3 High resolution performance of TESOS

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 $A_{{\rm tot}}(\alpha)$ denote the - circularly symmetric - complex transmission of the filtergraph as a function of angle of incidence, then the radial distribution $p_{\lambda}(\rho)$ of the complex field in the image plane at wavelength $\lambda$ is given with

 

$$p_{\lambda}(\rho) \,\, \,\, {\cal H} \left[ A_{{\rm tot}}(\alpha) \right]$$ = (5)

$$\,\, 2\pi \int_{0}^{F/2} A_{{\rm tot}}(\alpha) J_{0} \left( 2 \pi \frac{\alpha \rho}{\lambda} \right) {\rm d} \alpha .$$ =

${\cal H} \left[ \cdots \right]$ denotes the Hankel transform (Bracewell 1986) of its argument and J₀ is the zeroth-order Bessel function of the first kind. For light at wavelength $\lambda$, the monochromatic PSF is given with $I_{\lambda}(\rho) \,=\, \left\vert p_{\lambda}(\rho) \right\vert ^2$.

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 $\lambda_{\rm c}$, 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 $p_{\lambda}(\rho)$ with the width of the PSF $p_{0,\lambda}(\rho)$ for a uniformly illuminated pupil, we define the "equivalent radius'' $h_{\lambda}$ at wavelength $\lambda$ as

$$h_{\lambda} \,\, = \,\, \frac{\int_{0}^{\infty} p_{\lambda}(\rho) {\rm d} \rho} {p_{\lambda}(0)} ,$$ (6)

noting that the PSF has a maximum for $\rho = 0$. We then compute the relative equivalent radius $\epsilon_{\lambda}$ from the equivalent radius $h_{0,\lambda}$ for the case of a uniformly illuminated pupil as follows,

$$\epsilon_{\lambda} \,\, = \,\, \frac{h_{\lambda} - h_{0,\lambda}} {h_{\lambda} + h_{0,\lambda}} \cdot$$ (7)

Figure 5 shows $\epsilon_{\lambda}$ 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 4: Top: monochromatic point spread function as a function of wavelength for the low resolution F = 1/128 mode of TESOS (linear scale). Bottom: same as above, but for four decades of intensity in a log scale. The brightness scale represents values between 0 and 1. The horizontal scale is wavelength departure from $\lambda _{\rm c} = 500$ nm in pm. The vertical scale represents image position in units of $\lambda _{\rm c} / D$

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 $I(\rho)$ of a point source in the absence of any additional aberrations in the system is given as

$$I(\rho) \ = \ \int_{\lambda_{\rm c} - \Delta \lambda / 2}^{\l... ... \left\vert p_{\lambda}(\rho) \right\vert ^2 {\rm d} \lambda ,$$ (8)

where $S(\lambda)$ is the intensity of the point source at wavelength $\lambda$, $\lambda_{\rm c}$ is the wavelength of maximum transmission of the filtergraph and $\Delta \lambda$ is chosen such as to fully include the filtergraph passband. If the spectrum of the source is a continuum, S is a constant and can be taken outside the integral in Eq. (8) and the PSF will remain constant for small changes of $\lambda_{\rm c}$. The corresponding polychromatic PSF is shown in Fig. 6 along with the point spread function from a uniformly illuminated pupil. The filtergraph PSF differs from the uniformly illuminated pupil PSF mainly by the strong, fringed halo which is caused by the annular illumination of the pupil in the blue wing. The effect is particularly strong for the low resolution mode, where most of the intensity is actually in the wings of the PSF. The Strehl'sche Definitionshelligkeit (ratio of peak intensities of aberrated and unabberated PSFs) is as low as 0.16 in this case. The halo for the high resolution mode is much smaller and the Strehl ratio of the PSF is 0.81. The full width at half maximum (FWHM) of the PSF is independent of the magnification.

Figure 6: Top left: polychromatic PSF for spectrally flat sources at $\lambda _{\rm c} = 500$ nm. The PSF for low (solid) and high resolution modes (dashed), and the PSF for a uniformly illuminated pupil (dotted) are shown in a linear scale. Bottom left: same as above, but for three decades of intensity in a log scale. Top right: polychromatic PSF for the spectral line with the filtergraph tuned to line center (solid curve), to the blue wing (short dashed), and to the red wing (long dashed), for the low resolution mode (F = 1/128). Bottom right: same as above, but for the high resolution mode (F = 1/256). The horizontal scale is image position in units of $\lambda _{\rm c} / D$

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

Figure 8: Top: polychromatic PSF for spectrally flat sources at $\lambda _{\rm c} = 500$ nm for the low resolution mode with a linear scale. Solid line: PSF including pupil phase variation. Dashed line: PSF with pupil phase variation ignored. The PSF for a uniformly illuminated pupil is shown as the dotted line. Bottom: same as above, but for three decades of intensity in a log scale

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.