next previous
Up: Interferometric apodization of rectangular


Subsections

4 Application of interferometric apodization to stellar coronagraphy

4.1 Needs of entrance pupil apodization in stellar coronagraphy

The purpose of coronagraphy is to reduce the star diffracted light in the whole field to provide direct imaging capabilities of faint sources in the vicinity of the star. For the sake of clarity, we briefly recall some fundamental aspects of this otherwise very well-known technique. In Lyot's coronagraphy (Lyot 1930) applied to exoplanet detection, an opaque mask is set at the center of the stellar image. According to Babinet's theorem, the diffraction pattern of this dark mask appears in negative amplitude on an image of the entrance-pupil. There, a diaphragm (Lyot's stop) is set to get rid of the light diffracted outside the telescope aperture image. Inside this diaphragm, direct and diffracted light of the star are interfering destructively, while the light coming from the planet remains almost unaffected. However, because the smooth diffraction pattern of the mask cannot fit the flat impinging wavefront, the starlight is not fully cancelled. Typically, an integrated reduction factor of the order of 10-3 may be obtained using an optimized Lyot mask for a square aperture. The technique was improved by Roddier & Roddier (1997) and Guyon et al. (1999), replacing the opaque mask by a $\pi$ Phase Mask (PM): the performance can be enhanced while the principle remains the same.

Most of the star residual light appears at the edges of the pupil in the relay pupil plane. In the original technique, the Lyot's stop diameter is smaller than the pupil size to reduce this effect. A classical improved solution is to use optimized entrance pupil apodization. We examine here the effect of the apodized apertures permitted by the technique proposed in the previous section.

  \begin{figure}
\par\includegraphics[width=9cm,clip]{MS1697f4.eps} \end{figure} Figure 4:  Residual energy as a function of the mask size a and apodization parameter b for a square aperture. The value b=1 corresponds to full cosine apodization (Sect. 2). Higher values of b correspond to a partial apodization. For Lyot's coronagraphy several minima are obtained for different a and b values. See Table 2 for numerical values.


 

 
Table 2: Results of the analytical optimization for cosine apodizers and coronagraphy.
Technique Apodization Mask size: Residual energy Apodizer intensity transmission
  parameter: b $a\times L$ (square aperture) (square aperture)
R&R, $\cos$ (N=1) 2.160 0.848 $E=1.14 \times10^{-6}$ 70%
R&R, $\cos^2$ (N=2) 2.995 0.848 $E=3.30 \times10^{-8}$ 70%
Lyot, $\cos$ (N=1) 1.103 2.147 $E=6.67 \times10^{-4}$ 30%
Lyot, $\cos^2$ (N=2) 1.254 2.621 $E=2.95 \times10^{-5}$ 22%
Lyot 2nd min. (N=1) 1.056 4.040 $E=1.51 \times10^{-4}$ 30%
Lyot 2nd min. (N=2) 1.154 4.468 $E=4.15 \times10^{-6}$ 19%
ASA $\cos$ (N=1) 1     25%
ASA $\cos^2$ (N=2) 1     14%


For a rectangular aperture of size $L_x \times L_y$, the coronagraphic mask must be a rectangle of size $a_x \times
a_y$ proportional to the inverse of the aperture. Following the approach of Guyon & Roddier (2000) and Aime et al. (2001), the residual amplitude within the entrance aperture left by the coronagraphic process can be written as:

 
$\displaystyle \Psi(x,y)=P_X^{N}\left(x\right)P_Y^{N}\left(y\right)-\varepsilon\; F_X\left(x\right)F_Y\left(y\right)$     (10)

where $\varepsilon=1$ for Lyot's coronagraphy and $\varepsilon=2$ for Roddier's coronagraphy and $P_X^{N}\left(x\right)$ is the entrance apodized pupil of Eq. (9). The function $F\left(x\right)$ is the convolution product of the aperture and the diffraction pattern of the mask:
 
$\displaystyle F_X\left(x\right)=P_X^{N}\left(x\right)\ast \frac{\sin \pi a_x x}{\pi x}\cdot$     (11)

Analytic expressions for $F_X\left(x\right)$ are given in Appendix A. Corresponding values of ax, ayand bx, by depend on the sizes of Lx and Ly.

The case for entrance pupil apodization combined with coronagraphy appears clearly in Eq. (10): the idea is to optimize the pupil shape to achieve the best amplitude subtraction for the on-axis unresolved star. In other words we shall try to solve the equation $\Psi(x,y)=0$ within the aperture.

This solution can be approached very closely by using an iterative numerical algorithm as first proposed by Guyon & Roddier (2000): iterations are computed between the entrance and exit pupil planes, and the residual amplitude is negatively re-injected in the entrance pupil. The apodization functions so obtained do not resemble to the full cosine apodized aperture (b=1), but are very similar to the central part of a cosine arch.

  \begin{figure}
\par\includegraphics[width=8.8cm,clip]{MS1697f5.eps}\end{figure} Figure 5: Slices of the energy for different values of apodization parameter b (full lines correspond to the optimal b of Table 2).

We recently noticed that there is a formal solution to this problem (Aime et al. 2001, submitted) and that the cosine arch is indeed a very good approximation to a perfect apodization. In any case, as we shall see below, it permits the aperture to give very good stellar light rejection.

4.2 Residual energy for Lyot's and R&R's optimized apodized rectangular apertures.

We consider here a square aperture for which ax=ay=a and bx=by=b. A simple optimization criterion to test the efficiency of the coronagraphic experiment is to consider the integrated residual energy within the exit pupil, for the unresolved on-axis star. Note that since coronagraphy is a direct imaging technique, this criterion is rather pessimistic. Using minimisation techniques, the optimal mask size a and apodization parameter b can be numerically computed. The results are given in Table 2 for a square aperture, and a few details on the computations are given in Appendix B.

It may be also interesting to represent the variations of this energy as a two dimensional function of a and b. This is done in Fig. 4, for a square aperture, and may be more visible in Fig. 5 where slices of the functions of Fig. 4 are given.

Lyot's and R&R's techniques have quite different behaviours: there is a single well-defined global minimum for the PM technique, whereas successive several local minima exist for Lyot's coronagraphy as the mask size increases. The optimal b value is different if the solution is searched at the first or second minimum. This means that for a given shape of apodization, there is an optimal mask size, and is reciprocally so. The second optimal value gives a much better extinction ratio, at the expense of a reduced capability of detecting a planet very close to the star (larger a). These effects are visible in Figs. 45 and in Table 2.

Since most of the residual light is concentrated at the edges of the exit pupil, even with entrance pupil apodization, we can take advantage of the initial Lyot's method and use a smaller exit pupil than the entrance pupil. We have empirically tested a few Lyot's stop reductions $(90\%,80\%)$. The optimization results are different when carried out with a reduced Lyot's stop and can provide a better extinction inside the reduced exit pupil. However, a detailed study of the Signal to Noise Ratio (SNR) would be needed to determine the optimum exit pupil reduction.

A representation of the residual intensities $\vert\Psi(x,y)\vert^2$ of Eq. (10) is given in Fig. 6 for the optimal values of a and b. The residual field intensity (RFI) for the optimal cosine apodizations with coronography are illustrated in Fig. 7. These RFI expressions are not analytic and have been computed numerically. The results, summarized in Table 2, lead to the following comments:

4.3 PSF for the planet

A PSF for the planet can be computed if we assume that the coronagraphic mask (Lyot's mask or R&R's phase mask) has almost no effect on the re-imaged planet whatever its position in the field. For that we can use Eq. (2) and Eq. (3) i.e. the sum of two or three shifted sine cardinal functions. An illustration is given in Fig. 9. Note that here the smaller the PSF, the better the result, since the detection of the planet will be made with less background noise.

  \begin{figure}
\par\includegraphics[width=12cm,clip]{MS1697f6.eps}\end{figure} Figure 6: Top: entrance apodized pupils (amplitude transmission). Bottom: residual intensity in the exit pupil plane. From left to right: R&R with cosine, R&R with cosine squared, Lyot with cosine, Lyot with cosine squared.


next previous
Up: Interferometric apodization of rectangular

Copyright ESO 2001