5 Chromatic effects: Discussion on the possibility of broad-band observations

The interferometric technique we have described in Sect. 2 to obtain the cosine apodizations is wavelength sensitive. Since the size of the diffraction pattern in the focal plane increases with the wavelength, shifts and mask sizes should vary accordingly. If not, and such is the case that we analyse here, the parameters a and b are only optimized for a single wavelength $\lambda_0$ within the bandwidth.

The computations made previously can be used to compute chromatic effects. For that, we simply assume that at a wavelength $\lambda$ we are using inappropriate $a(\lambda)$ and $b(\lambda)$ values of the form:

 

$$b(\lambda)=\frac{\lambda}{\lambda_0}\;b_0, \quad a(\lambda)=\frac{\lambda_1}{\lambda}\; a_1$$ (12)

where a₀ and b₀ are the optimal values at $\lambda_0$.

For pure apodization, the resulting PSF is obtained integrating $\vert\Psi_b^{(N)}(x)\vert^2$ of Eq. (5) within the spectral window:

 

$$I_{\Delta\lambda}(x)=\int_{\lambda}^{\lambda+\Delta\lambda}\vert\Psi_{b(\lambda)}^{(N)}(x)\vert^2 \; {\rm d}\lambda.$$ (13)

In that relation, $b(\lambda)=\lambda/\lambda_0$ since b₀=1 (pure apodization). Examples of the results for different spectral windows are shown in Fig. 10. The chromatic effect degrades the result, as expected, but it is not as significant as one could fear. The curves can be compared to the monochromatic simulations in Fig. 3.

Figure 7: Residual Field Intensity in the final focal plane for a one-dimensional aperture. Top: PSF for a raw aperture without apodization nor coronagraphy (full line), Lyot's coronagraphy with a cosine apodizer (dash-dotted). Lyot's coronagraphy with a cosine squared apodizer (dotted). R&R's coronagraphy with a cosine apodizer (long-dashed)). Bottom: Lyot's coronagraphy is plotted at the second minimum (see Table 2 for the corresponding parameters).

A solution to the chromatic problem would be to use an achromatization lens-system which produces a magnification of the focal image proportional to $1/\lambda$ over the bandpass. These kind of achromatizers have been successfully tested for pupil plane interferometry (Wynne 1979; Roddier et al. 1980). In that case, the image of the planet will appear as a dispersed spectrum. For very wide bandwiths, such an achromatization technique would probably fail and a separation into several bands is certainly needed to adapt the achromatization in each band.

On the other hand, classical apodizers using absorbing devices also present technological challenges to realize the exact transmission function. The physical absorption process is wavelength dependent, and chromatic problems may also occur; multilayers may be needed to obtain an achromatic transmission over a wide bandpass. With the interferometric technique, the apodization function (cosine, cosine squared) is physically produced by the interference phenomenon itself and may be exactly obtained.

For PM coronagraphy, the first difficulty is to obtain a wavelength independent $\pi$ phase mask. Assuming that this result is obtained, we can compute the effect relative to the variation of the parameters a and b within the spectral bandwidth. Curves of Fig. 4 can be used for that: the residual intensity is the result of the integration along the parametric curve defined by equations $a(\lambda)$ and $b(\lambda)$ of Eq. (12). Examples are given in Fig. 11, for the one-dimensional case, where we consider that a perfect $\pi$phase mask is used.

The chromatic effect on the mask size is much more sensitive in the R&R case than in Lyot's case. This can be understood easily from Fig. 5. R&R's coronagraphy may give even worse results than Lyot if the chromatic effect on the mask size is not corrected. On the contrary, R&R coronagraphy is not very sensitive concerning the chromatism of the apodization, because the optimal apodization function corresponds to a very weak apodization (partial cosine arch)

Figure 8:   Residual intensity of the star in the focal plane for the 4 considered techniques: Top left: Lyot's coronagraphy with cosine apodization (at first minimum). Top right: Lyot's coronagraphy with cosine squared apodization (at first minimum). Bottom Left: R&R's PM with cosine apodization. Bottom right: R&R's PM with cosine squared apodization.

Figure 9: PSF for the planet (the effect of coronagraphy on the off-axis planet is assumed to be negligible), PSF for R&R with a cosine apodization (full line), PSF for R&R with a cosine squared apodization (long dash), PSF for Lyot with a cosine apodization (dashed line), PSF for Lyot with a cosine square apodization (dotted line)

Figure 10: Effect of increasing bandwidth on the PSF for the interferometric apodization (N=1) and (N=2): achromatic PSF, effect of a $10\%$ bandwidth, $20\%$ bandwidth,$30\%$ bandwidth.

Figure 11: Residual energy as a function of the bandwithexpressed in percents (this computation is made for the one-dimensional case). Top: effect of mask size chromatism andinterferometric apodization chromatism. R&R is much more sensitive to the mask size and can be worse than Lyot for a bandwidth over $10\%$. Bottom: effect of interferometric apodization alone (mask size chromatism corrected). The R&R's technique is less sensitive to the chromatism of the apodizer.