4 Comparison between square and cicular prolate apodized aperture with Lyot coronagraph

Figure 5:  Amplitude aperture transmission (left) and PSF (right), represented in Plane B, for Roddier & Roddier coronagraphy, and a circular (top) or square (bottom) aperture. The $\pi$ phase mask is shown as a black square and disk. Both aperture can perform a total nulling of the star light. Mask sizes are: 1.06/Dfor the disk diameter and 0.85/L=0.96/D for the length of the square. The circular aperture presents a slightly better telescope throughput ($73\%$ instead of $70\%$). The intensity scale for the PSF is not linear, but identical for the 2 figures.

In this section, we compare the technique of coronagraphy with prolate apodization for square or circular apertures of same collecting surface S, i.e. a square aperture of length $L=D~\sqrt{\pi}/2$.

With Roddier & Roddier coronagraphy, both shapes can perform a total extinction: in the case of a square aperture, the apodization solution consists of a product of linear prolate functions whilst for the circular aperture, the apodization is a circular prolate function. An illustration of the solutions for Roddier & Roddier is given in Fig. 5: the apodization (in amplitude) and the impulse response without coronagraph (or the response for a planet well outside the mask). The corresponding phase masks are also shown in the figures as a black disk and square. The surface of the circular mask (0.69/S)is slightly smaller than that of the square mask (0.72/S). The throughputs are similar, with again a slight advantage for the circular solutions ($73\%$ vs. $70\%$).

The comparison is also interesting for Lyot coronagraphy. Let us first remind the expression of the residual star intensity for a square aperture and a Lyot coronagraph, already given by Aime et al. (2002). We assume a symmetrical configuration for the square aperture of length L (square coronagraphic mask of size m with the same eigenvalues $\Lambda_x=\Lambda_y=\Lambda_{\rm s}$):

$$\vert{{{\Psi }_{\rm D}}(x,y)}\vert^2 = {\left(1-{\Lambda}^2_... ...ft(\frac{L}{m}~x\right)~ {\Phi }^2 \left(\frac{L}{m}~y\right).$$ (30)

The residual star intensity is then itself a product of linear prolate functions and the spatial intensity is not radially uniform in the final focal plane. Compared to the circular aperture, the square aperture produces a lower residual diffracted level along a diagonal cut but a higher level along an axial cut. A comparison of residual intensities of the two configurations is made in Figs. 6 and 7, for an arbitrary identical throughput of $30\%$. In that case, the mask sizes are the following: the diameter of the disk is 2.6/D and the length of the square is 2/L (or 2.26/D for the square aperture of same surface. The surface of the disk is slightly larger than that of the square (about 4%). The shapes of the residual intensities are quite different.

This can be explained easily as follows: comparing Eqs. (23) and (30), we see that the coronagraphic effect, for an on-axis point source, is simply the multiplication of the focal PSF without coronagraph by the factors ${\left( 1 - {\Lambda}_{\rm s}^2 \right) }^2$ and ${\left( 1 - {\Lambda }_{\rm c}\right) }^2$, respectively for a square and circular aperture. These values are different for a same throughput. This is clearly visible in Fig. 6 where the residual level intensity at the center of the pattern are $1.4\times10^{-3}$ for the square aperture and only $5.9 \times 10^{-4}$ for the circular one. Then the behavior of the residual light is given by the telescope PSF: the square aperture presents a lower residual intensity in four quadrants, but two strong diffraction patterns along its main axes.

Figure 6: Comparison of the residual star intensities (in log scale) for Lyot coronagraphy with prolate apodizations for square apertures (dashed lines) and circular apertures (full lines identical in the two figures). Curves are drawn for apertures of same surface and an identical throughput of 30%. Top: circular aperture and square aperture (axial cut). Bottom: circular aperture and square aperture (diagonal cut).

Figure 7: Comparison of the residual star intensities for Lyot coronagraphy with prolate apodizations for square apertures and circular apertures. The two apertures have the same surface, and the same intensity throughput $T=30\%$ of the apodizer. The square aperture has a lower residual intensity along a diagonal cut, but a higher residual intensity along an axial cut, than the circular aperture which is radially uniform.

The fractional wings energy, introduced above (Eq. (26)) can also be used as a criterion for a comparison between the residual diffracted levels for the circular and square apertures. The expressions for the fractional wings energy are respectively:

 

$${e_{\rm wings}^{(1)}} {\left( 1 - {\Lambda }_{\rm c} \right) }^3;$$ =

$${e_{\rm wings}^{(3)}} {\left( 1 - {\Lambda}_{\rm s}^2 \right) }^3.$$ = (31)

Note that the eigenvalues for the square ( $\Lambda_{\rm s}$) and circular ( $\Lambda_{\rm c}$) aperture are different for a same intensity throughput T. For example, for $T=50\%$, $\Lambda_{\rm s}=0.89$ and $\Lambda_{\rm c}=0.82$. For a given transmission T, we compare the two fractional wings energy of Eq. (31) in Fig. 8.

Figure 8: Comparison for Lyot coronagraphy with prolate apodizations between square apertures and circular apertures (same surface) as a function of the throughput of the entrance apodization. The curve represents the ratio between the fractional wings energy for the square and the circular apertures. The circular aperture gives a lower residual energy in the diffraction wings, especially with increasing eigenvalues (respectively decreasing throughput).