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 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
).
For an Airy disk the amount equals
82%. In the telecentric Fabry-Perot it ranges between
in the blue wing of the filter
profile to
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.
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
Figure 2:
Percentage of energy of the PSF contained in the image area within the
first dark Airy ring of a uniform illuminated aperture.
is the wavelength of peak the total filter
transmission
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 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
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.
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