Any variation of the complex field across the pupil modifies the point spread function (PSF)
at the position of the detector, which is readily computed from Fourier optics principles.
Let
denote the - circularly symmetric - complex transmission of the
filtergraph as a function of angle of incidence, then the radial distribution
of the complex field in the image plane at wavelength
is given with

denotes the Hankel transform (Bracewell 1986) of its argument and

Figure 4 shows the monochromatic PSF for a range of wavelengths across the transmission band of TESOS. The PSF is quite sharp but shows broad wings of fringes bluewards of the transmission peak. They occur when the monochromatic transmission within the pupil concentrates near the edge, representing an annulus of light, and in combination with the fluctuations of the optical phase. The latter effect was not taken into account by Beckers. The illuminated part of the pupil is concentrated near its center redwards of , reducing the effective telescope aperture and causing a broadening of the core of the PSF. This figure compares qualitatively with Fig. 1 of Beckers (1998a), except for the enhancement in the blue wings.

In order to be able to compare the width of the monochromatic PSF
with
the width of the PSF
for a uniformly illuminated pupil, we define
the "equivalent radius''
at wavelength
as

noting that the PSF has a maximum for . We then compute the relative equivalent radius from the equivalent radius for the case of a uniformly illuminated pupil as follows,

Figure 5 shows as a function of wavelength for both magnifications. It is seen that for the low magnification of TESOS, there is a variation of the PSF width of up to 50% compared with the uniformly illuminated pupil PSF, while for the high magnification the effect amounts to just a few %. It is this variation which causes the spurious velocity signals when filtergrams in different wings are compared.

Figure 5:
Relative equivalent radius of the point spread function the low resolution mode (top)
and high resolution mode(bottom) at a wavelength of 500 nm |

The *polychromatic* point spread function is calculated by taking the squared
modulus of the complex field distribution at each wavelength and then integrating
over the transmission band of the filtergraph (cf. Fig. 2).
The detected intensity distribution
of a point source in the absence
of any additional aberrations in the system is given as

where is the intensity of the point source at wavelength , is the wavelength of maximum transmission of the filtergraph and is chosen such as to fully include the filtergraph passband. If the spectrum of the source is a continuum,

Figure 7 shows the modulation transfer functions of TESOS which result from the polychromatic point spread functions of both magnifications. It is evident that the resolution of the instrument is seriously affected when the numerical aperture is large, as is the case for the low magnification. Note the high frequency tail which extends to the diffraction limit at a level of about 0.05. The high resolution mode with the less divergent beams suffers much less and differs only little from the theoretical case without pupil effects.

Figure 7:
Polychromatic modulation transfer function (MTF) for the low resolution
(solid line) and high resolution (dashed line) modes.
The dotted line represents the MTF for absence of pupil effects |

We computed the polychromatic point spread function without the phase variation in the pupil to be able to compare our treatment with Beckers' original results. Figure 8 presents the polychromatic PSF with and without the pupil phase variation for the low resolution mode. It is evident that ignoring the phase leads to much less pronounced wings and significantly underestimates the apodisations effect.

Copyright The European Southern Observatory (ESO)