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2 Pupil apodisation of TESOS

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 $\lambda = 500$ 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.

\par\includegraphics[width=8.5cm,clip]{ms9986f1.eps}\end{figure} Figure 1: Sketch of the optical layout of TESOS. L1 to L4: achromatic lenses. The telescope focal plane is reimaged by a system of a collimator and a camera lens (L1 and L2). The focal length of L2 equals its distance from Pupil 1, thus providing the telecentric beams in the intermediate image plane. A second system of collimator and camera lenses (L3 and L4) produces an image of the focal plane on the detector. FP1, FP2 and FP3 show the position of the three Fabry-Pérot etalons. The position of a prefilter wheel is in the same image space as the etalons is not indicated. The two magnifications with F = 1/128 and F = 1/256 are realized by an exchange of L2 and a mechanical rearrangement of the etalons

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

 \begin{displaymath}A\,\, = \,\,T^2 \frac{{1\, - \,R^2 \,\exp { - i\, \Psi }}}
{{1\, + \,R^4 \, - \,2R^2 \cos \Psi }} ,
\end{displaymath} (1)

where R is the amplitude reflectivity of the cavity surfaces, $T\,=\,1-R$ is the amplitude transmission of the surfaces, $\Psi$ is the optical phase delay of the cavity and $i \ = \ \sqrt{-1}$.

The phase delay for a wave with wavelength $\lambda$ and angle of incidence $\alpha$ is given with

 \begin{displaymath}\Psi \,\, = \,\,2\pi
\frac{2n\left( {d + \delta } \right)}{\lambda }\,\cos \alpha \ ,
\end{displaymath} (2)

where n is the index of refraction of the medium inside the cavity, d is the nominal width of the cavity, and $\delta$ is a tuning parameter which is adjusted such as to make the phase delay an integer multiple of $2 \pi$ for the desired center wavelength $\lambda_{\rm c}$ of the passband at $\alpha = 0$.

The cavity width of the widest etalon in TESOS is d1 = 1.3 mm, the others are narrower by factors of $\epsilon_2 = 0.617$ and $\epsilon_3 = 0.439$. The tuning parameters $\delta_1$, $\delta_2$ and $\delta_3$ 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 $\lambda _{\rm c} = 500$ nm and for $\alpha = 0^\circ $according to Eq. (1). The medium inside the cavities is air for which we assume n = 1. The separation of the FPIs is $20\, ...\, 30$ cm, so we disregard any interference effects between etalons.

\par\includegraphics[width=8.3cm,clip]{ms9986f2.eps}\end{figure} Figure 2: Spectral passband of TESOS for $\alpha = 0^\circ $ in linear (top) and logarithmic scales (bottom). Solid lines represent the transmission of the individual etalons, the dashed curve represents the combination of the three FPIs

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

$\displaystyle A_{{\rm tot}}\,\, = \,\, T^6
\left( \frac{{1\, - \,R^2 \,\exp \le...
... - i\,\Psi_1 } \right]}}
{{1\, + \,R^4 \, - \,2R^2 \cos \Psi_1 }} \right) \cdot$      
$\displaystyle \left( \frac{{1\, - \,R^2 \,\exp \left[ { - i\,\Psi_2 } \right]}}
{{1\, + \,R^4 \, - \,2R^2 \cos \Psi_2 }} \right) \cdot$     (3)
$\displaystyle \left( \frac{{1\, - \,R^2 \,\exp \left[ { - i\, \Psi_3 } \right]}}
{{1\, + \,R^4 \, - \,2R^2 \cos \Psi_3 }} \right)\ ,$      

where the phase delays $\Psi_1, \Psi_2, \Psi_3$ apply to the three FPIs, i.e.,
$\displaystyle \Psi_1 \,\,$ = $\displaystyle \,\,2\pi
\frac{{2n\left( {d_1 + \delta_1 } \right)}}{\lambda }\,\cos \alpha \ ,$  
$\displaystyle \Psi_2 \,\,$ = $\displaystyle \,\,2\pi
\frac{{2n\left( {\epsilon_2 d_1 + \delta_2 } \right)}}{\lambda }\,\cos \alpha \ ,$ (4)
$\displaystyle \Psi_3 \,\,$ = $\displaystyle \,\,2\pi
\frac{{2n\left( {\epsilon_3 d_1 + \delta_3 } \right)}}{\lambda }\,\cos \alpha \ .$  

The transmission computed in this manner applies to a single ray, representing a plane wavefront at wavelength $\lambda$, incident on the etalons under the same angle $\alpha$. 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 F of the beams at the intermediate image plane. This effect also broadens the transmission profile of the filtergraph and shifts the central wavelength slightly towards the blue by a factor of $\cos F/3$.

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 12.5 $ pm about $\lambda _{\rm c} = 500$ 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.

\par\includegraphics[width=8.8cm,clip]{ms9986f3a.eps}\par\includegraphics[width=8.8cm,clip]{ms9986f3b.eps}\end{figure} Figure 3: Top left: three decades of intensity in a log scale as a function of wavelength across the pupil diameter for F = 1/128. The horizontal scale is the wavelength departure from $\lambda _{\rm c} = 500$ nm in pm. The vertical scale represents pupil position in units of pupil diameter. Top right: intensity distribution inside the pupil in a linear scale at the wavelength indicated by the vertical line in the left panel. The white circle on the right shows the edge of the pupil. Bottom: wrapped pupil phase as a function of wavelength across the pupil diameter (left) and within the pupil (right). The scale represents values between $\pm 180^\circ $

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