**1.2. Limitations of conventional adaptive optics**

Performance of AO systems is limited by a number of factors among which wavefront cor‐ rectors performance have a strong impact. First, WFC's have a limited number of degreesof-freedom. For example, the number of control channels of a deformable mirror seldom exceeds a few tens across its aperture. This limitation affects the spatial scale of the wave‐ front features the WFC can compensate and prevents the system from mitigating high-order aberrations (i.e. aberrations with small spatial features). This constrain is especially critical for optical systems with aperture diameter *D*≫*r* 0, where *r* 0 is the Fried parameter [4]. An‐ other restrictive feature of WFC's is the limited amplitude of the wavefront phase they can compensate. This limitation prevents in parts conventional AO systems to be effective under strong (deep) turbulence conditions, which are typical for optical systems operating over long and/or near-horizontal (slant) atmospheric propagation paths. Finally, the limited tem‐ poral response of WFC's may prevent them from providing compensation at rates that ex‐ ceed the rate of aberration changes.

ces along the light of sight – an approach referred to as multi-conjugate AO (MCAO) [9-11]. Using multiple guide stars distributed within the field-of-view has also been explored [12]. Although these approaches have been shown to be effective, they both result in significant increase of system complexity and cost. Post-processing techniques have been investigated but they usually assume knowledge of the point spread function (PSF) for several values of

Digital Adaptive Optics: Introduction and Application to Anisoplanatic Imaging

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

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In the remainder of this chapter we introduce an alternative approach to conventional adap‐ tive optics – referred to as digital adaptive optics (DAO) – which alleviates the need for physical WFC devices and their corresponding real-time control hardware, and relieves the system from the limitations associated to them (see section 1.2). In section 2 we present the approach used in DAO systems and discuss their limitations. The DAO technique is then applied to an anisoplanatic imaging scenario and results of numerical analysis are presented

The notional schematic in Figure 1 shows the sequential steps required for obtaining a com‐ pensated image using the digital adaptive optics approach. Two major steps of the process

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

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

(**r**) + *φturb*(**r**) (2)

*φin*(**r**)=*φobj*

the field angle *θ* [13].

in section 3. Finally section 4 draws conclusions.

**2. Digital adaptive optics**

**Step 1: Optical field measurement**

sensing techniques for DAO systems.

two components:

**2.1. General approach**

are as follow:

Although technological developments have been providing WFC's with higher spatial reso‐ lution, increased dynamical range and bandwidth, an effect known as anisoplanatism which is reviewed briefly in the next section remains a fundamental limitation for adaptive optics compensation.

### **1.3. Anisoplanatism**

Conventional AO systems typically require a reference beam (guide star) that is used to probe the atmospheric turbulence and provide an optical signal to the WFS [5]. However, the light arising from different directions within the scene does not experience the same at‐ mospheric turbulence aberrations (propagation through volume turbulence) [6]. This causes AO performance to vary spatially across the field-of-view (FOV) with best image quality achieved for directions near the reference beam and over a small angular subtense in the or‐ der of the isoplanatic angle *θ*<sup>0</sup> [7]. The isoplanatic angle depends on the turbulence strength profile *Cn* 2 (*z*) where *z* is the altitude, and is given by

$$\Theta\_0 = \frac{58.1 \times 10^{-3} \lambda^{\prime 65}}{\prod (\sec \theta\_z)^{83} \int\_0^L \mathbb{C}\_n^{-2}(z) z^{\frac{5 \times 3}{2}} dz} \tag{1}$$

where *θz* is the Zenith angle of observation and *λ* is the wavelength [4]. Even under condi‐ tions of weak turbulence *θ*0 is usually small and remains in the order of a few microradians to a few tens of microradians. The isoplanatic angle is especially narrow for near-ground and near-horizontal propagation paths (i.e. high and nearly constant *Cn* <sup>2</sup> values). Anisopla‐ natism degrades the performance of AO systems as the angular separation *θ* (known as field angle) between the reference beam and points on the object increases [8].

A number of techniques have been developed to mitigate the effect of anisoplanatism such as using multiple WFS's and WFC's located in optical conjugates of planes at various distan‐ ces along the light of sight – an approach referred to as multi-conjugate AO (MCAO) [9-11]. Using multiple guide stars distributed within the field-of-view has also been explored [12]. Although these approaches have been shown to be effective, they both result in significant increase of system complexity and cost. Post-processing techniques have been investigated but they usually assume knowledge of the point spread function (PSF) for several values of the field angle *θ* [13].

In the remainder of this chapter we introduce an alternative approach to conventional adap‐ tive optics – referred to as digital adaptive optics (DAO) – which alleviates the need for physical WFC devices and their corresponding real-time control hardware, and relieves the system from the limitations associated to them (see section 1.2). In section 2 we present the approach used in DAO systems and discuss their limitations. The DAO technique is then applied to an anisoplanatic imaging scenario and results of numerical analysis are presented in section 3. Finally section 4 draws conclusions.
