**Step 2: Digital image synthesis**

In this second step of the DAO technique, a digital processing technique is used to synthe‐ size a compensated image from the complex-field measurement *A*(**r**) provided by the CFS (see step 1). As illustrated in Fig. 1 this step includes compensation of the measured complex field *A*(**r**) using phase function *φDAO*(**r**) and results in compensated field *ADAO*(**r**)= *A*(**r**)exp { - *jφDAO*(**r**)}. This phase conjugation step using numerical phase function *φDAO*(**r**) can be regarded as the digital equivalent of phase conjugation using a wavefront corrector such as a deformable mirror or a liquid crystal phase modulator which are used in conventional AO systems. We refer to this step as digital wavefront compensation (DWFC).

The compensated complex-field *ADAO*(**r**) is then used to synthesize image *IDAO*(**r**). Image quality of *IDAO*(**r**) hence depends on phase *φDAO*(**r**) applied at the DWFC step. The quality of image *IDAO*(**r**) is assessed by computing an image quality (sharpness) metric *J*, and an algo‐ rithm is used to optimize metric *J*, leading to an image *IDAO*(**r**) with improved quality. De‐ tails about the image formation and optimization process are presented in section 2.4.

#### **2.2. Comparison between conventional and digital AO system operations**

For a conventional AO system to operate successfully wavefront compensation (conjuga‐ tion) is required to be performed during time *τAO* <*τat* where *τat* is the characteristic time of atmospheric turbulence change (i.e. under "frozen" turbulence conditions). Such a system is 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 *φobj*

128 Adaptive Optics Progress

a compensated image.

(**r**) is the phase component related to the object of interest (scene) and *φturb*(**r**) is

(**r**) which is used to synthesize

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

In this second step of the DAO technique, a digital processing technique is used to synthe‐ size a compensated image from the complex-field measurement *A*(**r**) provided by the CFS (see step 1). As illustrated in Fig. 1 this step includes compensation of the measured complex field *A*(**r**) using phase function *φDAO*(**r**) and results in compensated field *ADAO*(**r**)= *A*(**r**)exp { - *jφDAO*(**r**)}. This phase conjugation step using numerical phase function *φDAO*(**r**) can be regarded as the digital equivalent of phase conjugation using a wavefront corrector such as a deformable mirror or a liquid crystal phase modulator which are used in conventional AO systems. We refer to this step as digital wavefront compensation (DWFC).

The compensated complex-field *ADAO*(**r**) is then used to synthesize image *IDAO*(**r**). Image quality of *IDAO*(**r**) hence depends on phase *φDAO*(**r**) applied at the DWFC step. The quality of image *IDAO*(**r**) is assessed by computing an image quality (sharpness) metric *J*, and an algo‐ rithm is used to optimize metric *J*, leading to an image *IDAO*(**r**) with improved quality. De‐

For a conventional AO system to operate successfully wavefront compensation (conjuga‐ tion) is required to be performed during time *τAO* <*τat* where *τat* is the characteristic time of atmospheric turbulence change (i.e. under "frozen" turbulence conditions). Such a system is

tails about the image formation and optimization process are presented in section 2.4.

**2.2. Comparison between conventional and digital AO system operations**

the images produced by the system and (2) preserve phase*φobj*

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

**Step 2: Digital image synthesis**

$$
\tau\_{AO} = \tau\_{WFC} + \tau\_{WFS} + \tau\_{cont} \tag{3}
$$

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

**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.

Although digital AO systems are based on the same principle of phase conjugation than conventional AO, compensation is implemented in a difference manner. DAO systems em‐ ploy numerical techniques and do not use physical wavefront corrector devices such de‐ formable or segmented mirrors. This has the advantage of alleviating the need for WFC's and their real-time control hardware, two elements that impacts significantly the cost and complexity of conventional AO systems.

sors suitable for DAO applications. Another important criterion is the computational cost of the wavefront reconstruction algorithm as it impacts the speed of operation of the sensor. In this regard the MAPR sensor might be suitable for DAO applications since it is capable of providing high-resolution measurements under conditions of strong intensity scintillation (so called scintillation-resistant) and in the presence of branch points [38,39]. It yields an average Strehl ratio exceeding 0.9 for scintillation in‐

However, as the selection of wavefront sensing techniques suitable for DAO-based imaging requires further investigations, we assume in the remainder of this chapter that the complex amplitude of the incident optical field *Ain*(**r**) (see Fig. 1) is known and that *A*(**r**)= *Ain*(**r**). DAO image synthesis and compensation techniques based on measurement *A*(**r**) are provided in

As a result of anisoplanatism (see section 1.3) image quality varies significantly across the field-of-view of the system and image compensation based on phase conjugation (section 2.1) is effective only over a small angular extent with size related to the isoplanatic angle *θ*0. An approach for performing efficient DAO compensation is to apply the technique locally over image regions that are nearly isoplanatic. We present in this section a block-by-block (mosaic) post-processing technique in which the DAO approach is applied sequentially to regions (blocks) Ω *<sup>j</sup>* and the resulting image consists in the combination of compensated im‐ age regions Ω *<sup>j</sup>* into a single image corresponding to the entire FOV (region Ω). Consider an

> |**r** - **r** *<sup>j</sup>* |2 2*ω*<sup>Ω</sup> <sup>2</sup> ) 8

where **<sup>r</sup>** *<sup>j</sup>* defines the position of the *<sup>j</sup> th* image region and *ω*<sup>Ω</sup> denoted its size. Image synthe‐

<sup>2</sup> <sup>≤</sup>1.75 and *<sup>D</sup>* /*r*<sup>0</sup> <sup>≤</sup>12 and reconstruction

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

131

Digital Adaptive Optics: Introduction and Application to Anisoplanatic Imaging

, (4)

is performed based on the measurement *A*(**r**) of

*akSk* (**r**), (5)

<sup>2</sup> <sup>≤</sup>1.25 and *<sup>D</sup>* /*r*<sup>0</sup> <sup>≤</sup>8, and 0.8 for *σ<sup>I</sup>*

**2.4. Anisoplanatic image synthesis and compensation**

defined by function

sis and DAO compensation over region Ω *<sup>j</sup>*

**Step 1: Digital wavefront correction**

*M*<sup>Ω</sup> *<sup>j</sup>*

A wavefront corrector phase function *φDAO*(**r**) is represented as

(**r**)=exp (-

the optical field in the pupil of the system and consists of the following steps:

*φDAO*(**r**)= ∑

*k*=1 *NDAO*

where {*Sk* (**r**)} is a set of response functions for the digital wavefront corrector, **a**={*ak* } is the vector of commands sent to the DWFC, and *NDAO* is the number of control

is computationally efficient as a result of the parallel nature of the algorithm.

dex values *σ<sup>I</sup>*

the next section.

image region Ω *<sup>j</sup>*

In a DAO system measurements of the input complex-field *Ain*(**r**) are performed during *τCFS* <*τat* (i.e. in real-time) while image formation and compensation of turbulence-in‐ duced phase aberrations are performed as *post-processing* steps as illustrated in Fig. 2(b). Since the post-processing step is not required to be performed at high speed, it can be achieved using standard computation techniques such as a PC, which simplifies imple‐ mentation of DAO systems. The time delay associated with image synthesis and compen‐ sation using such techniques may be suitable for some applications. However, for applications where real-time DAO operation is critical, the post-processing step may be implemented on a dedicated high-speed hardware.

## **2.3. Wavefront sensing techniques for DAO systems**

In conventional AO systems the spatial resolution of the wavefront sensor output (i.e. the spacing between data points) is related to the spatial resolution of the wavefront corrector (e.g. spacing between deformable mirror actuators). Sensing of the incoming wavefront aberrations with high spatial frequency does not provide better AO performance if the cor‐ rector device is unable to match this spatial resolution. In the other hand image quality in DAO systems is directly related to the spatial resolution of the wavefront measurement. DAO systems hence require high resolution wavefront sensing capabilities.

Although the resolution yielded by wavefront sensors typically used in adaptive optics such as Shack-Hartmann [1,14] or curvature sensors [15,16] does not exceed a few tens to a couple hundred data points across the system's aperture, a number of wavefront sensing techniques capable of providing high resolution outputs exist. Among them some are potentially suitable for DAO systems including: phase retrieval from sets of pupil and focal plane intensity distributions [17-19], phase diversity [20,21], schlieren techniques and phase contrast techniques [22,23] such as the Zernike filter [24-26] and the Smartt point-diffraction interferometer [27-30]. Approaches based on holographic re‐ cording of the wavefront have also been used successfully [31-33]. The recently devel‐ oped sensor referred to as multi-aperture phase reconstruction (MAPR) sensor [34] uses a hybrid approach between the Shack-Hartmann and Gerchberg-Saxton [17] tech‐ niques to provide high-resolution measurements and is also a candidate for DAO sys‐ tem implementation.

A growing number of applications now require operation over near-horizontal or slant atmospheric paths. These propagations scenarios are characterized by moderate to strong intensity scintillation [35-37]. This means that in addition to high resolution re‐ quirements, robustness to high scintillation levels is a critical criterion for selecting sen‐ sors suitable for DAO applications. Another important criterion is the computational cost of the wavefront reconstruction algorithm as it impacts the speed of operation of the sensor. In this regard the MAPR sensor might be suitable for DAO applications since it is capable of providing high-resolution measurements under conditions of strong intensity scintillation (so called scintillation-resistant) and in the presence of branch points [38,39]. It yields an average Strehl ratio exceeding 0.9 for scintillation in‐ dex values *σ<sup>I</sup>* <sup>2</sup> <sup>≤</sup>1.25 and *<sup>D</sup>* /*r*<sup>0</sup> <sup>≤</sup>8, and 0.8 for *σ<sup>I</sup>* <sup>2</sup> <sup>≤</sup>1.75 and *<sup>D</sup>* /*r*<sup>0</sup> <sup>≤</sup>12 and reconstruction is computationally efficient as a result of the parallel nature of the algorithm.

However, as the selection of wavefront sensing techniques suitable for DAO-based imaging requires further investigations, we assume in the remainder of this chapter that the complex amplitude of the incident optical field *Ain*(**r**) (see Fig. 1) is known and that *A*(**r**)= *Ain*(**r**). DAO image synthesis and compensation techniques based on measurement *A*(**r**) are provided in the next section.

#### **2.4. Anisoplanatic image synthesis and compensation**

Although digital AO systems are based on the same principle of phase conjugation than conventional AO, compensation is implemented in a difference manner. DAO systems em‐ ploy numerical techniques and do not use physical wavefront corrector devices such de‐ formable or segmented mirrors. This has the advantage of alleviating the need for WFC's and their real-time control hardware, two elements that impacts significantly the cost and

In a DAO system measurements of the input complex-field *Ain*(**r**) are performed during *τCFS* <*τat* (i.e. in real-time) while image formation and compensation of turbulence-in‐ duced phase aberrations are performed as *post-processing* steps as illustrated in Fig. 2(b). Since the post-processing step is not required to be performed at high speed, it can be achieved using standard computation techniques such as a PC, which simplifies imple‐ mentation of DAO systems. The time delay associated with image synthesis and compen‐ sation using such techniques may be suitable for some applications. However, for applications where real-time DAO operation is critical, the post-processing step may be

In conventional AO systems the spatial resolution of the wavefront sensor output (i.e. the spacing between data points) is related to the spatial resolution of the wavefront corrector (e.g. spacing between deformable mirror actuators). Sensing of the incoming wavefront aberrations with high spatial frequency does not provide better AO performance if the cor‐ rector device is unable to match this spatial resolution. In the other hand image quality in DAO systems is directly related to the spatial resolution of the wavefront measurement.

Although the resolution yielded by wavefront sensors typically used in adaptive optics such as Shack-Hartmann [1,14] or curvature sensors [15,16] does not exceed a few tens to a couple hundred data points across the system's aperture, a number of wavefront sensing techniques capable of providing high resolution outputs exist. Among them some are potentially suitable for DAO systems including: phase retrieval from sets of pupil and focal plane intensity distributions [17-19], phase diversity [20,21], schlieren techniques and phase contrast techniques [22,23] such as the Zernike filter [24-26] and the Smartt point-diffraction interferometer [27-30]. Approaches based on holographic re‐ cording of the wavefront have also been used successfully [31-33]. The recently devel‐ oped sensor referred to as multi-aperture phase reconstruction (MAPR) sensor [34] uses a hybrid approach between the Shack-Hartmann and Gerchberg-Saxton [17] tech‐ niques to provide high-resolution measurements and is also a candidate for DAO sys‐

A growing number of applications now require operation over near-horizontal or slant atmospheric paths. These propagations scenarios are characterized by moderate to strong intensity scintillation [35-37]. This means that in addition to high resolution re‐ quirements, robustness to high scintillation levels is a critical criterion for selecting sen‐

DAO systems hence require high resolution wavefront sensing capabilities.

complexity of conventional AO systems.

130 Adaptive Optics Progress

implemented on a dedicated high-speed hardware.

**2.3. Wavefront sensing techniques for DAO systems**

tem implementation.

As a result of anisoplanatism (see section 1.3) image quality varies significantly across the field-of-view of the system and image compensation based on phase conjugation (section 2.1) is effective only over a small angular extent with size related to the isoplanatic angle *θ*0. An approach for performing efficient DAO compensation is to apply the technique locally over image regions that are nearly isoplanatic. We present in this section a block-by-block (mosaic) post-processing technique in which the DAO approach is applied sequentially to regions (blocks) Ω *<sup>j</sup>* and the resulting image consists in the combination of compensated im‐ age regions Ω *<sup>j</sup>* into a single image corresponding to the entire FOV (region Ω). Consider an image region Ω *<sup>j</sup>* defined by function

$$M\_{\Omega\_{\hat{j}}}(\mathbf{r}) = \exp\left[ \left( \cdot \frac{|\mathbf{r} - \mathbf{r}\_{j}|^{2}}{2\omega\_{\Omega}^{2}} \right)^{8} \right] \tag{4}$$

where **<sup>r</sup>** *<sup>j</sup>* defines the position of the *<sup>j</sup> th* image region and *ω*<sup>Ω</sup> denoted its size. Image synthe‐ sis and DAO compensation over region Ω *<sup>j</sup>* is performed based on the measurement *A*(**r**) of the optical field in the pupil of the system and consists of the following steps:

#### **Step 1: Digital wavefront correction**

A wavefront corrector phase function *φDAO*(**r**) is represented as

*φDAO*(**r**)= ∑ *k*=1 *NDAO akSk* (**r**), (5)

where {*Sk* (**r**)} is a set of response functions for the digital wavefront corrector, **a**={*ak* } is the vector of commands sent to the DWFC, and *NDAO* is the number of control channels of the DWFC. The corrector phase function *φDAO*(**r**) is then applied to the complex field amplitude *A*(**r**) provided by the CFS and results in compensated field *ADAO*(**r**) given by

$$A\_{\rm DAO}(\mathbf{r}) = A(\mathbf{r}) \exp\left[-\left.j\varphi\_{\rm DAO}(\mathbf{r})\right]\right].\tag{6}$$

where *γ* (*n*)

image quality.

>0 is a gain coefficient, *δJ*<sup>Ω</sup> *<sup>j</sup>*

perturbations of control vector {*δak*

is defined from the common criterion

the vector obtained at the last iteration: **a**<sup>Ω</sup> *<sup>j</sup>*

results in a set of control vectors {**a**Ω<sup>j</sup>

**3. Performance analysis**

**3.1. Numerical model**

*Aprop*(**r**, *z* =0)= *Iobj*

(*n*)

<sup>|</sup>*<sup>J</sup>* <sup>Ω</sup> *<sup>j</sup>* (*n*) - *<sup>J</sup>* <sup>Ω</sup> *<sup>j</sup>* (*n*-1) |

}.

being imaged remains in the compensated field: arg *ADAO*(**r**) =*φobj*

1/2(**r**)exp *<sup>j</sup>φsurf* (**r**) , where *Iobj*

*J* Ω *j*

=**a** (*n*=*Nit*)

gion Ω can be achieved by repeating sequentially steps 1 through 4 for each region Ω *<sup>j</sup>* and

In an ideal compensation scenario the DWFC phase *φDAO*(**r**) resulting from the optimization process would compensate exactly the turbulence-induced phase aberration so that *φDAO*(**r**)=*φturb*(**r**) [see Eq. (2)]. In this ideal case only phase information related to the object

In this section performance of DAO systems is analyzed using a numerical simulation and

Performance was evaluated from an ensemble of digitally-generated random complex-fields used as input optical waves to the DAO system. For each realization of the input field the complex amplitude in the DAO system pupil plane *Ain*(*r*) (see Fig. 1) was obtained using the conventional split-operator approach for simulating wave optics propagation through a vol‐ ume of atmospheric turbulence [41]. At the beginning of the propagation path (*z* =0) we used a monochromatic optical field with complex amplitude

ing imaged and *φsurf* (**r**) is a random phase function uniformly distributed in - *π*, *π* and *δ*correlated in space. The term *φsurf* (**r**) is used to model the "optically rough" surface of the object. The optical field complex amplitude at the end of the propagation path (*z* = *L* ) was utilized as the DAO system input field: *Ain*(**r**)= *Aprop*(**r**, *z* = *L* ). Optical inhomogeneities along the propagation path were modeled with a set of 10 thin random phase screens correspond‐

results for various system configurations and turbulence strengths are discussed.

channels. The control channel updates are repeated until convergence of vector **a** toward a small vicinity of the stationary state. The number of iterations *Nit* required for convergence

(*n*)

*ε*(*n* = *Nit*) =

and the resulting control vector for compensation of region Ω *<sup>j</sup>*

is the metric response to small-amplitude random

(*n*) ≤*ε*0≪1, (10)

, denoted **a**<sup>Ω</sup> *<sup>j</sup>*

(**r**) is the intensity distribution of the object be‐

. Optimization over the entire image re‐

, corresponds to

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

133

(**r**) and leads to optimal

} applied simultaneously to all *NDAO* DWFC control

Digital Adaptive Optics: Introduction and Application to Anisoplanatic Imaging

#### **Step 2: Image synthesis**

The compensated field *ADAO*(**r**) in Eq. (6) is used for synthesis of the DAO compensat‐ ed image *IDAO*(**r**). The DAO image is computed using the Fresnel approximation as follows [25]:

$$I\_{DAO}(\mathbf{r}) = \left| \frac{1}{\frac{1}{\lambda Z} \int\_{-\infty}^{+\infty} A\_{DAO}(\mathbf{r'}) \exp\left[ -j \frac{k}{2} \left( \frac{|\mathbf{r'}|^2}{F} - \frac{|\mathbf{r} - \mathbf{r'}|^2}{L\_{\parallel}} \right) \right] d\mathbf{r'} \right|^2 \tag{7}$$

where *k* is the wave number, *F* is the focal length of the digital lens and *<sup>L</sup> <sup>i</sup>* =(1 / *<sup>F</sup>* - <sup>1</sup> / *<sup>L</sup>* )-1 is the distance between the digital lens plane and the image plane. The term *L* denotes the distance between the lens and the object plane of interest.

#### **Step 3: Local image quality metric computation**

The quality of the synthesized image *IDAO*(**r**) in Eq. (7) depends on the command vector **a** applied to the digital WFC [Eq. (5)]. Vector **a** can be considered as a parameter controlling the quality of image *IDAO*(**r**). Improving image quality in the region Ω *<sup>j</sup>* can be achieved by optimizing a sharpness metric *J*Ω<sup>j</sup> given by

$$J\_{\Omega\_{\\_j}}(\mathbf{a}) = \frac{\iint\_{D4O}^2 \langle \mathbf{r} \rangle \mathcal{M}\_{\Omega\_{\\_j}}(\mathbf{r}) d\mathbf{r}}{\iint\_{D4O} \langle \mathbf{r} \rangle \mathcal{M}\_{\Omega\_{\\_j}}(\mathbf{r}) d\mathbf{r}} \tag{8}$$

Where *M*<sup>Ω</sup> *<sup>j</sup>* (**r**) is the function defining region Ω *<sup>j</sup>* [see Eq. (4)]. Note that although the intensi‐ ty-squared sharpness metric is commonly used, other criteria could be used to assess image quality such as a gradient-based metric or the Tenengrad criterion.

#### **Step 4: Image quality metric optimization**

Optimization of metrics *J*<sup>Ω</sup> *<sup>j</sup>* can be achieved using various numerical techniques. We con‐ sider for example metric optimization based on the stochastic parallel gradient descent (SPGD) control algorithm [40]. In accordance with this algorithm command vector **a** update rule is given at each iteration *n* by the following procedure:

$$a\_k^{(n+1)} = a\_k^{(n)} + \gamma^{(n)} \delta J\_{\Omega\_j} \, ^{(n)} \delta a\_k^{(n)} \text{ for } k = 1, \dots, N\_{\text{DAO}}.\tag{9}$$

where *γ* (*n*) >0 is a gain coefficient, *δJ*<sup>Ω</sup> *<sup>j</sup>* (*n*) is the metric response to small-amplitude random perturbations of control vector {*δak* (*n*) } applied simultaneously to all *NDAO* DWFC control channels. The control channel updates are repeated until convergence of vector **a** toward a small vicinity of the stationary state. The number of iterations *Nit* required for convergence is defined from the common criterion

$$\varepsilon\left(\mu = \mathcal{N}\_{it}\right) = \frac{\left| \begin{smallmatrix} I\_{\left<\Omega\right>}^{\left<\alpha\right>} \ \cdot \ I\_{\left<\Omega\_{\left<\mu\right>}^{\left<\alpha\right>}}^{\left<\alpha\right>} \end{smallmatrix} \right|}{I\_{\left<\Omega\right>}^{\left<\alpha\right>}} \leq \varepsilon\_0 \ll 1,\tag{10}$$

and the resulting control vector for compensation of region Ω *<sup>j</sup>* , denoted **a**<sup>Ω</sup> *<sup>j</sup>* , corresponds to the vector obtained at the last iteration: **a**<sup>Ω</sup> *<sup>j</sup>* =**a** (*n*=*Nit*) . Optimization over the entire image re‐ gion Ω can be achieved by repeating sequentially steps 1 through 4 for each region Ω *<sup>j</sup>* and results in a set of control vectors {**a**Ω<sup>j</sup> }.

In an ideal compensation scenario the DWFC phase *φDAO*(**r**) resulting from the optimization process would compensate exactly the turbulence-induced phase aberration so that *φDAO*(**r**)=*φturb*(**r**) [see Eq. (2)]. In this ideal case only phase information related to the object being imaged remains in the compensated field: arg *ADAO*(**r**) =*φobj* (**r**) and leads to optimal image quality.
