2 PSF reconstruction: Formalism

2.1 The adaptive optics system

The adaptive optics system consists of three major parts: the wavefront sensor (WFS), the real-time processor and the deformable mirror (DM). Figure 1 gives a schematic view of an AO system. The incident turbulent wavefront is corrected by the DM, then splitted into two parts. One part forms the image on a detector, the other one falls on the WFS which measures the degree of the residual wavefront deformation. A real-time processor transforms the WFS measurements into a signal which is sent to the deformable mirror to readapt the surface of the mirror to the turbulent wavefront.

Figure 1: Schematic view of an adaptive optics system

2.2 Modal description of the phase

 

 

Table 1: Notations used in this paper

AO	adaptive optics
WFS	wavefront sensor
DM	deformable mirror
	mirror space
	turbulent phase
	mirror phase configuration
	residual phase
	phase projected on
	phase perpendicular to
N	number of mirror modes
	wavefront sensor measurements
	WFS measurement noise
	turbulent modal coefficients
	residual modal coefficients
	modal mirror commands
	remaining error
	measurement noise projected onto
	the modes
	modal gains
	correction transfer function
	close loop transfer function
	noise transfer function
D	interaction matrix between mirror
	and WFS
D+	control matrix
	operator describing the WFS

For astronomical applications the near-field approximation holds (Roddier 1981) which means that the amplitude of the light complex field can be considered as constant over the telescope's pupil. Hence, only the phase variations will affect the quality of the image. It is useful to express the phase on a basis of eigenmodes ,

$$\Phi \left( \vec{r},t\right) =\sum ^{\infty }_{i=1}z_{i}\left( t\right) Z_{i}\left( \vec{r}\right) \; .$$ (1)

The most common basis of modes are the Zernike polynomials (Noll 1976) and the Karhunen-Loeve functions (Wang & Markey 1978). Another basis of eigenfunctions is obviously the basis of mirror modes which of course is finite and can therefore only express the low-frequency part of the turbulent phase (for frequencies typically less than the inverse of the distance between two actuators). We will call this basis . is the phase function generated by the mirror commands m_i(t) and the mirror modes :

$$\phi _{{\rm m}}(\vec{r},t)=\sum _{i=1}^{N}m_{i}(t)\, M_{i}(\vec{r})\, \, \, \, \, \, \, M_{i}(\vec{r})\in {\cal M}\, ,$$ (2)

where N is the number of the mirror modes. The residual phase, the phase after the AO correction, is given by:

$$\Phi _{\epsilon }\left( \vec{r},t\right) =\Phi _{\epsilon _{\... ...,t\right) +\Phi _{{\rm a}_{\bot }}\left( \vec{r},t\right) \, ,$$ (3)

where

$$\Phi _{\epsilon _{\parallel }}\left( \vec{r},t\right) =\Phi _... ...eft( \vec{r},t\right) -\Phi _{{\rm m}}\left( \vec{r},t\right)$$ (4)

is the low-frequency part of the turbulent phase , partially corrected by the system:

$$\phi _{\epsilon _{\parallel }}(\vec{r},t)=\sum _{i=1}^{N}\epsilon _{i}(t)\, M_{i}(\vec{r})$$ (5)

and is the high-frequency part of the turbulent phase which is not affected by the AO correction at all. Let be the projection of onto :

$$\phi _{{\rm a}_{\parallel }}(\vec{r},t)=\sum _{i=1}^{N}a_{i}(t)\, M_{i}(\vec{r})\, .$$ (6)

We can express in vector form:

$$\vec{\epsilon }_{\parallel }(t)=\vec{a}_{\parallel }(t)-\vec{m}(t)\, ,$$ (7)

where , and . For a "perfect'' correction, we would have . Table 1 summaries the notations used in this paper.

The ADONIS system possesses two mirrors for the wavefront correction: a plan mirror for the tip/tilt correction and a deformable mirror with 52 actuators (piezo-stack elements). Due to invisible and redundant modes, only 50 modes are corrected (Gendron & Léna 1994).

2.2.1 The wavefront sensor

The wavefront sensor (WFS) of the ADONIS system is a Shack-Hartmann device. It is a grid of sub-lenses placed in the conjugate plane of the telescope pupil. Due to the partial shielding by the secondary mirror, the system uses only 32 out of 49 sub-apertures. Each lens forms a spot on a detector whose location depends on the average phase gradient over the sub-aperture,

$$\vec{w}(t)\propto \int _{{\rm sub}}\nabla \Phi _{\epsilon }\left( \vec{r},t\right) {\rm d}\vec{r}\, .$$ (8)

The relationship between the WFS measurements and is supposed to be linear. The interaction matrix D describes the relationship between the low-order modal coefficients and the slope measurements . The expression for the WFS measurements is then given by:

$$\vec{w}(t)=D\, \vec{\epsilon }(t)+{\cal W}[\phi _{{\rm a}_{\perp }}(\vec{r},t)]+\vec{n}_{w}(t)\, .$$ (9)

The first term of the right-hand side describes the WFS measurements due to the low-order residual phase , the second term is the contribution of the high-order non-corrected phase to the WFS measurements and the last term is the measurement noise. The symbol stands for the operator describing the WFS. ADONIS actually uses two WFS cameras, the RETICON for high-flux sources ( magnitudes) and the EBCCD for low-flux sources (8-13 magnitudes).

2.2.2 The control loop

From the WFS measurements , the low-order residual phase estimate is calculated from a least-square fit of the equation , which leads to:

$$\widehat{\vec{\epsilon }}(t)=D^{+}\vec{w}(t)\, ,$$ (10)

where D⁺ is the generalized inverse of D, i.e. the control matrix,

$$D^{+}=\left( D^{T}\, D\right) ^{-1}\, D^{T}\: .$$ (11)

Using the Eqs. (9) and (10) we find the following relation:

$$\widehat{\vec{\epsilon }}\left( t\right) =\vec{\epsilon }\left( t\right) +\vec{r}\left( t\right) +\vec{n}\left( t\right),$$ (12)

where is the modal error due to the contribution of the high-order phase to the WFS measurements, which is interpreted as a combination of spatial aliasing and cross-correlation (Hermann 1981; Southwell 1982):

$$\vec{r}\left( t\right) =D^{+}\, {\cal W}[\phi _{\perp }(\vec{r},t)]\,.$$ (13)

We call it remaining error (Dai 1996). is the WFS measurement noise propagated on the modes:

Figure 2: Scheme of the PSF reconstruction method

$$\vec{n}\left( t\right) =D^{+}\, \vec{n}_{w}\left( t\right) \: .$$ (14)

If the contribution of the noise and the high-order phase to the WFS measurements would be zero, then , and the new mirror commands would be . Since this is not the case, the components of are multiplied by a gain g with values between 0 and 1 in order to reduce the error contribution to the residual phase variance. While the zonal correction consists in taking the same gain for all modes, the modal control, as applied in the ADONIS system, determines a specific gain for each mode (Gendron & Léna 1994). This has the advantage to take into account that the relative magnitude between the variance of the turbulent coefficients and the modal noise is different for each mode.

The following equation will help to understand the modal control. It describes the wavefront correction in the Fourier domain (Gendron & Léna 1994):

$$\begin{array}{ll} \widetilde{\epsilon _{i}}\left( \nu \right... ...nu \right) \widetilde{n_{i}}\left( \nu \right) \, , \end{array}$$ (15)

where , and are the correction transfer function, the close loop transfer function and the noise transfer function, respectively. The noise transfer function is similar to the close loop transfer function which is a low-pass frequency filter. The correction transfer function is a high-pass frequency filter equal to: . The system bandwidth is defined by the frequency for which drops beneath -3dB. It corresponds roughly to the lowest frequency transmitted by and decreases as the gain is reduced. Hence, for high gains, reduces efficiently the turbulent modes a_i, while the remaining error r_i and the noise contribution n_i are not filtered. In order to reduce their contribution, we have to decrease the gains. But then, the turbulent modes are less corrected. The modal control consists in determining for each mode the gain which minimizes both the contribution from the turbulent modes and the contribution from the remaining error and the measurement noise.

  
2.3 The long-exposure image

The partially corrected long-exposure optical transfer function (OTF) in adaptive optics is given by (Conan 1995; Véran 1997)

$$\begin{array}{ll} \left\langle {\rm OTF}(\vec{\rho }/\lambda ... ...\vec{r}+\vec{\rho })\, {\rm d}\vec{r}\, ,\textrm{ } \end{array}$$ (16)

where S is the pupil's surface area and is the complex pupil function which is inside the pupil and 0 otherwise. The term represents static aberrations not corrected by the AO system. They are essentially low-frequency aberrations which arise after the splitting of the light beam, either in the optical path of the science camera or in the optical path of the WFS. Some hypotheses are introduced to simplify the last equation. If is a random variable of Gaussian statistics, we have the following relation:

$$\left\langle {\rm e}^{i\, \phi _{\epsilon }(\vec{r},t)}\, {\r... ...}^{-\frac{1}{2}D_{\phi _{\epsilon }}(\vec{r},\vec{\rho })}\, ,$$ (17)

where

$$D_{\phi _{\epsilon }}(\vec{r},\vec{\rho })=\left\langle \left... ... _{\epsilon }(\vec{r}+\vec{\rho },t)\right] ^{2}\right\rangle$$ (18)

is the phase structure function of . The phase structure function depends on . We will replace it by , its average over the pupil's aperture (Conan 1995; Véran et al. 1997a). Then, we can write:

$$\left\langle {\rm OTF}(\vec{\rho }/\lambda )\right\rangle ={\... ...vec{\rho })}\, {\rm OTF}_{{\rm sta}}(\vec{\rho }/\lambda )\, .$$ (19)

The last term of this equation is the instrumental optical transfer function,

$${\rm OTF}_{{\rm sta}}(\vec{\rho }/\lambda )=\frac{1}{S}\int P... ...ec{r})\, P_{{\rm o}}(\vec{r}+\vec{\rho })\, {\rm d}\vec{r}\, ,$$ (20)

which can, for example, be calibrated by measuring an artificial source in close loop.

Since , we can write the mean phase structure function as:

$$\bar{D}_{\phi _{\epsilon }}(\vec{\rho })=\bar{D}_{\phi _{\eps... ...silon _{\parallel }}\phi _{{\rm a}_{\perp }}}(\vec{\rho })\, ,$$ (21)

where is the mean structure function of , the mean structure function of and a crossed term which we will neglect in our calculation (Véran et al. 1997a). The long-exposure optical transfer function can then be written as the product of three terms:

$$<\! {\rm OTF}\!\! \left(\frac{\vec{\rho }}{\lambda }\right)\... ..._{{\rm sta}}\!\! \left(\frac{\vec{\rho }}{\lambda }\right)\! .$$ (22)

  
2.3.1 Estimation of the low-order phase structure function

Figure 3: Variance of the turbulent mode coefficients (continuous line) and variance of the remaining error (dashed line) versus the mode number i, assuming Kolmogorov turbulence and

We estimate from:

$$\bar{D}_{\phi _{\epsilon _{\parallel }}}(\vec{\rho })=\sum _{... ...N}\, <\epsilon _{i}\, \epsilon _{j}>\, U_{ij}(\vec{\rho })\, ,$$ (23)

where

$$\begin{array}{ll} U_{ij}(\vec{\rho })= & \\ & \\ \frac{\in... ...)\, P(\vec{r}+\vec{\rho })\, {\rm d}\vec{r}}\cdot & \end{array}$$ (24)

is the ideal pupil function taking the value 1 inside the aperture and 0 outside. The covariance of the residual mode coefficients , found with the help of Eqs. (12) and (15) and after some calculation, is given by:

$$\begin{array}{ll} <\epsilon _{i}\, \epsilon _{j}>~= & <\hat{... ...{\rm r}_{i}{\rm r}_{j}}(\nu )\right] \, {\rm d}\nu \end{array}$$ (25)

where S_xx represents the power spectrum of the random variable xand S_xy the cross power spectrum of the random variables xand y. is a high-pass frequency filter. Thus, if we assume that the AO bandwidth is higher than , the highest cut-off frequency of the temporal power spectra of the turbulent mirror modes, we can neglect the two last terms in Eq. (25) (Véran et al. 1997a) and get:

$${\cal C}_{\epsilon \epsilon }={\cal C}_{\hat{\epsilon }\hat{\epsilon }}-{\cal C}_{nn}+{\cal C}_{rr\, ,}$$ (26)

where stands for the covariance matrix. We determine the covariance matrix in Eq. (26) from the WFS measurements and the control matrix D⁺, from the measurement noise and D⁺, and by simulation of the adaptive optics system in the presence of Kolmogorov turbulence. For this, we use a program written by F. Rigaut (Rigaut et al. 1994). To fix the ideas, Fig. 3 shows, as dashed line, the variance of the remaining error compared to the variance of the turbulent mode coefficients for a Kolmogorov turbulence, both for D/r₀=1. The total variance of the remaining error is

$$\sum \sigma _{{\rm r}_{i}}^{2}\approx 0.0025\, \left( \frac{D}{r_{0}}\right) ^{5/3}\, ,$$ (27)

where D is the diameter of the telescope (be careful not to confuse with the interaction matrix D) and r₀ is Fried's parameter.

2.3.2 Estimation of the high-order phase structure function

With this AO simulation package we also estimate the high-order phase structure function for D/r₀=1. The actual phase structure function has then to be calibrated by . For large distances, saturates to with:

$$\sigma _{{\rm a}_{\perp }}^{2}\approx 0.012\, \left( \frac{D}{r_{0}}\right) ^{5/3}\, .$$ (28)

The scheme in Fig. 2 summarizes the PSF reconstruction process.