2 Theoretical treatment

The propagation of light waves through the turbulent atmosphere is described by the so-called parabolic partial differential equation (Tatarski 1968). For the weak-turbulence conditions which take place under astronomical observations, this equation can be solved by the Rytov method (Tatarski 1968) which is actually the conventional perturbation method but applied to the logarithm of the field rather than to the field itself. If one places the X and Y axes of the three-dimensional Cartesian coordinate system at the aperture plane and considers the initial monochromatic plane wave propagating through the turbulence along the Z-axis, the classic Rytov solution is given by Tatarski (1968):

$$\begin{eqnarray} \Psi _{1}\left( {\vec \rho}\right) &=&\chi _{1}\left( {\vec \r... ... i{\vec\kappa}\cdot {\vec \rho }+\frac{\kappa ^{2}z}{2ik}\right) ,\end{eqnarray}$$ (1)

where $\chi _{1}$ and S₁ denote the log-amplitude and the phase fluctuations in the aperture plane, respectively, k is the wavenumber, L is the propagation length, g_n is the spectrum of fluctuations of the refractive-index random field, and ${\vec \rho}$ denotes the two-dimensional vector at the aperture plane.

The Rytov solution for the off-axis propagation was obtained in Vitrichenko et al. (1984). It follows straightforwardly if one applies the same Rytov method, but for the initially tilted plane wave. Assuming that this wave has the propagation unit vector $\begin{array} {c} \vec{n}=\left( \cos \alpha {,}\cos \beta ,{\cos}\gamma \right) ,\end{array}$ where $\alpha ,\beta$, and $\gamma$ are the angles between the vector $\vec{n}$ and the axes X, Y, and Z, respectively, one can get (Vitrichenko et al. 1984)

$$\begin{eqnarray} \Psi _{2}\left( {\vec\rho}\right) &=& \nonumber\end{eqnarray}$$

$$\begin{eqnarray} ik\int_{0}^{L}{\rm d} z\int {\rm d}^{2}\vec\kappa g_{n}\left( z... ...{2}z}{2ik} -i{\vec\kappa}\cdot{\vec{n}_{\bot }}z\right),\nonumber \end{eqnarray}$$

$$\begin{eqnarray} {\vec {n}_{\bot }} &=&\left(\sin \gamma \cos \alpha {,}\sin \gamma \cos \beta \right) .\end{eqnarray}$$ (2)

The derivation of Eq. (2) is practically the same as for the along-axis propagation, so we do not describe it here referring the reader to Tatarski (1968) for details. The only difference is that the tilted wave is considered as the initial condition for the Rytov solution of parabolic equation.

Equations (1, 2) are sufficient to calculate main statistical quantities which are of interest in isoplanatic-related problems. Let us apply these results to our problem.

Let $\gamma$ be the separation angle between the two stars (1 and 2), observed through the atmosphere. To simplify the mathematical derivations, we assume that the star 1 is on the telescope axis, but the generalization of results for the off-axis location of the star 1 is straightforward. Then, let us suppose that the phase fluctuations S₂ are perfectly measured and that this measurement is used as the compensation signal to perform the adaptive correction for the star 1. We are interested in calculation of the long-exposure Strehl ratio to be reached for the star 1 in the case of perfect phase measurement and correction. Under these conditions the residual phase distortion $S_{\rm R}$of the star 1 is $\begin{array} {c} S_{\rm R}=S_{1}-S_{2}\end{array}$. The phase fluctuations S₁ and S₂ of the stars 1 and 2 at the aperture plane are determined from Eq. (1) and Eq. (2), respectively, and the associated residual structure function $D_{\rm S_{R}}$is expressed as

$$D_{\rm S_{R}}\left( {\vec\rho _{1}},{\vec\rho _{2}}\right) \... ...ft( z\right)\! \int\! { \rm d}^{2}\vec\kappa^{-11/3} \nonumber$$

$$\begin{eqnarray} \times\left\{ 1-\exp \left[ i{\vec\kappa}\cdot\left( {\vec\rho ... ...1-\cos \left( {\vec\kappa}\cdot{\vec{n}_{\bot }}z\right) \right] ,\end{eqnarray}$$ (3)

where C_n² denotes the refractive-index structure characteristic, and the isotropic Kolmogorov spectrum $\Phi _{n}\left( \kappa ,z\right) =0.033~C_{n}^{2}\left( z\right) \kappa ^{-11/3}$ of refractive-index fluctuations has been used. Also, we have used the approximated equality $\begin{array} {c} 1+\exp \left( \frac{\kappa ^{2}z}{ik}\right) \approx 2,\end{array}$ which, as it has been shown in Tatarski (1968), holds with a high accuracy for the light wave propagation through the weak-turbulent atmosphere. A procedure of the statistical average for the quantities $g_{n}\left( z, {\vec\kappa}\right)$ which has been applied to get Eq. (3) from Eqs. (1, 2) is described in details in Tatarski (1968). Actually, to perform this average, it is sufficient to know only two equalities:
$$\begin{align} &\left\langle g_{n}\left( z_{1},{\vec\kappa _{1}}\right) g_{n}^{*}... ... \delta \left( {\vec\kappa _{1}}+ {\vec\kappa _{2}}\right) ,\nonumber\end{align}$$
where the sign $\left( *\right)$ stands for the complex conjugation, $\delta$ denotes the Dirac delta-function, and $\Phi _{n}$ is the spectrum of the refractive-index fluctuations.

As one can see from Eq. (3), $D_{\rm S_R}\left( {\vec\rho _1}, {\vec\rho _2}\right)$ is an anisotropic function that is not convenient for the calculation of the Strehl ratio. So, we calculate its isotropic approximation as follows. Introducing the polar coordinates at the aperture plane, we can express the scalar product ${\vec\kappa }\cdot{\vec{n}_{\bot }}$ as

$$\begin{eqnarray} {\vec\kappa}\cdot{\vec{n}_{\bot }}=\kappa \sin \gamma \cos \left( \varphi -\theta \right) ,\end{eqnarray}$$ (4)

where $\gamma$ is the separation angle between the stars, $\varphi$ is the polar angle of projection of the propagation vector $\vec{n}$ on the aperture plane, and $\theta$ is the polar angle of the vector ${\vec\kappa}$.

Substituting Eq. (4) into Eq. (3), averaging $D_{\rm S_R}\left( {\vec\rho _1}, {\vec\rho _2}\right)$ over the angle $\varphi$, evaluating the integrals over $\theta$ and $\kappa$, and taking into account that for the most cases of practical concern $\sin \gamma \approx \gamma$, we get the following isotropic approximation for the residual structure function:

$$\begin{eqnarray} D_{\rm S_{R}}\left( r=\left\vert {\vec\rho _{1}}-{\vec\rho _{2}... ...=5.83k^{2}\int_{0}^{L}{\rm d} zC_{n}^{2}\left( z\right) \nonumber \end{eqnarray}$$

$$\begin{eqnarray} &\times\left\{ r^{5/3}+\left( \gamma z\right) ^{5/3}\right. \no... ...frac{4r\gamma z}{ \left( r+\gamma z\right) ^{2}}\right] \right\} ,\end{eqnarray}$$ (5)

where ₂F₁ denotes the Gauss hypergeometric function.

In the general case, the long-exposure Strehl ratio SR is expressed through the four-times integral. However, if the residual structure function is isotropic, this expression is reduced to (Tatarski 1968)

$$\begin{eqnarray} SR &=&\frac{16}{\pi }\int_{0}^{1}{\rm d}\xi \xi \left( \arccos ... ...\exp \left[ -\frac{1}{2}D_{\rm S_{R}}\left( \xi D\right) \right] ,\end{eqnarray}$$ (6)

where D is the telescope diameter.

As one can see from Eqs. (5, 6), one needs to know the C_n² profile to calculate the Strehl ratio of interest. In what follows we perform the calculations considering both the analytical Hufnagel model of C_n² profile Hufnagel (1974) and the experimental data obtained recently at SPM (Mexico).