2 Long-exposure PSF in off-axis adaptive system

Let S₁ and S₂ be the turbulence-induced phase fluctuations at the aperture associated with the observed and the guide star, respectively. In the case of perfect adaptive correction the residual phase fluctuation $S_{\rm R}$ associated with the observed star is $S_{\rm R}=S_{1}-S_{2}$. Since in the astronomical observations the turbulence-induced amplitude fluctuations are practically negligible (Roddier 1981), the long-exposure PSF P of the observed star can be written as:  

$$P \left( {\vec r} \right) = \left( \frac{kA}{2\pi f} \right)... ...c r}\cdot \left( \vec{\rho}_1- \vec{\rho}_2 \right) \right\} ,$$ (1)

where ${\vec r}$ denotes the two-dimensional vector in the focal plane, k is the wavenumber, f is the focal length of the telescope, G denotes that the integration is performed over the telescope aperture, and

$$D_{\rm R} \left( \vec{\rho}_1, \vec{\rho}_2 \right) =\left\l... ...left( \vec{\rho}_2 \right) \right]^{2} \right\rangle \nonumber$$

is the residual structure function of phase. Because we are interested to consider only the properties of the turbulence-induced PSF, it is assumed in derivation of Eq. (1) that the telescope itself has diffraction-limited image quality.

To calculate $D_{\rm R}$, let us introduce the three-dimensional Cartesian coordinate system in such a way that the observed star is on the Z-axis, the propagation vector ${\vec n}$ of the initially plane wave produced by the guide star at the upper boundary of the atmosphere lies in the XZ plane, and XY plane is the aperture plane. Under these conditions and for the case of Kolmogorov turbulence, $D_{\rm R}$ can be expressed as (Voitsekhovich et al. 1998):  

$$D_{\rm R} \left( \vec{\rho}=\vec{\rho}_1- \vec{\rho}_2 \righ... ...rt \vec{\rho}-\vec{n}_{\perp}z \right\vert^{5/3} \right]\!\! ,$$ (2)

where L is the propagation distance, C_n² denotes the refractive index structure characteristic, $\rho$ and $n_{\bot }$ stand for the modulus of the corresponding vectors, and ${\vec n}_{\perp}$ is the projection of the propagation vector ${\vec n}$ on the aperture plane. For the small angular star separations $\gamma$ which are of interest in astronomical applications, the vector $\vec{n}_{\bot }$ can be expressed as

$$\vec{n}_{\perp}\approx (\gamma,0), \nonumber$$

where $\gamma$ is given in radians.

As one can see from Eq. (2), the residual structure function is an anisotropic but still a homogeneous function of the vector $\vec{\rho }$.For any fixed $\vec{n}_{\bot }$ (or, in other words, for any fixed separations $\gamma$ between the stars), it can be considered in polar coordinates as a function of two arguments: the module $\rho$ and the polar angle $\varphi$ of the vector $\vec{\rho }$:  

$$D_{\rm R} (\rho,\varphi) = 5.83k^2\int_0^L {\rm d}z C_n^2(z)... ...^{5/3}- \frac{1}{2}r_{+}^{5/3}-\frac{1}{2}r_{-}^{5/3} \right],$$ (3)

where $r_{\pm }=\sqrt{\rho ^2+\gamma ^2z^2\pm 2\rho \gamma z\cos \varphi}$.

Taking into account the homogeneity of $D_{\rm R}$ and applying an approach similar to that used in Tatarski (1968) for the integration of four-dimensional isotropic functions, we can reduce the four-times integral (1) to the two-times one as:  

$$P({\vec r}) = \left(\frac{kA}{2\pi f}\right)^2\frac{D^2}{2}... ...ht\} \cos \left[ \frac kfr\rho \cos (\varphi-\theta) \right] ,$$ (4)

where D is the telescope diameter, and r and $\theta$ are the modulus and the polar angle of the vector ${\vec r}$, respectively.

Equation (4) allows for a direct interpretation in the framework of linear incoherent imaging systems. The PSF P is the Fourier transform of the overall optical transfer function (OTF) which is the product of the diffraction limited telescope OTF $H_{\rm T}$

$$H_{\rm T}(\rho) = \left(\frac{2}{\pi}\right) {\rm circ} \le... ...\frac{\rho}{D}\sqrt{1-\frac{\rho^{2}}{D^{2}}} \right] \nonumber$$

and the residual atmospheric OTF $H_{\rm C}$

$$H_{\rm C} (\rho,\varphi)= \exp \left\{ -\frac{1}{2}D_{\rm R} (\rho,\varphi) \right\}. \nonumber$$

Applying to Eq. (4) the convolution theorem, the long-exposure PSF can be described as the bidimensional convolution of the telescope diffraction-limited PSF (Fourier transform of $H_{\rm T}$) with the Fourier transform of $H_{\rm C}$. Note that while the telescope diffraction-limited PSF is rotationally symmetric and has a fixed size, the Fourier transform of $H_{\rm C}$ will in turn show a variable size and a variable degree of anisotropy, depending on $\gamma$ and on the profile C_n²(z) of refractive-index structure characteristic.

In general grounds, for big enough values of $\gamma$, the size and shape of the long-exposure PSF will be determined by the residual uncorrected atmospheric turbulence, since in those cases $H_{\rm C}$ has a noticeably smaller width in the frequency space than $H_{\rm T}$. In particular, for very big $\gamma$ the long-exposure PSF is expected to be wide and nearly symmetrical due to the progressive decorrelation between S₁ and S₂. On the other hand, in the region of small $\gamma$, the long-exposure PSF also becomes nearly symmetrical since the off-axis perfect correction is efficient enough as to nearly compensate the phase distortions affecting S₁. In this zone the resulting long-exposure PSF size and shape are determined mainly by the diffraction-limited telescope PSF. Between those two regions, a point of maximum anisotropy of the long-exposure PSF is expected to be found.

From the physical point of view the effect of anisotropic imaging arises because there is no rotational symmetry in the system formed by the guide and the observed stars. This lack of rotational symmetry results in anisotropic cross-correlations between the phase fluctuations produced by both stars. Since the quality of off-axis correction is determined by the degree of cross-correlations, this quality will also depend on the direction. The size of the corrected image will be different in different directions, giving rise to the anisotropic effect.