3 Departing from the ideal case

Departure from the ideal case occurs in two scenarios: OPD is not zero and complex transmission up to the recombination pupil plane is not centro-symmetric (which relates to both modulus and phase of the complex transmission function).

3.1 Departing from zero OPD

Since the effect of non-zero OPD is a non-zero weighting of the total energy from the on-axis source, we readily have a specification of the tolerance regarding OPD. Using w for the residual energy and w₀ for the energy available when no extinction occurs, we have, using Eq. (1) and the centro-symmetry condition:

$${w}=\frac{w_0}{2}.\left[1-\cos\left(2\pi.\frac{d}{\lambda}\right)\right].$$ (10)

Selecting a given maximum value M for $\frac{w}{w_0}$ we have to control OPD so as to keep $\frac{d}{\lambda}<\frac{\sqrt{M}}{\pi}\cdot$

A value of M=10^-4 maintains $\frac{d}{\lambda}$ < $\frac {1}{400}$. This precision implies that servo-loop of OPD sould be performed. A rms precision of $\frac{1}{200}$ is currently achieved in the visible (Reynaud & Delaire [1993]) and corresponds to better than $\frac{1}{600}$ at $2.2~\mu$m. In addition, monitoring OPD variations during the recording of images with suitable exposure-time allows a selection of "best exposures'' in terms of lower values of M.

3.2 Departing from centro symmetry

In this situation the condition $P(\xi)=P(-\xi)$ no longer applies. Therefore complex amplitudes at the recombination step no longer mutually destroy (Eq. 1), and some energy from the central source is able to reach the image plane.

Departure related to modulus of the complex transmission relates to essentially obscuration and vignetting in the pupil plane (telescope spider for example), and it is ruled out by using a suitable mask in an intermediate pupil plane, restoring a centro-symmetric distribution for the modulus.

Departure related to phase of the complex transmission results from phase distorsions accumulated up to the recombination pupil plane. Phase distribution conveys both deterministic and stable distorsions (aberrations, misadjustment, etc.) and rapidly changing random distortions (atmospheric turbulence). It must be noted that phase distortions described by an even function do not prevent the removal of the on-axis source. This means for example that a defocus and aberrations such as spherical aberration, astigmatism and chromatic aberration still allow the extinction process to work fully. However, they tend to degrade images of the companion, which decreases detection capabilities. On the contrary, such "odd'' aberrations as coma and distorsion lead to an incomplete destructive interference for the on-axis source. This problem also concerns aberrations which could be different in each arm of the device. Indeed, aberrations and misadjustments can be reduced to an acceptable level by proper optical design and control. Thus we will limit our analysis to the case of turbulence, which for ground-based operation is much more severe than the residual instrumental aberrations. This analysis is based on the description of the phase distorsions over the aperture in terms of a weighted sum of Zernike polynomials (Born &Wolf [1970]; Noll [1976]; Roddier [1990]).

Any "odd'' distortion generates a spurious halo, a residual volcano-shaped light distribution around the center of image plane. Extension of this halo remains roughly the one of the focal patch. Note that the trouble is not the halo itself but fluctuations of its shape, whether they result from photon noise or from instabilities of the intensity distribution (speckle noise), this latter being largely dominant for ground-based observations. Noise is not uniformly distributed in the focal plane, since it depends on the local illumination level within the halo.

We consider the case of an on-axis point-like source, with intensity $\Omega_{0}$ leading to the incident field $\psi(\xi)=\omega_{0}$ with $<\vert\omega_{0}\vert^{2}>$ = $\Omega_{0}$. Our starting point is the following Equation describing the complex amplitude in recombining pupil plane with OPD set to zero:

 

$$Q(\xi) \protect\sqrt{R.T}. \Pi\left(\frac{\xi}{D}\right).\omega_{0}.[\exp(i.\Phi(\xi))$$ =

$$-\exp(i.\Phi(-\xi))]$$ (11)

where $\Phi(\xi)$ describes the phase distortions in the recombination pupil plane. Those turbulence-induced distorsions are random and gaussian (central limit theorem). To evaluate the unwanted integrated residual energy we have to work with $\begin{array}[c]{c}% E(\xi)=\ <\vert~\exp(i.\Phi(\xi))-~\exp(i.\Phi(-\xi))\vert^{2}>\!\cdot \end{array}$ Thanks to the Gaussian statistics and using the phase structure function $D_{\Phi}(\xi)$ = $<\vert\Phi(\mu)-\Phi(\mu+\xi)\vert^{2}>$(Roddier [1981]), the normalized integrated energy versus the parameter the parameter p= D/r₀ is given by:

$${w}(p)=\frac{{w}_{0}}{2}.\left\{1-2.\int\nolimits_{0}^{1}\exp[-3.44(p.y)^{5/3}]\,\textrm{\ }y\textrm{ }{\rm d}y\right\}$$ (12)

with ${w}_{\mathrm{0}}=4.R.T.\Omega_{0}.S$ as stated previously (Eq. 5).

This expression shows that the use of real time compensation of wavefront distorsions is mandatory, since even for p=1 we have a residual energy larger than 0.3 w.

3.2.1 Use of real-time correction of phase distortions

In the following analysis we consider the use of real-time wavefront compensation by adaptive optics for both the integrated residual energy and the residual energy distribution. The first provides only a useful numerical figure to summarize the performance relating to extinction and serves to compare various situations. The second is needed to evaluate the Signal to Noise Ratio (SNR) pertaining to the detection of a companion, since this SNR depends on the location of the companion in the image plane via the level of residual light at this location. So, we need to express analytically the distribution in the image plane and not only its integral. A convenient way to express analytically this distribution is to use the small phase distortions hypothesis, leading to $% \begin{array}[c]{c}% \exp(i.\Phi(\xi))\approx1+i.\Phi(\xi) \end{array}$. The validity of this assumption will be checked later, but initially, we can refer to the fact that the largest part of the phase distortion effects comes from the tip-tilt of the wavefront (Roddier [1990]). Since this defect is the first which is compensated for in any adaptive optics device, we believe our assumption is relevant.

We first describe the amplitude at the recombination step in the pupil plane, then we evaluate the integrated residual energy and finally we express the residual light distribution in the image plane.

To describe the corrected amplitude at the recombination pupil plane, our starting point is Eq. (11) but since we consider real-time correction of phase distorsions we now deal with a residual phase distribution $\Phi_{\rm r}(\xi)=\Phi(\xi)-\Phi_{\rm c}(\xi)$ where $\Phi(\xi)$ is the phase function and $\Phi_{\rm c}(\xi)$ represents the applied corrections. Assuming that $\Phi_{\rm r}$ is small leads to a new expression for the amplitude $Q(\xi)$:

$$Q(\xi)=\omega_{0}.\sqrt{R.T}.i.\Pi\left(\frac{\xi}{D}\right).[~\Phi_{\rm r}(\xi)-~\Phi _{\rm r}(-\xi)]$$ (13)

showing that only the odd part of $\Phi_{\rm r}(\xi)$ is to be taken into account. The even part is automatically corrected "by design''. Introducing $\Phi _{\rm or}$ for odd residual phase distorsions over the pupil of diameter D (the expression of the amplitude transmission $\Pi(\frac{\xi}{D})$ is now absorbed in $\Phi _{\rm or}$) we write:

$$Q(\xi)=\omega_{0}.2.\sqrt{R.T}.i.\Phi_{\rm or}(\xi).$$ (14)

Thus, describing $Q(\xi)$ we still need to describe $\Phi_{\rm or}(\xi)$. It is usual to describe a phase distribution $\Phi(\xi)$ over a pupil of radius $R_{\rm p}$ by a weighted sum of Zernike polynomials (Noll [1976]; Roddier [1990]; Conan [1995])

$$\Phi(q,\phi)=\sum\limits_{j}aj\textrm{ }Zj\left(\frac{q}{R_{\rm p}},\phi\right).$$ (15)

Here we use the notation introduced by Noll, where j is a synthetic index based on classical indexes n and k (Born & Wolf [1970]) undergoing usual restrictions (n integer,$\vert k\vert\leq n$, n-|k| = even):

$$\begin{eqnarray*}Z_{\rm evenj}~=~\sqrt{n+1}.R_{nk}(q).\sqrt{2}.\cos(k.\phi)\hfil... ...m}&k\neq0 \\ Z_{j}=\sqrt{n+1}.R_{n0}(q)\hfill&\hspace*{1cm}&k=0 \end{eqnarray*}$$

the R_nk are radial polynomials, defined for $% \begin{array}[c]{c}% 0\leq q\leq1 \end{array}$, by the relation:

R_nk(q)=

$$\sum_{s=0}^{(n-k)/2}\frac{(-1)^{s}(n-s)!}{s!\textrm{ }% [(n+k)/2-s]!\textrm{ }[(n-k)/2-s]!}\textrm{ }q^{n-2s}.%$$ (16)

Using the pupil distribution $W(\frac{q}{R_{\rm p}})$ giving $\frac{1}{\pi}$ when $% \begin{array}[c]{c}% q\leq R_{\rm p}% \end{array}$ and else zero, the following orthogonality relation holds (where we use $R_{\rm p}$ =D/2 and the Kronecker symbol $\delta_{jj\prime}$)

$$\int_{0}^{2\pi}\int_{0}^{R_{\rm p}}Z_{j}\left(\frac{q}{R_{\rm... ...,\phi\right)\textrm{ }q{\rm d}q{\rm d}\phi=\delta_{jj\prime}.%$$ (17)

For a given phase distribution $\Phi$, the weighting coefficients a_j are retrieved by the relation:

$$a_{j}=\frac{1}{R_{\rm p}^{2}}.\int_{0}^{2\pi}\int_{0}^{R_{\rm... ...\frac{q}{R_{\rm p}},\phi\right)\textrm{ }q{\rm d}q{\rm d}\phi.$$ (18)

Applying corrections up to j=J, leads to the residual phase distribution:

$$\Phi_{\rm r}(q,\phi)=\Phi(q,\phi)-\sum_{j=1}^{J}a_{j}\textrm{ }Z_{j}\left(\frac{q}{R_{\rm p}% },\phi\right).$$ (19)

This expression is equivalent to a sum of the a_j.Z_j running from j=J+1 up to j infinite. In our concept, only the odd part of the residual phase distribution is to be counted, so that in $\Phi _{\rm or}$ (Eq. 15) some j's are taken out of this infinite summation.

Let us point out that restricting the summation to the odd part of $\Phi_{\rm r}$ does not mean that we restrict the sum to odd values of j, but rather that we restrict it to odd values of n, the radial polynomial order (and therefore to odd values of k). This, for example, automatically eliminates such defects as defocus, 3rd order astigmatism, 3rd order spherical, etc. as already mentioned.

If we want to take into account the case of incomplete correction we write:

$$\Phi_{\rm or}(q,\phi)=\sum_{j}p_{j}\,\textrm{\ }a_{j}\textrm{ }Z_{j}\left(\frac{q}{R_{\rm p}% },\phi\right)$$ (20)

where we have introduced the weighting parameter p_j (presence of the defect of index j) to which is assigned the value 0 when correction is fully effective for j, the value 1 when order is not corrected and an intermediate value when correction is not fully effective. For values of jcorresponding to even phase defects, p_j automatically has the value 0.

3.2.2 Integrated residual energy under adaptive optics correction

When correction is performed up to index J, the residual energy w_J in image plane can also be found by integrating the intensity in pupil plane:

$$w_{J}=\int<\vert Q(q,\phi)\vert^{2}>q{\rm d}q{\rm d}\phi.$$ (21)

Let us disregard for a while the automatic cancellation of even phase distortions, and let us introduce a coefficient $\varepsilon_{j}$(effectiveness of the correction) defined by $\varepsilon_{j}^{2}=1-p_{j}^{2}%$. Taking into account the weighted Zernike sum and the orthogonality relation we have:

$$w_{J}=4.R.T.\Omega_{0}.\pi.R_{\rm p}^{2}.\left[<\Phi^{2}>-\sum_{j}\varepsilon_{j}% ^{2}<\vert a_{j}\vert^{2}>\right].$$ (22)

For a correction device working up to J, we have a residual energy g_Jnormalized on the total energy collected on the star (given by ${w_{0}}= 4.R.T.\Omega_{0}.\pi.R_{\rm p}^{2}$) so we have:

$$g_{J}=~<\Phi^{2}>-\sum_{j}\varepsilon_{j}^{2}<\vert a_{j}\vert^{2}>\cdot$$ (23)

This expression is quite similar to the one derived by Noll ([1976]) for the residual variance $\Delta_{J}$ of the phase over the pupil, when correction is achieved up to (and including) j=J. The departure from Noll's coefficients comes from the energy removal efficiency $\varepsilon_{j}^{2}$ applied to each term of the sum. Correcting up to the radial order N corresponds to a correction up to $J=\frac{(N+1).(N+2)}{2}$. The value N=7 has been effectively reached on a real device (Conan [1995]) which corresponds to J=36that we adopt to evaluate a realistic residual energy. For J greater than 10 (N>3) the residual variance can be approximated by $% \begin{array}[c]{c}% \Delta_{J}=0.2944.J^{-a}.(\frac{D}{r_{0}})^{5/3}% \end{array}$ with $a=\frac{\sqrt{3}}{2}$ (Noll 1976) that we can use to evaluate the residual energy g_J resulting from complete corrections up to the mode J. Since distortions with even radial orders are automatically corrected, the g_J's are always less than the Noll's $\Delta_{J}$'s when applied corrections are fully effective as shown in Fig. 6. For a complete correction with J=36 we have $% \begin{array}[c]{c}% g_{36}\approx0.006(\frac{D}{r_{0}})^{5/3}\!.% \end{array}$

So, for a 150 cm telescope diameter and a r₀ of 10 cm (leading to 53 cm at 2.2 $\mu$m) we have $g_{36}\approx0.034$ at 2.2 $\mu$m, which corresponds to a rejection by $\ 3.67$ magnitudes of the unwanted energy, integrated over the whole image plane.

In the more realistic case of incompletely achieved corrections, we use values of $\varepsilon_{j}^{2}$ taken from numerical simulations based on measured performance of a real servo-correction device (Conan [1995]). Values for $\varepsilon_{j}^{2}$ are shown in Table 1 (for even orders n, efficiencies $\varepsilon_{j}^{2}$ are automatically set to unity, they are not shown beyond n=7 that is j=36)

 

 

Table 1: Correction efficiencies used for radial orders up to n=7 (that is j=36). Even orders remain fully corrected beyond orders displayed in the table

j	1	2, 3	4, 5, 6	7-10	11-15	16-21	22-28	29-36
n	0	1	2	3	4	5	6	7
${\scriptsize\varepsilon}_{j}^{2}$	1	0.995	1	0.9	1	0.7	1	0.6

In this case, we define a figure g_J^u similar to g_J (u for incomplete) and we have $% \begin{array}[c]{c}% g_{36}^{u}\approx0.016(\frac{D}{r_{0}})^{5/3}.% \end{array}$ In the same situation as above we have now an integrated rejection of roughly 2.6 magnitudes.

Figure 6: Noll's residual variances (solid line) and normalised integrated residual energy in the image plane of the coronagraph versus J, the index of the upper corrected mode. Dotted line: incomplete correction. Dashed line: complete correction. This curve constantly runs below the Noll's variances, due to the automatic correction of even modes

It must be kept in mind that these numerical figures pertain to the total unwanted energy in the image plane, not to the one found at the location of an image of companion. So they must not be interpreted as the limiting magnitude difference that our coronagraph is able to work with. A more relevant evaluation of detection capabilities needs the energy distribution in the image plane to be expressed.

3.2.3 Energy distribution in the image plane

It is more convenient here to come back to the expression $% \begin{array}[c]{c}% \ Q(\xi)=\omega_{0}.\sqrt{R.T}.2.i.\sum\limits_{j}p_{j}\textrm{ }a_{j}\textrm{ }Z_{j}(\xi) \end{array}$ where p_j traces the presence of the defect associated with j and where we have an infinite sum.
The residual energy distribution $I_{\rm halo}$in the image plane (polar coordinates $\rho$ and $\theta$) is given by the square modulus of the FT of $Q(q,\phi)$ and we have (using $\hat{Z}$ for the FT of Z):

$$I_{\rm halo}(\rho,\theta) =4.RT.\Omega_{0}.\pi^{2}.R_{\rm p}^{4}.$$

$$.\sum_{j;j\prime}p_{j}\textrm{ }p_{j\prime} <a_{j}.a_{j\prime}^{\... ...}_{j}(R_{\rm p}.\rho,\theta)\textrm{ }\hat{Z}_{j}^{\ast}(R_{\rm p}.\rho,\theta)$$ (24)

and the residual energy normalized on the collected energy is:

$$g_{\rm halo}(\rho,\theta) =\sum_{j;j\prime}p_{j}\textrm{ }p_{j\prime}% <a_{j}.a_{j\prime}^{\ast}>.$$

$$.\hat{Z}_{j}(R_{\rm p}.\rho,\theta)\textrm{ }\hat{Z}_{j}^{\ast }(R_{\rm p}.\rho,\theta)$$ (25)

which is more conveniently expressed in using n and k instead of j:

 

$$g_{\rm halo}(\rho,\theta) =\sum_{nk;n\prime k\prime}p_{n}\textrm{ }p_{n\prime }<a_{nk}.a_{n\prime k\prime}^{\ast}>.$$

$$\textrm{ \ \ \ \ \ \ \ \ }.\hat{Z}_{nk}(R_{\rm p}.\rho,\theta)\textrm{ }\hat {Z}_{n\prime k\prime}^{\ast}(R_{\rm p}.\rho,\theta)).$$ (26)

To evaluate $g_{\rm halo\textrm{ }}$ we use for $\hat{Z}_{nk}$ the expression (Born & Wolf [1970]):

$$\hat{Z}_{nk}(R_{\rm p}.\rho,\theta) =\pi.R_{\rm p}^{2}.\sqrt{n+1}.(-1)^{(n-k)/2}% .i^{-k}.$$

$$\textrm{ \ }.\frac{2.J_{n+1}(2\pi.R_{\rm p}.\rho)}{2\pi.R_{\rm p}.\rho}% .\exp(i.k.\theta)$$ (27)

and for < $a_{nk}.a_{n\prime k\prime}^{\ast}>$ according to Noll ([1976]) as above, we use:

$$<a_{nk}.a_{n\prime k\prime}^{\ast}>=\frac{0.046}{\pi.2^{5/3}}.\left(\frac {D}{r_{0}}\right)^{5/3}.$$

$$\textrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }.\sqrt{n+1}.\sqrt {n\prime+1}.(-1)^{(n+n\prime-2.k)/2}.\delta_{kk\prime}.$$

$$\textrm{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }.\int\rho^{-14/3}\textrm{ }J_{n+1}(2\pi.\rho)\textrm{ }J_{n\prime+1}(2\pi\rho)\textrm{ }{\rm d}\rho$$ (28)

where it is apparent that only remain terms with $k=k^{\prime}$.

Figure 7 shows examples of radial profiles of the residual halo when full correction is applied up to radial orders 3, 5, 7, 9 successively (the higher the order the fainter the halo). Taking into account corrections beyond order 9 would induce fainter and fainter contributions of the halo, raising it further and further from the center of the field.

This behaviour is interesting not only for reducing the level of unwanted energy but also for cleaning progressively from the center, which improves the close-sensing capabilities as early as the lower order is corrected.

Figure 7: Radial profiles of residual intensity distribution $g_{\rm halo}$ for $\frac{D}{r_{0}}({\rm vis})=10$ (with imaging in K) normalized on the maximum of the Airy pattern without coronagraph, in the cases of correction of radial orders n's up to 3, 5, 7, 9. The higher the n, the lower the profile and the wider the central darkening. Horizontal scale: Angular distance from the center of the field with Airy radius as the unit