**2. General analysis geometry**

In the development that follows, we will employ the geometry shown in Figure 1, which is introduced by Whiteley et al. (Whiteley, Roggemann et al. 1998). This geometry shows two apertures, including sensing aperture and compensation aperture, whose position vectors are given by *r* → *<sup>s</sup>* and *r* → *<sup>c</sup>*. Two optical sources, including target and beacon, are located by posi‐ tion vectors *r* → *<sup>t</sup>* and *r* → *<sup>b</sup>*, respectively. The position vectors of the two apertures and the two sources share a fixed coordinate system. A vertical atmospheric turbulence layer, located at altitude z, is also shown in Figure 1.

The projected separation of the aperture centres in this turbulence layer is given by

$$\dot{\vec{s}}\_z = \dot{\gamma}\_z \dot{\vec{d}} + (\dot{\gamma}\_z \cdot \alpha\_z) \dot{\vec{r}}\_s + A\_{tcz} \dot{\vec{r}}\_t \cdot A\_{bsz} \dot{\vec{r}}\_b \tag{1}$$

where *d* <sup>→</sup> =*r* → *<sup>c</sup>* - *r* → *<sup>s</sup>* is the distance of two apertures, *Absz* and *Atcz* are the layer scaling factors given by *Absz* = *z* - (*r* → *<sup>s</sup>* ∙ *z* ^) / (*<sup>r</sup>* → *<sup>b</sup>* - *r* → *<sup>s</sup>*)∙ *z* ^ and *Atcz* <sup>=</sup> *<sup>z</sup>* - (*<sup>r</sup>* → *<sup>c</sup>* ∙ *z* ^) / (*<sup>r</sup>* → *<sup>t</sup>* - *r* → *<sup>c</sup>*)∙ *z* ^ , while *αz* and *γ<sup>z</sup>* are propagating factors of beacon and target, and defined by *α<sup>z</sup>* =1- *Absz*, and *γ<sup>z</sup>* =1- *Atcz*.

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems http://dx.doi.org/10.5772/54602 193

**Figure 1.** General geometry of the adaptive optical system

ture of new AO applications and the concept of anisoplanatism, and the associated analysis methods must be expanded to treat these new systems so their performance may be proper‐

Although anisoplanatism takes many forms, it can be quantified universally by the correla‐ tive properties of the turbulence-induced phase. Therefore, instead of investigating a partic‐ ular form of anisoplanatism, this paper concentrates on constructing a unified approach to analyse general anisoplanatic effects and their effects on the performance of AO systems. For the sake of brevity, we will consider only the case of classic single-conjugate AO systems and not consider the case of a multi-conjugate AO system (Ragazzoni, Le Roux et al. 2005).

In section 2 the most general analysis geometry with two spatially-separated apertures and two spatially-separated sources is introduced. In section 3, we introduce the transverse spec‐ tral filtering method which will be used to develop the unified approach for anisoplanatism in this chapter and the general expression of the anisoplanatic wave-front variance will be introduced. In section 4, some special geometries will be analysed. Under these special geo‐ metries, the scaling laws and the related characteristic quantities widely used in the AO field, such as Fried's parameter, the Greenwood frequency, the Tyler frequency, the isoplan‐ atic angle, the isokinetic angle, etc., can be reproduced and generalized. In section 5, two specific AO systems will be studied to illustrate the application of the unified approach de‐ scribed in this chapter. One of these systems is an adaptive-optical bi-static lunar laser rang‐ ing system and the other is an LGS AO system where, besides the tip-tilt components, the defocus is also corrected by the NGS subsystem. Simple conclusions are drawn in section 6.

In the development that follows, we will employ the geometry shown in Figure 1, which is introduced by Whiteley et al. (Whiteley, Roggemann et al. 1998). This geometry shows two apertures, including sensing aperture and compensation aperture, whose position vectors

sources share a fixed coordinate system. A vertical atmospheric turbulence layer, located at

^ and *Atcz* <sup>=</sup> *<sup>z</sup>* - (*<sup>r</sup>*

are propagating factors of beacon and target, and defined by *α<sup>z</sup>* =1- *Absz*, and *γ<sup>z</sup>* =1- *Atcz*.

The projected separation of the aperture centres in this turbulence layer is given by

→ *<sup>s</sup>* + *Atczr* → *<sup>t</sup>* - *Abszr* →

<sup>→</sup> <sup>+</sup> (*γ<sup>z</sup>* - *<sup>α</sup>z*)*<sup>r</sup>*

*<sup>c</sup>*. Two optical sources, including target and beacon, are located by posi‐

*<sup>b</sup>*, respectively. The position vectors of the two apertures and the two

*<sup>s</sup>* is the distance of two apertures, *Absz* and *Atcz* are the layer scaling factors

→ *<sup>c</sup>* ∙ *z* ^) / (*<sup>r</sup>* → *<sup>t</sup>* - *r* → *<sup>c</sup>*)∙ *z*

*<sup>b</sup>* (1)

^ , while *αz* and *γ<sup>z</sup>*

ly assessed.

192 Adaptive Optics Progress

**2. General analysis geometry**

altitude z, is also shown in Figure 1.

→ *<sup>s</sup>* ∙ *z* ^) / (*<sup>r</sup>* → *<sup>b</sup>* - *r* → *<sup>s</sup>*)∙ *z*

*s* → *<sup>z</sup>* =*γzd*

are given by *r*

tion vectors *r*

where *d*

<sup>→</sup> =*r* → *<sup>c</sup>* - *r* →

given by *Absz* = *z* - (*r*

→ *<sup>s</sup>* and *r* →

→ *<sup>t</sup>* and *r* →

Under some hypotheses, these expressions can be further simplified. We suppose two aper‐ tures are at the same altitudes and select the centre of the sensing aperture as the origin of coordinates. We express the positions of target and beacon with the zenith angle and alti‐ tude as*(θ → <sup>t</sup>, L)* and *(θ* → *<sup>b</sup>, H)*, respectively. We notice that in studying anisoplanatic effects, the offsets angular is very small in general (Welsh and Gardner 1991), i.e., *θ* <sup>→</sup> ≪1, then Eq. (1) is well approximated by

$$
\vec{s}\_z = \gamma\_z \vec{d} + z\vec{\Theta} \tag{2}
$$

where *θ* <sup>→</sup> =*θ* → *<sup>t</sup>* - *θ* → *<sup>b</sup>* is the angular separation between target and beacon. At the same time, the propagating factors can be simplified to *α<sup>z</sup>* =1- *z* / *H* , and *γ<sup>z</sup>* =1- *z* / *L* .

Further, if we consider delayed-time (*τ*) of the compensating process, then the projected separation can be expressed as

$$
\vec{s}\_z = \gamma\_z \vec{d} + z \vec{\Theta} + \vec{v}\_z \tau \tag{3}
$$

where *v* → *<sup>z</sup>* is the vector of wind velocity in this turbulent layer.

The above is the most general geometric relationship of AO systems. Depending on the conditions of application, more simple geometry can often be used to consider the aniso‐ planatism of AO systems. Some examples are showed in Figure 2. When the target is suf‐ ficiently bright, wave-front perturbation can be measured by directly observing the target. Thus an ideal compensation can be obtained and no anisoplanatism exists. This case is showed in Figure 2(a). In general, the target we are interested in is too dim to provide wave-front sensing, another bright beacon in the vicinity of the target must be used, as depicted in Figure 2(b). In this case, the so-called angular anisoplanatism exits (Fried 1982). In more general cases, a naturally existed object (NGS) cannot be find ap‐ propriately, to use AO systems, artificial beacons (LGS) must be created to obtained the wave-front perturbations (Happer, Macdonald et al. 1994; Foy, Migus et al. 1995). Then so called focal anisoplanatism (Buscher, Love et al. 2002; Muller, Michau et al. 2011) ap‐ pears because of an altitude difference between LGS and target, as depicted in Figure 2(c). Figure 2(d) illustrates that a special anisoplanatism will be induced when a distribut‐ ed source is used as the AO beacon because it is different from a pure point source (Stroud 1996). Distributed beacons are often occurred, for example, a LGS will wander and expand as a distributed source because of the effects of atmospheric turbulence when the laser is projected upward from the ground (Marc, de Chatellus et al. 2009). In Figure 2(e), the anisoplanatism induced by a separation of the wave-front sensing and compensation aperture is illustrated. With many applications, such as airborne lasers, the separated apertures are indispensable because of the moving platform (Whiteley, Rogge‐ mann et al. 1998). Figure 2(f) illustrates a hybrid case, in which many anisoplanatic ef‐ fects coexist at the same time.

**3. Transverse spectral filtering method and general expressions of**

*<sup>L</sup> dz*(*∫*Φ(*κ*

whose explicit form can be determined by the corresponding physical processes.

<sup>→</sup> , *z*)=0.033*Cn*

*<sup>σ</sup>* <sup>2</sup> =0.4147*πk*<sup>0</sup>

in which radial and angular integration can be written respectively as:

*<sup>F</sup>* (*κ*, *<sup>z</sup>*)= <sup>1</sup>

*IF* (*z*)=*∫* 0

Sasiela and Shelton developed a very effective analytical method to solve the problem of wave propagating in atmospheric turbulence (Sasiela 2007). This method uses Rytov's weak fluctuation theory and the filtering concept in the spatial-frequency domain for coordinates transverse to the propagation direction. In the most general case, the variance of a turbu‐ lence-induced phase-related quantity for the propagating waves, when diffraction is ignor‐

<sup>→</sup> , *z*) *f* (*κ*

where *L* is the propagation distance and k0 is the space wave number, which when related

fluctuated refractive-index at the plane vertical to the direction of wave propagation and

For the atmospheric turbulence, the two-dimensional transverse power spectrum of fluctu‐

propagation path, and *g*(*κ*) is the normalized spectrum. If *g*(*κ*)=1, then the classic Kolmog‐

Now substitute Eq. (5) into Eq. (4), and sequentially perform the integration of wave vector

<sup>→</sup> =(*κ*, *φ*) at the angular and radial components (Sasiela and Shelton 1993), then the variance

2 *∫* 0 *<sup>L</sup> dzCn*

<sup>2</sup>*<sup>π</sup> ∫* 0 <sup>2</sup>*<sup>π</sup> f* (*κ*

To evaluate the integral Eq. (6), the expression of the filter function must be given. We will introduce the anisoplanatic filter function for general geometry illustrated in Figure 1. The

<sup>→</sup> , *z*)*dκ*

<sup>→</sup> , *z*) is the transverse spectral filter function related to this calculated quantity,

2(*z*) is refractive-index structure parameter which is allowed to vary along the

<sup>→</sup> , *z*) is two-dimensional transverse power spectrum of

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

2(*z*)*g*(*κ*)*κ* -11/3 (5)

2(*z*)*IF* (*z*) (6)

<sup>→</sup> , *z*)*dφ* (8)

<sup>∞</sup>*F*(*κ*, *z*)*g*(*κ*)*κ* -8/3*dκ* (7)

<sup>→</sup> ) (4)

http://dx.doi.org/10.5772/54602

195

**corrected (anisoplanatic) wave-front variance**

*<sup>σ</sup>* <sup>2</sup> =2*πk*<sup>0</sup> 2 *∫* 0

Φ(*κ*

ed, can be written as:

*κ*

<sup>→</sup> =(*κ*, *φ*); *f* (*κ*

where *Cn*

reduces to

*κ*

rov spectrum is obtained.

to wavelength *λ* by *k*<sup>0</sup> =2*π* / *λ*; Φ(*κ*

ated refractive-index can generally be written as:

All these special anisoplanatic effects are degenerated cases and can be analysed under gen‐ eral geometry. In the following section, we will construct the general formularies of aniso‐ planatic variance under the most general geometry.

**Figure 2.** Some special cases of geometry and anisoplanatism. (a) ideal compensation, where the target is also used as the beacon; (b) angular anisoplanatism; (c) focal anisoplanatism; (d) extended beacon; (e) separated apertures; (f) hy‐ brid beacon - many anisoplanatic effects existing at the same time.
