (ii) the very different PSFs for up- and downward motions, different both in the image core and in the far wings.

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Figure 3: Point-spread-functions in velocity images for Doppler shifts corresponding to +100 m/s (full drawn line), -100 m/s (dashed line) and 0 m/s (dotted line). The heavy lines correspond to the full telecentric configuration, the thin line to the (artificially) uniform illuminated pupil


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