3 Gain of the adaptive optics system

The Fried parameter gives a measure of the maximum resolution attainable in uncompensated long-exposure images. Thus, the point spread function (PSF) size is proportional to $\lambda/r_0$. As compensation increases the image becomes a bright core surrounded by a speckled halo (Goodman 1985; Hardy 1998; Smithson et al. 1988). The halo width, $\omega_0$, can be obtained, equivalently to Eq. (7.20) in Roddier (1981), from:

$$% \int\limits_{\rm image}I_{\rm halo}(\vec{x})\,\mbox{d}\vec{x}= \frac{\pi}{4}\omega_0^2I_{\rm halo}(\vec{0}).$$ (8)

The left part of this equation is the total energy at the halo. To estimate its value, we will use the following approximated expression for the optical transfer function (Goodman 1985)

$$% {\rm OTF} {\rm OTF}_{\rm TEL}$$ =

$$\times\left[\exp(-\Delta_j)+\exp(-\Delta_j) \left(\exp(\gamma\Delta_j)-1 \right)\right].$$ (9)

As the separation between points becomes arbitrarily large, the autocorrelation function $\gamma$ tends to zero and, hence, the first term is the asymptote to which the OTF falls, whereas the second represents the rise above that asymptote. Assuming that the OTF of the original telescope is much wider than the second term, an approximated expression for the PSF becomes:

$$% {\rm PSF} {\rm PSF}_{\rm TEL}\exp(-\Delta_j)$$ =

$$+\mathcal{F}\left[\exp(-\Delta_j)\left(\exp(\gamma\Delta_j)-1 \right)\right].$$ (10)

The first term (coherent energy) can be interpreted as a diffraction-limited core of the PSF and the second a much broader halo. Thus, the coherent energy is approximately a fraction $\exp(-\Delta_j)$ of the total energy, and, consequently, the total energy in the halo is $1-\exp(-\Delta_j)$ times the total energy, $E_{\rm T}$. Then the first term of Eq. (8) can be estimated as:

$$% \int\limits_{\rm image}I_{\rm halo}(\vec{x})\,\mbox{d}\vec{x}= E_{\rm T}[1-\exp(-\Delta_j)].$$ (11)

On the other hand, for high degrees of correction (phase error lower than 1 rad rms) $\omega_0=\lambda f/2l_{\rm corr}$ is the average diameter of the halo (where f is the focal length of the system). Hence, the halo mean intensity ($I_{\rm h}$) can be estimated as the ratio between the total energy in the halo $E_{\rm T}[1-\exp(-\Delta_j)]$ and the halo area:

$$% I_{\rm h}=\frac{E_{\rm T}[1-\exp(-\Delta_j)]}{\pi(\lambda f/2l_{\rm corr})^2}\cdot$$ (12)

Then the mean intensity in the coherent peak is estimated as:

$$% I_{\rm c}=\frac{E_{\rm T}\exp(-\Delta_j)}{\pi(\lambda f/2D)^2} \cdot$$ (13)

Now, in an experiment where the energy incoming from the star is E_*and the energy from the planet $E_{\rm o}$, the ratio between the intensity at the star and the peak of the planet will be:

$$% \frac{R}{G}=\frac{E_*[1-\exp(-\Delta_j)]} {\pi(\lambda f/2l... ...corr})^2} \frac{\pi(\lambda f/2D)^2}{E_{\rm o}\exp(-\Delta_j)}$$ (14)

where $R=E_*/E_{\rm o}$ and G is the gain of the adaptive optics. Now, G can be written as an explicit function of the residual phase variance and $l_{\rm corr}$ as:

$$% G=\frac{\exp(-\Delta_j)}{1-\exp(-\Delta_j)} \left( \frac{D}{l_{\rm corr}} \right)^2.$$ (15)

Hence, this expression allows us to estimate the gain of the adaptive system using the theoretical values of the residual phase variance $\Delta_j$ and of the correlation length. It can be approximated by:

$$% G \approx \frac{1}{\Delta_j} \left( \frac{D}{l_{\rm corr}} \right)^2.$$ (16)

This approximated expression is similar to that obtained by Angel (1994) $G \approx (1/\Delta_j)[(D/\Delta x)^2]$. The main difference is that we determine the halo width from the correlation length $l_{\rm corr}$, instead of $\Delta x$, which is the projection of the actuator size over the telescope aperture. However, in the high correction range that is considered in this paper, $l_{\rm corr}$ and $\Delta x$ have identical evolutions and represent the greatest scale in which turbulence is corrected. Figure 2 shows how the gain evolves as a function of the degree of compensation for a series of D/r₀ values.

Figure 2: System gain as a function of the number of actuators for D/r0=20 (short-dashed curve), 30 (solid curve) and 40 (long-dashed curve). Values obtained by Angel, for D/r0=20, are also shown for comparison (dots)

It can be seen that values obtained using Eq. (15) are slightly lower than those predicted by Angel's expression. Our method is more accurate because it is derived from the PSF model, instead of an approximate estimate.

Finally, we will compare the integration time required using gain values derived from Eq. (15) with that obtained by Angel. The comparison will be performed considering the ideal compensation system proposed by Angel (1994). The compensation system uses 10⁴ actuators at a frequency higher than 2 kHz and scintillation is not considered (it is assumed that it has been previously compensated). From the Angel's expression relating the integration time T, the signal-to-noise ratio S/N and the system gain, it is possible to obtain an expression for the integration time as a function of the residual phase variance:

$$% T=\Delta t_{\rm opt}\frac{S}{N}\frac{R^2}{G^2}$$ (17)

where the system gain G can be estimated from Eqs. (15) or (16). A typical value of the optimal duration of the adaptive correction is $\Delta t_{\rm opt}=0.35$ ms. We will consider that the planet is detectable for a S/N=5 and the planet luminosity is 10⁹ times that of the star. Figure 3 shows the exposure time evolution as a function of the degree of compensation for a series of D/r₀ values. The integration time is in order of magnitude similar to that predicted by Angel. As an example with a telescope of 3 m and a Fried parameter of 10 cm a planet could be detected in a 3 hours integration. These values are too optimistic because an ideal compensation process is considered.

In actual experiments the residual phase variance can be different from that theoretically expected due to a series of noises. In this case, the actual phase variance can be obtained from the experimental estimate of the Strehl ratio either using the Marechal approximation $1-\sigma_\phi^2 \approx SR$, which provides quite good estimates for high compensation levels, or resolving a more accurate expression recently proposed (Cagigal & Canales 2000b)

$$% \sigma_\phi^2=3.44 \left\{0.286 j^{-0.362} \left[ \frac{SR-... ... {1-\exp(-\sigma_\phi^2)} \right]^{-1/2} \right\}^{5/3}\!\!\!.$$ (18)

This expression has the advantage that it remains valid for the whole range of compensation degrees. Hence the method to obtain the system gain will provide good results even in noisy conditions. Still there are some error sources not taken into account as scintillation or residual temporal decorrelation errors, that would require a more detailed analysis.

Figure 3: Integration time to give an S/N ratio of 5 for R=109 and D/r0=20 (short-dashed curve), 30 (solid curve) and 40 (long-dashed curve)