4 Expected detection capabilities

Detection capabilities primarily relate to the detection of a faint companion and are governed by Signal to Noise Ratio (SNR in the following).

We consider the case of a companion off-axis by$\,$ $\beta_{0}$ with respect to an on-axis source, both sources being unresolved. We note No and Nc for the number of collected photo-events respectively for on-axis source (magnitude mo) and companion (magnitude $mc=mo+\Delta m$). Using $\eta$ the quantum efficiency, $t_{\rm opt}$ the optical throughtput, (4.RT) the beamsplitting factor and $F_{\rm r}(\lambda)$ the reference flux (zero magnitude at wavelength $\lambda$), S the collecting area, $\Delta\lambda$ the working spectral bandwidth centered at $\lambda$, and $\tau$ the exposure-time we have for a single frame:

$$No=[\eta.4RT.t_{\rm opt}].\frac{F_{\rm r}(\lambda)}{h\nu}.10^{-0.4mo}.\Delta \lambda.S.\tau$$ (29)

$$Nc=No.10^{-0.4\Delta m}.%$$ (30)

The coronagraph gives two twin-images of the companion. For a given $\beta_{0}$, the shape of each image is roughly given by the central lobe of the normalized Airy distribution $A(\beta-\beta_{0})$ weighted by the spatial response of the coronagraph ${w}(\beta)={w}(\rho)$ with $\rho=\vert\beta\vert$ (Eq. 5).

Taking into account the two twin-images, the "signal'' for Mexposures is given by:

$$\mathrm{signal}(\beta_{0})=\frac{Nc}{4}.\sqrt{2}.\sqrt{M}.\int_{O}{w}(\beta)\textrm{ }A(\beta-\beta_{0})\textrm{\ d}\beta$$ (31)

where $\int_{O}$ means sum over the central lobe (that is a disk of radius $1.22\frac{\lambda}{D}$).

The noise originates in the fluctuations of the unwanted illumination in the image plane comprising residual light from incomplete extinction of the on-axis source ($\beta$-dependant) and the contribution of the background. Adding the detector noise contribution we have for a given pixel at $\beta$

$$\mathrm{pixnoise}(\beta)=\left[ \mathrm{var}(no(\beta)+\mathrm{var}_{\rm bg}(\beta)+\mathrm{ron}^{2}\right] ^{1/2}%$$ (32)

where $\mathrm{var}(no(\beta))$ and $\mathrm{var}_{\rm bg}(\beta)$ are the variances of the number of photo-events counted by the pixel at $\beta$, from the residual halo and from the background respectively. Detector contribution is represented by the variance ron² of the readout noise.

The background illumination is from a grey body of emissivity $\mu$and temperature T seen within a solid angle $\omega_{\rm p}$ by a pixel. The corresponding variance of background induced photoevents is:

$$\mathrm{var}_{\rm bg}(\beta)=\frac{\eta.\mu.L_{\lambda}(T).\Delta\lambda.\omega_{\rm p}.s.\tau }{h\nu}%$$ (33)

where $L_{\lambda}(T)$ is the Planck's function and s the area covered by a pixel.

The noise from the residual illumination is a double stochastic process based on Poisson statistics and Rayleigh statistics. From the Mandel's formulae (Goodman [1985]) the associated variance includes two terms: one reflects the Poisson noise at a given level of light $<no(\beta)>$ the other reflects the fluctuations of this level (speckle noise). So we have:

$$\mathrm{var}(no(\beta))=<no(\beta)>+[<no(\beta)>]^{2}.%$$ (34)

For ground based observations the speckle noise is the dominant one, except at a sufficiently low level of light.

The distribution $no(\beta)$ is given by: $\ no(\beta)=g(\beta).No$, where $g(\beta)$ describes the normalized residual energy (Eq. 26).

To evaluate the total noise we have to consider the sum of variances for the pixels covered by the image of the companion. We can only do that directly, if the fluctuations do not correlate from one pixel to another one, which is questionable considering the speckle noise contribution. Indeed, the exposure-time for a single frame is typically much larger than the coherence-time of the atmosphere, so that a correlation between adjacent pixels is likely to be destroyed. Moreover working at large bandwidth leads to an additional decorrelation.

Therefore the SNR for a companion off-axis by $\beta_{0}$ with respect to an unresolved on-axis source can be written as ($\beta$ is a vector and bstands for $\mathrm{var}_{\rm bg}(\beta)$):

$$\mathrm{SNR}(\beta_{0})=$$

$$\frac{Nc}{4}.\frac{\sqrt{2}.\sqrt{M}.\int_{O}{w}(\beta)\textr... ...\mathrm{ron}^{2}% ]\textrm{ }{\rm d}\beta\right] ^{1/2}}\cdot%$$ (35)

The dependance on $\beta_{0}$ for the noise stands in the summation area, that is a disk of radius $a_{0}=1.22\frac{\lambda}{D}$ centered at $\beta_{0}$. This drives us to insert into the integrands the weighting function (pil-box) $\Pi(\frac{\beta-\beta0}{a_{0}})$. Figure 8 and Fig. 9 show examples of expectable detection capabilities (at $RSB(\beta_{0})=5$ for K₀= 5) in terms of detected magnitude differences $\Delta m$ between on-axis source and companion, when adaptive optics provides real time corrections up to radial modes 1, 3, 5, 7, 9. Figure 8 and Fig. 9 refer to complete (fully efficient) and incomplete correction respectively. The following values of instrumental parameters corresponding to the PUEO adaptive optics device already in operation (Beuzit, private communication, [PUEO] internet site) and expected observation conditions for our coronagraph.

Working wavelength $\lambda=2.2~\mu$m, bandwidth $\Delta\lambda=0.5$ $\mu$m, reference flux $F_{\rm r} (\lambda)=3.9$ 10^-14 W cm $^{-2}\mu{\rm m}^{-1}$, telescope diameter D=3.6 m, optical throughput $t_{\rm opt}=0.1$, exposure time (single frame) $\tau=0.1$ s, number of frames M=5000, Fried's parameter r₀=30 cm (that is an $r_{0}(2.2~\mu$m) of 140 cm), R=T=0.5, quantum efficiency $\eta= 0.65$, ron=20 e  pix^-1 exposure^-1, background temperature T=280 K, background emissivity $\mu=0.2$, field of view $\omega_{\rm p}$ corresponding to the central lobe of the Airy pattern of the collecting aperture.

Figure 8: Expected detectable magnitude difference $\Delta m$ for the companion versus its angular distance to the on-axis source, in the case of COMPLETE CORRECTIONS (full efficiency) up to radial modes 1, 3, 5, 7, 9 (the higher the mode, the higher the $\Delta m$). D=12.r0(vis) (with imaging in K). Other values used for observation parameters are given in the text. Unit for angular distance is the Airy radius

Figure 9: Similar to Fig. 8 but here in the case of INCOMPLETE CORRECTIONS up to radial modes 1, 3, 5, 7. Values used are the same as for Fig. 8. Note the change in vertical scale (0 to 10)

From these profiles, it is apparent that even in the case of incomplete corrections (at a level likely to be nowadays negative) detection capabilities of our Achromatic Interfero Coronagraph are worth considering in the study of the stellar environment and similar topics. For example, clear detection of a companion as close as half the Airy radius, exhibiting a magnitude difference $\Delta m$ of the order 6 appears as a reasonable goal in rather ordinary observation conditions and with limited integration time (500 s here). Let us point out that the given estimates correspond to raw data and do not include the significant increase of performance usually brought about by appropriate data processing. Moreover, improvements relating to the correction capabilities of the adaptive optics devices (cut-off frequencies, number of actuators, etc.) are still in progress and the relevant profiles are likely to be close to the ones of the full correction case. The major limitation encountered in the correction process comes from the tilt effect, however there are serious hopes for much better capabilities. Thus, owing to the result of the present theoretical analysis it is conceivable that companions with (roughly speaking) $\Delta m$ up to 12 would be detectable from raw data, as suggested by profiles in Fig. 8. Moreover the reported values of $\Delta m$ can be increased by observing simultaneously (on the same camera) in two separate and adjacent spectral bands covering the total bandwidth of work, which relaxes the problem of the reference source and the possible random variations of the point spread function (Racine et al. [1999]). Of course an additional gain is achievable by using a longer integration time (number of frames) and by using a longer exposure-time for each single frame when allowed by atmospheric seeing and adaptive optics capabilities.

The large $\Delta m$ appearing within the Airy radius in the case of complete corrections (Fig. 8) reflects the behaviour of the residual halo, whose central part becomes larger and larger and darker and darker as the quality of the correction becomes better and better, the ultimate limitation to detection being then essentially photon noise. The bump appearing in the $\Delta m$ profiles about one Airy radius reflects the bump in the field response to extinction (Fig. 3).