**Step 1: Optical field measurement**

The front-end of the DAO system consists of an optical reducer and an optical sensor refer‐ red to as complex-field sensor (CFS). The CFS provides simultaneous measurements of the optical field wavefront phase and intensity distributions in its pupil plane, denoted *φ*(**r**) and *I*(**r**) respectively. This sensor is referred to as a complex field sensor since the complex am‐ plitude of optical field *A*(**r**) can be represented in the form *A*(**r**)= |*A*(**r**)|exp { *jφ*(**r**)}, where |*A*(**r**)| = *I* 1/2(**r**), and both phase *φ*(**r**) and amplitude |*A*(**r**)| functions can be obtained from the sensor measurements. The reducer is used for re-imaging of the DAO system pupil onto the CFS pupil so that *A*(**r**)≈ *Ain*(*M* **r**) where *Ain*(**r**) denotes the complex amplitude of the opti‐ cal field entering the DAO system. The term *M* is a scaling factor associated with the beam reducer and **r**={*x*, *y*} designates a coordinate vector in the system pupil plane. To simplify notation, we assume *M* =1 and *A*(**r**)≈ *Ain*(**r**). Section 2.3 provides details about complex field sensing techniques for DAO systems.

As a result of propagation through atmospheric turbulence the wavefront phase *φin*(**r**) re‐ ceived at the DAO system's pupil (and measured by the CFS sensor) can be separated into two components:

$$
\varphi\_{in}(\mathbf{r}) = \varphi\_{obj}(\mathbf{r}) + \varphi\_{turb}(\mathbf{r}) \tag{2}
$$

where *φobj* (**r**) is the phase component related to the object of interest (scene) and *φturb*(**r**) is the turbulence-induced phase term which needs to be compensated. The second step of the DAO process aims to (1) compensate phase aberrations *φturb*(**r**) which degrade the quality of the images produced by the system and (2) preserve phase*φobj* (**r**) which is used to synthesize a compensated image.

referred to as *real-time* system and its temporal response results of the combination of the individual response time of each element of the AO feedback loop shown in Fig. 2(a) so that:

where *τWFC*, *τWFS* and *τcont* correspond respectively to the temporal response of the WFC, WFS and controller devices. Bandwidth requirements hence apply to each element of the

**Figure 2.** Block diagram identifying keys components of (a) a conventional AO system and (b) a digital AO system. While conventional AO requires both wavefront sensing and wavefront compensation to be performed in real-time, digital AO requires only complex-field sensing to be realized in real-time. Subsequent digital image formation and

compensation can be performed as a post-processing step.

feedback loop and drive in part the cost of AO systems.

*τAO* =*τWFC* + *τWFS* + *τcont* (3)

Digital Adaptive Optics: Introduction and Application to Anisoplanatic Imaging

http://dx.doi.org/10.5772/54108

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**Figure 1.** Notional schematic of a digital adaptive optics system.
