**1. Introduction**

[161] Starikov, F. A., Aksenov, V. P., Izmailov, I. V., et al. (2007). Wave front sensing of an

[162] Atuchin, V. V., Soldatenkov, I. S., Kirpichnikov, A. V., et al. (2004). Multilevel kino‐ form microlens arrays in fused silica for high-power laser optics. *Proc. SPIE*, 5481,

[163] Leach, J., Keen, S., Padgett, M., Saunter, C., & Love, G. D. (2006). Direct measurement of the skew angle of the Poynting vector in helically phased beam. *Optics Express*,

[164] Starikov, F. A., Kochemasov, G. G., Kulikov, S. M., et al. (2007). Wave front recon‐ struction of an optical vortex by Hartmann-Shack sensor. *Optics Letters*, 32(16),

[165] Starikov, F. A., Aksenov, V. P., Atuchin, V. V., et al. (2009). Correction of vortex laser beams in a closed-loop adaptive system with bimorph mirror. *Proc. SPIE*, 7131, 1-8.

[166] Starikov, F. A., Aksenov, V. P., Atuchin, V. V., et al. (2007). Wave front sensing of an optical vortex and its correction in the close-loop adaptive system with bimorph mir‐

[167] Bokalo, S. Yu., Garanin, S. G., Grigorovich, S. V., et al. (2007). Deformable mirror based on piezoelectric actuators for the adaptive system of the Iskra-6 facility. *Quan‐*

[168] Garanin, S. G., Manachinsky, A. N., Starikov, F. A., & Khokhlov, S. V. (2012). Phase correction of laser radiation with the use of adaptive optical systems at the Russian Federal Nuclear Center- Institute of Experimental Physics. *Optoelectronics, Instrumen‐*

[169] Starikov, F. A., Kochemasov, G. G., Koltygin, M. O., et al. (2009). Correction of vortex laser beam in a closed-loop adaptive system with bimorph mirror. *Optics Letters*,

[170] Soskin, M. S., Gorshkov, V. N., Vasnetsov, M. V., Malos, J. T., & Heckenberg, N. R. (1997). Topological charge and angular momentum of light beams carrying optical

optical vortex. *Proc. SPIE*, 634, 1-8.

43-46.

190 Adaptive Optics Progress

2291-2293.

14(25), 11919-11923.

ror. *Proc. SPIE*, 6747, 1-8.

34(15), 2264-2266.

*tum Electronics*, 37(8), 691-696.

*tation and Data Processing*, 48(2), 134-141.

vortices. *Phys. Rev. A*, 56(5), 4064-4075.

To improve the quality of a laser beam propagating in atmospheric turbulence or to improve the resolution of turbulence-limited optical systems, adaptive optics (AO) (Hardy 1998; Ty‐ son 2011) has been developed. In classical AO systems, the compensation is realized by realtime detection of the turbulence-induced perturbations from a source (beacon) using a wave-front sensing device and then removing them by adding a conjugated item on the same path using a wave-front compensating device.

However, the perturbations caused by the beacon and the target may not be the same, so when the perturbations measured by the beacon are used to compensate the perturbations caused by the target, the compensation performance is degraded. These effects are referred to as anisoplanatism (Sasiela 1992). Anisoplanatic effects are present if there is a spatial sep‐ aration between the target and beacon (Fried 1982), a spatial separation between the wavefront sensing and compensating apertures (Whiteley, Welsh et al. 1998), when time delays in the system cause the beacon phase and the target phase to be only partially corrected due to atmospheric winds or motion of the system components (Fried 1990) or when the beacon and target have different properties such as distributed size (Fried 1995; Stroud 1996) or wavelength (Wallner 1977), and so on.

Conventionally, all kinds of anisoplanatic effects are studied individually, assuming that they are statistically uncorrelated, and the total effects are obtained by summing them all to‐ gether when necessary (Gavel, Morris et al. 1994). This conventional approach has a rich his‐ tory dating back to the earliest days of AO technology and has obtained many good results. But this approach is very limited, because for actual applications of AO systems, many kinds of anisoplanatic effects exist simultaneously and are dependent on each other (Tyler 1994). It is increasingly obvious that these methods are inadequate to treat the diverse na‐

© 2013 Chen and Chang; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Chen and Chang; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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‐ ly assessed.

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.

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

*<sup>t</sup>, L)* and *(θ*

separation can be expressed as

well approximated by

<sup>→</sup> =*θ* → *<sup>t</sup>* - *θ* → →

tude as*(θ →*

where *θ*

where *v* →

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‐

<sup>→</sup> + *zθ*

Further, if we consider delayed-time (*τ*) of the compensating process, then the projected

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

<sup>→</sup> + *zθ* <sup>→</sup> + *v* →

offsets angular is very small in general (Welsh and Gardner 1991), i.e., *θ*

*s* → *<sup>z</sup>* =*γzd*

propagating factors can be simplified to *α<sup>z</sup>* =1- *z* / *H* , and *γ<sup>z</sup>* =1- *z* / *L* .

*s* → *<sup>z</sup>* =*γzd*

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

*<sup>b</sup>, H)*, respectively. We notice that in studying anisoplanatic effects, the

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

*<sup>b</sup>* is the angular separation between target and beacon. At the same time, the

<sup>→</sup> ≪1, then Eq. (1) is

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

193

<sup>→</sup> (2)

*<sup>z</sup>τ* (3)
