Figure 1 presents a sketch of the principle of the TESOS filtergraph.
TESOS uses two magnifications, a low resolution mode with 0.25 arcsec/pixel and
an *F* number of *F* = 1/128 of the beam at the position of the etalons,
and a high resolution mode with 0.11 arcsec/pixel and *F* = 1/256.
The low resolution mode is far from resolving the diffraction limit of
0.15 arcsec at
nm, while the high resolution mode remains within a factor
of 1.4 from critically sampling the diffraction-limited point spread function of the VTT.
The pupil apodisation effects are roughly four times less in the high resolution mode
compared to the low resolution mode due to the smaller numerical aperture.

The transmission *A* for the complex electromagnetic field of a single FPI is given
with the Airy function (see e.g., Tolansky 1948),

where

The phase delay for a wave with wavelength
and angle of incidence
is given with

where

The cavity width of the widest etalon in TESOS is *d*_{1} = 1.3 mm, the others are
narrower by factors of
and
.
The tuning parameters ,
and
are computer controlled
to make the maxima of the spectral channels coincide at any desired wavelength
within a range from 430 nm to 750 nm.
The superposition results in a 2.5 pm wide overall passband, the spectral channels
remain incommensurable for a spectral range of about 3 nm.
Interference filters with 1 nm passbands select the desired superposition
of channels within the region of interest in the spectrum.
The passbands of the three etalons are scanned successively across a spectral line
with typical step sizes of 2.5 pm (equivalent to 1.5 km s^{-1}).
A filtergram is taken at each wavelength position simultaneously with an image of the
observed region in the continuum; the latter serves for precise registration of the
filtergrams during data analysis.

Figure 2 shows the individual passbands of the FPIs as well as their
superposition for a central wavelength
nm and for
according to Eq. (1).
The medium inside the cavities is air for which we assume *n* = 1.
The separation of the FPIs is
cm, so we disregard any
interference effects between etalons.

We compute the combined complex field transmission by multiplication of the individual
transmissions of the etalons.
If we take the same reflectivity *R* for all surfaces, we obtain

where the phase delays apply to the three FPIs, i.e.,

The transmission computed in this manner applies to a single ray, representing a plane wavefront at wavelength , incident on the etalons under the same angle . Inspection of Fig. 1 reveals that a plane wave transmitted by the etalons corresponds to light which emerges from a point in Pupil 1. Since the phase delay (Eq. 2) varies with the cosine of the angle of incidence, the overall transmission peak wavelength shifts slightly across the pupil. The result is a variation in transmission across the pupil for monochromatic light which depends on the numerical aperture

Figure 3 shows in the left panels the logarithm of the monochromatic intensity
and the phase across the pupil diameter as seen through the FPIs, and as a function of
wavelength for
pm about
nm and for *F* = 1/128(the effects for *F* = 1/256 are similar, but considerably smaller).
The right panels show the variation of intensity and phase at a fixed position in wavelength
corresponding to the shifted peak of the transmission profile, across the
two-dimensional pupil.
When the wavelength is scanned from the blue to the red side of the passband, the
intensity first increases near the pupil edges as a ring of light which shrinks as the
wavelength increases until the light patch disappears at the pupil center redwards of the
peak transmission.

Copyright The European Southern Observatory (ESO)