**5. Two examples for hybrid anisoplanatism**

To illustrate the application of the unified approach described in this chapter, we will study two special AO systems as examples in this section. In these examples many anisoplanatic effects exist at the same time, so no analytical solution for anisoplanatic variances can be ob‐ tained - only numeric results.

To calculate the anisoplanatic variances, we use the Hufnagel-Valley model:

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems http://dx.doi.org/10.5772/54602 209

$$\mathcal{L}\_n^{-2}(z) = A \exp\left(-\frac{z}{10^7}\right) + \frac{2.7}{10^{16}} \exp\left(-\frac{z}{1500}\right) + \frac{5.94}{10^3} \left(\frac{w}{27}\right) \left(\frac{z}{10^5}\right)^{10} \exp\left(-\frac{z}{10^3}\right) \tag{68}$$

where w is the pseudo-wind, and the altitude *z* expressed in meters. The turbulence strength is usually changed by a variation of the *w* term or *A*, the parameter to describe the turbu‐ lence strength at the ground. At the same time, the modified von Karman spectrum

$$\log \text{(\(\kappa\) = \[1 + (\kappa\_o/\kappa)^2\]^{-11/6}} \exp[-(\kappa/\kappa\_i)^2] \tag{69}$$

will be use. Where *κo* and *κ<sup>i</sup>* are the space wave numbers corresponding to the outer scale and the inner scale of the atmospheric turbulence, respectively. To consider the effect of time-delay, we use the Bufton wind model

$$v\_z = v\_g + 30 \exp\left[-(z - 9400)^2 / 4800^2\right] \tag{70}$$

Where *vg* is the wind speed on the ground.

Where *μ*<sup>0</sup>

duced to

*σϕ*

*σT* <sup>2</sup> <sup>=</sup>*σ*1,1

<sup>2</sup> =0.5327 *θ<sup>r</sup>*

+

*de* ={*k*<sup>0</sup>

spectrum, i.e., using Eq. (12) and *g*(*κ*)=1.

ton and tip-tilt) as follows: *σ<sup>P</sup>*

**5. Two examples for hybrid anisoplanatism**

law for angular anisoplanatism.

<sup>2</sup> =0.3799 *<sup>D</sup>* 5/3*μ*0.

tained - only numeric results.

<sup>2</sup> 0.0569*μ*<sup>0</sup>

**4.5. The anisoplanatism induce by an extended beacon**

proximately as

208 Adaptive Optics Progress

is the mth upper turbulence moment, defined by *μm*

<sup>+</sup> + 0.5*μ*5/3

*Fϕ*(*κ*, *z*)= 1 - *Gs*

*Fn*,*m*(*κ*, *z*)=*CmNn*

This is the same result as that obtained in other research (Sasiela 1994).

sider the effect of turbulence above the beacon, the effective diameter can be expressed ap‐

We now consider the anisoplanatic effect induced by a distributed beacon and neglect other anisoplanatic effects. Let *d* =0, *θ* =0, *τ* =0, and *γ<sup>z</sup>* =*α<sup>z</sup>* =1, then Eq. (18) and Eq. (23) are re‐

2(*κ*) 1 - *Gs*

Substituting above two equations into Eq. (6) and performing the integration, the anisoplan‐ atic variance of the total phase and its Zernike components can be obtained. Below we give the corresponding results for a Gaussian distributed beacon and Kolmogrov's turbulent

For the total phase, the integration can easily be obtained. The result is

mospheric isoplanatic angle. Obviously, the result is similar to the classic *5/3* power scaling

For Zernike component *Z(m,n)*, we consider the limit case of very big *θr*, i.e., *θrz* ≫*D*. From Eq. (82), the approximate results expanding to the second order turbulence moment can be obtained. Here, we only list the first two components (i.e., the anisoplanatic variances of pis‐

To illustrate the application of the unified approach described in this chapter, we will study two special AO systems as examples in this section. In these examples many anisoplanatic effects exist at the same time, so no analytical solution for anisoplanatic variances can be ob‐

To calculate the anisoplanatic variances, we use the Hufnagel-Valley model:

5/3*μ*5/3 =(0.3608 *θ<sup>r</sup>* / *<sup>θ</sup>*0)5/3, here *μm* is the mth turbulence moment, and *θ*0 is at‐

<sup>2</sup> <sup>=</sup>*σ*0,0

<sup>2</sup> =0.5327*θ<sup>r</sup>*


<sup>+</sup> =*∫ H* <sup>∞</sup>*Cn*


2(*z*)*z mdz*. So when con‐

/ *H* <sup>2</sup> }-3/5 (65)

(*κ*, *z*) <sup>2</sup> (66)

(*κ*, *z*) <sup>2</sup> (67)

5/3 *<sup>μ</sup>*5/3 - 0.4369 *<sup>D</sup>* 5/3*μ*0 and

#### **5.1. An adaptive-optical bi-static Lunar Laser Ranging (LLR) system**

Although the technique of Lunar Laser Ranging (LLR) is one of most important methods to modern astronomy and Earth science, it is also a very difficult task to develop a suc‐ cessful LLR system (Dickey, Bender et al. 1994). One of the main reasons is that the qual‐ ity of the outgoing laser beams deteriorates sharply due to the effect of atmospheric turbulence, including the wandering, expansion, and scintillation. To mitigate these ef‐ fects of atmospheric turbulence and improve the quality of laser beams, one can use AO systems to compensate the outgoing beams (Wilson 1994; Riepl, Schluter et al. 1999). In this section we will study the anisoplanatism of a special adaptive optical bi-static LLR system in which the receiving aperture is also used to measure the turbulence-induced wave-front and the outgoing beam is compensated by the conjugated wave-front meas‐ ured by this aperture. It is a concrete application of the unified approach described in this paper.

For this special AO system, two apertures and the useful point-like beacon (Aldrin, Collins, et al.) and the targets (Apollo 11, Apollo 15, et al.) are separated, so the anisoplanatism is hybrid. Let *Gs* (*κ*, *z*)=1, *L* =*H* (=3.8×10<sup>8</sup> *m*), and denote respectively the offset distance and angle of apertures and sources as d and *θ*, then the anisoplanatic filter function in altitude z are reduced to

$$F\_{\phi}(\kappa, z) = 2 \mathbf{\tilde{1}} \mathbf{1} \mathbf{-} J\_0(\mathbf{s}\_z \kappa) \mathbf{\tilde{1}} \tag{71}$$

$$F\_{n,m}(\kappa, z) = 2C\_m N\_n^2(\gamma\_z \kappa) \mathbf{[1 - J\_0(s\_z \kappa)]} \tag{72}$$

where *γ<sup>z</sup>* =1- *z* / *L* , *sz* =*γzd* + *zθ* + *vzτ*, and the corrected time delay has been considered. Us‐ ing above Equations, the variances can be computed easily, but the results can be expressed by higher transcendental functions with no simpler expressions existing.

but it will not decrease when the separated distance is increased to a certain scale. This is because the effective distances *dn*;eff are smaller at larger orders, as has been showed in Fig‐ ure 3. A similar conclusion can be drawn for the offset angle of sources from Figure 6(b) and

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211

In Figure 6 (d), the relationship between anisoplanatic variance and turbulence intensity are showed for two wavelengths (*λ=532nm* and *1064nm*) and two corrected orders (*n=2* and *5*). In this case, all three anisoplanatic effects (angular, time-delayed and that induced by sepa‐ rated apertures) exist at the same time and the corresponding parameters are selected as

A laser beacon is insensitive to full-aperture tilt because the beam wanders on both the up‐ ward and the downward trips through the atmosphere, so currently when using LGS AO systems other NGS subsystems are usually used to sense and correct wave-front tilt. All oth‐ er Zernike modes except tip-tilt can be corrected by LGS subsystems, but the corrected per‐ formance is limited by the focal anisoplanatism. Besides tip-tilt, the defocus (or focus) mode is another main component of the turbulence-induced phase and decreasing the focal aniso‐ planatism of the defocus component is very important (Esposito, Riccardi et al. 1996; Ney‐ man 1996). In this section, we consider the performance of a special kind of LGS AO system, in which, besides the overall tilt, the focus mode can also be sensed and corrected by the NGS subsystems. Using this special LGS AO system, the focal anisoplanatism of the defocus

We concentrate on the relationship between the focal and angular anisoplanatism of the de‐ focus mode, and neglect the effects induced by time-delay and separated aperture. We also neglect the correlation between LGS and NGS subsystem, and suppose them to be statisti‐ cally independent of each other. Then the anisoplanatic filter functions for the NGS subsys‐

(*γzκ*) 1-2*Gs*,*<sup>N</sup>* (*κ*, *z*)*J*0(*κzθ<sup>N</sup>* ) + *Gs*,*<sup>N</sup>*

<sup>2</sup> (*κ*, *<sup>z</sup>*) - 2*Nn*(*γzκ*)*Nn*(*αzκ*)*Gs*,*<sup>L</sup>*

<sup>2</sup> (*κ*, *z*) (73)

<sup>2</sup> (*κ*, *z*) (74)

<sup>2</sup> (*κ*, *z*) (75)

<sup>2</sup> (*κ*, *<sup>z</sup>*) *<sup>J</sup>*0(*κzθ<sup>L</sup>* ) (76)

*Fϕ*,*<sup>N</sup>* (*κ*, *z*)=1-2*Gs*,*<sup>N</sup>* (*κ*, *z*)*J*0(*κzθ<sup>N</sup>* ) + *Gs*,*<sup>N</sup>*

While for the LGS subsystem, under the LGS beacon, the results reduce to

(*αzκ*)*Gs*,*<sup>L</sup>*

*Fϕ*,*<sup>L</sup>* (*κ*, *z*)=1-2*N*0((*γ<sup>z</sup>* - *αz*)*κ*)*Gs*,*<sup>L</sup>* (*κ*, *z*)*J*0(*κzθ<sup>L</sup>* ) + *Gs*,*<sup>L</sup>*

2

for the time delay of the correcting process from Figure 6(c).

**5.2. A special LGS AO system: Defocus corrected by the NGS subsystem**

*d* =5*cm*, *θ* =2'', *τ* =2*ms*.

mode can be reduced further.

*Fn*,*m*;*<sup>N</sup>* (*κ*, *z*) = *CmNn*

2

(*γzκ*) + *Nn*

2

Those above the LGS beacon are same as Eq. (17) and Eq. (24).

tem are reduced to

*Fn*,*m*;*<sup>L</sup>* (*κ*, *z*)=*C<sup>m</sup> Nn*

In Figure 6, we show the anisoplanatic variances when turbulence-induced wave-fronts are compensated to different Zernike orders.

**Figure 6.** The anisoplanatic variances for LLR AO system, *D=1.2m*. (a) The relationship between residual phase var‐ iance and separated distances for four different compensational orders at λ=532nm and *r*0*=10cm*; (b) the relationship between residual phase variance and offset angles; (c) the relationship between residual phase variance and corrected time-delays for four different compensational orders at λ=532nm and *r*0*=10cm*; (d) the relationship between residual phase variance and turbulent intensities for two different compensational orders (n=2 and 5) and two different wave‐ lengths (λ=532nm and 1064nm)

In the first three sub-graphs, the relationships between the anisoplanatic variances and some important parameters (separation distance of apertures, offset angle of sources, time-delay of the correcting process) are also showed respectively. From Figure 6(a), we can see the var‐ iances usually monotonously increase with the separated distance. We can also see that in‐ creasing the corrected order the variance will decrease when the separated distance is small, but it will not decrease when the separated distance is increased to a certain scale. This is because the effective distances *dn*;eff are smaller at larger orders, as has been showed in Fig‐ ure 3. A similar conclusion can be drawn for the offset angle of sources from Figure 6(b) and for the time delay of the correcting process from Figure 6(c).

In Figure 6 (d), the relationship between anisoplanatic variance and turbulence intensity are showed for two wavelengths (*λ=532nm* and *1064nm*) and two corrected orders (*n=2* and *5*). In this case, all three anisoplanatic effects (angular, time-delayed and that induced by sepa‐ rated apertures) exist at the same time and the corresponding parameters are selected as *d* =5*cm*, *θ* =2'', *τ* =2*ms*.

#### **5.2. A special LGS AO system: Defocus corrected by the NGS subsystem**

where *γ<sup>z</sup>* =1- *z* / *L* , *sz* =*γzd* + *zθ* + *vzτ*, and the corrected time delay has been considered. Us‐ ing above Equations, the variances can be computed easily, but the results can be expressed

In Figure 6, we show the anisoplanatic variances when turbulence-induced wave-fronts are

**Figure 6.** The anisoplanatic variances for LLR AO system, *D=1.2m*. (a) The relationship between residual phase var‐ iance and separated distances for four different compensational orders at λ=532nm and *r*0*=10cm*; (b) the relationship between residual phase variance and offset angles; (c) the relationship between residual phase variance and corrected time-delays for four different compensational orders at λ=532nm and *r*0*=10cm*; (d) the relationship between residual phase variance and turbulent intensities for two different compensational orders (n=2 and 5) and two different wave‐

In the first three sub-graphs, the relationships between the anisoplanatic variances and some important parameters (separation distance of apertures, offset angle of sources, time-delay of the correcting process) are also showed respectively. From Figure 6(a), we can see the var‐ iances usually monotonously increase with the separated distance. We can also see that in‐ creasing the corrected order the variance will decrease when the separated distance is small,

by higher transcendental functions with no simpler expressions existing.

compensated to different Zernike orders.

210 Adaptive Optics Progress

lengths (λ=532nm and 1064nm)

A laser beacon is insensitive to full-aperture tilt because the beam wanders on both the up‐ ward and the downward trips through the atmosphere, so currently when using LGS AO systems other NGS subsystems are usually used to sense and correct wave-front tilt. All oth‐ er Zernike modes except tip-tilt can be corrected by LGS subsystems, but the corrected per‐ formance is limited by the focal anisoplanatism. Besides tip-tilt, the defocus (or focus) mode is another main component of the turbulence-induced phase and decreasing the focal aniso‐ planatism of the defocus component is very important (Esposito, Riccardi et al. 1996; Ney‐ man 1996). In this section, we consider the performance of a special kind of LGS AO system, in which, besides the overall tilt, the focus mode can also be sensed and corrected by the NGS subsystems. Using this special LGS AO system, the focal anisoplanatism of the defocus mode can be reduced further.

We concentrate on the relationship between the focal and angular anisoplanatism of the de‐ focus mode, and neglect the effects induced by time-delay and separated aperture. We also neglect the correlation between LGS and NGS subsystem, and suppose them to be statisti‐ cally independent of each other. Then the anisoplanatic filter functions for the NGS subsys‐ tem are reduced to

$$F\_{\phi,N}(\kappa, z) = 1 - 2G\_{s,N}(\kappa, z)I\_0(\kappa z \Theta\_N) + G\_{s,N}^2(\kappa, z) \tag{73}$$

$$F\_{n,m;N}(\kappa,z) = \mathbb{C}\_m \text{N}\_n^2(\mathbb{V}\_z \kappa) \mathbf{I} \mathbf{1} \text{ -} \mathbf{2} \mathbf{G}\_{s,N}(\kappa,z) \mathbf{J}\_0(\kappa z \boldsymbol{\Theta}\_N) + \mathbf{G}\_{s,N}^2(\kappa,z) \mathbf{J} \tag{74}$$

While for the LGS subsystem, under the LGS beacon, the results reduce to

$$F\_{\phi,L} \text{ (\(\kappa, z\)} = 1 \text{ - 2N}\_0 \{ (\gamma\_z \cdot \alpha\_z) \kappa \} \text{G}\_{s,L} \text{ (\(\kappa, z\)} \text{J}\_0 \{ \kappa z \Theta\_L \} + \text{G}\_{s,L}^2 \text{ (\(\kappa, z\)} \tag{75}$$

$$F\_{\boldsymbol{\eta},\boldsymbol{\eta};\boldsymbol{L}}\left(\boldsymbol{\kappa},\boldsymbol{z}\right) = \mathbb{C}\_{\boldsymbol{m}} \Big[ \mathrm{N}\_{\boldsymbol{\eta}}^{2} \{ \boldsymbol{\gamma}\_{z} \boldsymbol{\kappa} \} + \mathrm{N}\_{\boldsymbol{\eta}}^{2} \{ \boldsymbol{a}\_{z} \boldsymbol{\kappa} \} \mathrm{G}\_{s,\boldsymbol{L}}^{2} \left(\boldsymbol{\kappa},\boldsymbol{z}\right) - 2 \mathrm{N}\_{\boldsymbol{\eta}} \{ \boldsymbol{\gamma}\_{z} \boldsymbol{\kappa} \} \mathrm{N}\_{\boldsymbol{\eta}} \{ \boldsymbol{a}\_{z} \boldsymbol{\kappa} \} \mathrm{G}\_{s,\boldsymbol{L}}^{2} \left(\boldsymbol{\kappa},\boldsymbol{z}\right) I\_{0} \{ \boldsymbol{\kappa} \boldsymbol{z} \boldsymbol{\theta}\_{\boldsymbol{L}} \} \right] \quad \text{(76)}$$

Those above the LGS beacon are same as Eq. (17) and Eq. (24).

In above equations, *γz*, *θ<sup>N</sup>* , and *Gs*,*N* (or *αz*, *θ<sup>L</sup>* , *Gs*,*<sup>L</sup>* ) are main related parameters of aniso‐ planatic effect, and they are the propagating factor, the offset angle, and the filter function of the NGS (or LGS), respectively. Here we have supposed that the altitude of NGS is same as that of the target.

Using these filter functions, the effective anisoplantic variance for this particular LGS AO system can be calculated and expressed as follows:

$$\sigma\_{\rm eff}^2 = \left(\sigma\_{1,1;N}^2 + \sigma\_{2,0;N}^2\right) + \left[\sigma\_{\phi,L}^2 \cdot \left(\sigma\_{0,0;L}^2 + \sigma\_{1,1;L}^2 + \sigma\_{2,0;L}^2\right)\right] \tag{77}$$

In this equation, the first two items in parentheses are the contribution of the NGS subsys‐ tem, describing the anisoplanatism of tip-tilt and defocus modes respectively. While the items in brackets are the contribution of the LGS subsystem, and the four items are the var‐ iance of the total phase, the piston, the tip-tilt and the defocus mode, sequentially. As a com‐ parison, the effective anisoplanatic variance for a usual LGS AO system, in which only tiptilt mode can be sensed and corrected by the NGS subsystem, can be expressed as:

$$
\sigma\_{\rm eff}^2 = \sigma\_{1,1;N}^2 + \left[\sigma\_{\phi,L}^2 \cdot \left(\sigma\_{0,0;L}^2 + \sigma\_{1,1;L}^2\right)\right] \tag{78}
$$

**Figure 7.** (a) Anisoplanatism of distributed beacon; (b) angular anisoplanatism; (c) the focal anisoplanatism (below the beacon); (d) focal anisoplanatism (above the beacon); (e) focal anisoplanatism (sum); (f) effective variance.

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213

In Figure 7(c) and (d), the components of anisoplanatic variance below and above the bea‐ con, are given respectively. The values for the total phase and its first three components (pis‐ ton, tilt and defocus) are showed altogether. In Figure 7(e), the variances for the total phase, the piston and tip-tilt removed phase, and the piston and tip-tilt and defocus removed phase, are showed respectively. When the altitudes of the beacon are more than *20 km* the variances are almost the same as the results of the NGS. In Figure 7(f), the effective aniso‐ planatic variances expressed by Eq. (77) are showed for three different offset angles of NGS. For the special LGS AO system, the anisoplanatic variance of defocus comes from NGS subsystem and not from the LGS subsystems as usual LGS AO systems. In Figure 8, we com‐ pare the values of these two variances and the relationship between the altitude of LGS and the offset angle of NGS. The transverse coordinates are magnitudes of the variances. The solid line describes the change of the defocus variances with the altitude of LGS and the alti‐ tude of LGS is showed in the left longitudinal coordinates. Similarly, the dotted line de‐ scribes the change of the defocus variances with the offset angle of NGS and the offset angle

From this figure the value of the LGS altitude and the NGS offset angle, having the same value of the variance, can be read directly and some operational conclusions can be drawn.For example, for a Rayleigh LGS (with an altitude of *10km* to *20km*) the anisoplanatic

NGS with the offset angle between 8'' and 9''. Similarly, the sodium LGS (with altitude of

, same as that for a

of NGS is showed in the right longitudinal coordinates.

variance of the focus component has the value between *0.08* to *0.1 rad*<sup>2</sup>

Obviously, for this special LGS AO system, the contribution of the defocus mode to the ef‐ fective anisoplanatic variance comes from the NGS system, i.e., *σ*2,0;*<sup>N</sup>* <sup>2</sup> , while for a usual LGS AO system, it comes from the LGS subsystem.

Below we give some numerical results. We mainly study the changes of the anisoplanatic variance with some control parameters, including the altitudes (*L* and *H*), the offset angles (*θN* and *θ<sup>L</sup>* ), and the angular width (for Gaussian sources: *θr*,*N* and *θr*,*<sup>L</sup>* ) of the NGS and LGS sources. Some typical results are showed in the figures below. In our calculation, the altitude of the target L is selected as 500*km* (a typical value for a LEO satellite), and the wavelength as 1.315*μm*.

In Figure 7(a) and Figure 7(b), the changes of the anisoplanatic variance with the angular widths and the offset angles of the beacons are given. In this case the invalid piston compo‐ nent of variance has been removed. In these figures, we also compare the values for three different altitudes of beacons, including a NGS (*H=L=500km*) and two kinds of LGSs with al‐ titude *H=15km* and *H=90km* respectively. It is easy to see that the variances generally in‐ crease with the offset angles and the angular widths of the beacons. But there is some minor difference for the beacon size: the variance first decrease as beacon size increases, then it in‐ creases. We can also see that the changes are more obvious when the altitudes of the beacons are larger, for example, we can see the variance changes from *0.1* to *1.6 rad*<sup>2</sup> when the offset angles changes from *0* to 10*''* for NGS, but there are nearly no changes for *15km* Rayleigh LGS, as showed in Figure 7(b).

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems http://dx.doi.org/10.5772/54602 213

In above equations, *γz*, *θ<sup>N</sup>* , and *Gs*,*N* (or *αz*, *θ<sup>L</sup>* , *Gs*,*<sup>L</sup>* ) are main related parameters of aniso‐ planatic effect, and they are the propagating factor, the offset angle, and the filter function of the NGS (or LGS), respectively. Here we have supposed that the altitude of NGS is same as

Using these filter functions, the effective anisoplantic variance for this particular LGS AO

<sup>2</sup> - (*σ*0,0;*<sup>L</sup>*

In this equation, the first two items in parentheses are the contribution of the NGS subsys‐ tem, describing the anisoplanatism of tip-tilt and defocus modes respectively. While the items in brackets are the contribution of the LGS subsystem, and the four items are the var‐ iance of the total phase, the piston, the tip-tilt and the defocus mode, sequentially. As a com‐ parison, the effective anisoplanatic variance for a usual LGS AO system, in which only tip-

tilt mode can be sensed and corrected by the NGS subsystem, can be expressed as:

<sup>2</sup> - (*σ*0,0;*<sup>L</sup>*

Obviously, for this special LGS AO system, the contribution of the defocus mode to the ef‐

Below we give some numerical results. We mainly study the changes of the anisoplanatic variance with some control parameters, including the altitudes (*L* and *H*), the offset angles (*θN* and *θ<sup>L</sup>* ), and the angular width (for Gaussian sources: *θr*,*N* and *θr*,*<sup>L</sup>* ) of the NGS and LGS sources. Some typical results are showed in the figures below. In our calculation, the altitude of the target L is selected as 500*km* (a typical value for a LEO satellite), and the

In Figure 7(a) and Figure 7(b), the changes of the anisoplanatic variance with the angular widths and the offset angles of the beacons are given. In this case the invalid piston compo‐ nent of variance has been removed. In these figures, we also compare the values for three different altitudes of beacons, including a NGS (*H=L=500km*) and two kinds of LGSs with al‐ titude *H=15km* and *H=90km* respectively. It is easy to see that the variances generally in‐ crease with the offset angles and the angular widths of the beacons. But there is some minor difference for the beacon size: the variance first decrease as beacon size increases, then it in‐ creases. We can also see that the changes are more obvious when the altitudes of the beacons are larger, for example, we can see the variance changes from *0.1* to *1.6 rad*<sup>2</sup> when the offset angles changes from *0* to 10*''* for NGS, but there are nearly no changes for *15km* Rayleigh

<sup>2</sup> <sup>+</sup> *σϕ*,*<sup>L</sup>*

fective anisoplanatic variance comes from the NGS system, i.e., *σ*2,0;*<sup>N</sup>*

<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*1,1;*<sup>L</sup>*

<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*1,1;*<sup>L</sup>*

<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*2,0;*<sup>L</sup>*

<sup>2</sup> ) (77)

<sup>2</sup> ) (78)

<sup>2</sup> , while for a usual LGS

that of the target.

212 Adaptive Optics Progress

system can be calculated and expressed as follows:

<sup>2</sup> <sup>+</sup> *<sup>σ</sup>*2,0;*<sup>N</sup>*

*σ*eff <sup>2</sup> <sup>=</sup>*σ*1,1;*<sup>N</sup>*

AO system, it comes from the LGS subsystem.

wavelength as 1.315*μm*.

LGS, as showed in Figure 7(b).

<sup>2</sup> ) <sup>+</sup> *σϕ*,*<sup>L</sup>*

*σ*eff <sup>2</sup> =(*σ*1,1;*<sup>N</sup>*

**Figure 7.** (a) Anisoplanatism of distributed beacon; (b) angular anisoplanatism; (c) the focal anisoplanatism (below the beacon); (d) focal anisoplanatism (above the beacon); (e) focal anisoplanatism (sum); (f) effective variance.

In Figure 7(c) and (d), the components of anisoplanatic variance below and above the bea‐ con, are given respectively. The values for the total phase and its first three components (pis‐ ton, tilt and defocus) are showed altogether. In Figure 7(e), the variances for the total phase, the piston and tip-tilt removed phase, and the piston and tip-tilt and defocus removed phase, are showed respectively. When the altitudes of the beacon are more than *20 km* the variances are almost the same as the results of the NGS. In Figure 7(f), the effective aniso‐ planatic variances expressed by Eq. (77) are showed for three different offset angles of NGS.

For the special LGS AO system, the anisoplanatic variance of defocus comes from NGS subsystem and not from the LGS subsystems as usual LGS AO systems. In Figure 8, we com‐ pare the values of these two variances and the relationship between the altitude of LGS and the offset angle of NGS. The transverse coordinates are magnitudes of the variances. The solid line describes the change of the defocus variances with the altitude of LGS and the alti‐ tude of LGS is showed in the left longitudinal coordinates. Similarly, the dotted line de‐ scribes the change of the defocus variances with the offset angle of NGS and the offset angle of NGS is showed in the right longitudinal coordinates.

From this figure the value of the LGS altitude and the NGS offset angle, having the same value of the variance, can be read directly and some operational conclusions can be drawn.For example, for a Rayleigh LGS (with an altitude of *10km* to *20km*) the anisoplanatic variance of the focus component has the value between *0.08* to *0.1 rad*<sup>2</sup> , same as that for a NGS with the offset angle between 8'' and 9''. Similarly, the sodium LGS (with altitude of 90km) correspond to the offset angle of NGS between 2'' and 3''. It is also easy to see that the variance is a monotonically increasing function of the NGS offset angle and a almost monot‐ onically decreasing function of the LGS altitude. Therefore, if the NGS offset angle is smaller or 0'' (such as directly imaging of a bright satellite) using NGS to correct the defocus compo‐ nent, the variance is smaller. Otherwise, when the NGS offset angle is larger (for example, when projecting laser beams to a LEO satellite, the advance angle about 10'' must be consid‐ ered) using sodium LGS to correct the defocus the variance is smaller.

results, which prove the effectiveness and universality of the general formula constructed in this chapter. We also give some numerical results of hybrid anisoplanatism under some

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

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

215

Here we give some expressions describing the integrations of the anisoplanatic filter func‐ tion *Fn*,*m*(*κ*, *z*) for the Zernike mode *Z(m,n)* with respect to the radial component of the wave

Further, if |*sz*|≫*γD*, an asymptotic series of small parameter |*γzD* /*sz*| can be found. When

<sup>6</sup> ) - - <sup>2</sup> <sup>|</sup>*γz<sup>D</sup>* / *sz*|2*<sup>n</sup>*+1/3 *<sup>Γ</sup>*( <sup>11</sup>

> *<sup>γ</sup>z<sup>D</sup>* <sup>|</sup><sup>2</sup> )-

<sup>6</sup> - *<sup>n</sup>*) <sup>2</sup>*nΓ*(2 <sup>+</sup> *<sup>n</sup>*) <sup>2</sup>

(*<sup>n</sup>* - <sup>5</sup> 6 )2 <sup>n</sup> <sup>+</sup> <sup>2</sup> +

2-13/3

(*γz<sup>D</sup>* / *sz* )2 } (79)

*<sup>γ</sup>z<sup>D</sup>* | are

more complex geometry.

<sup>∞</sup>*d<sup>κ</sup> Fn*,*m*(*κ*, *<sup>z</sup>*)*<sup>κ</sup>* -8/3

**Expressions used in section 4.1.- 4.3.**

(*γzκ*) 1 - *J*0(*szκ*)

<sup>6</sup> )(*γzD*)5/3 <sup>2</sup>1/3

*<sup>Γ</sup>*(-*<sup>n</sup>* <sup>+</sup> <sup>11</sup> / 6)*<sup>Γ</sup>* 2(*<sup>n</sup>* <sup>+</sup> 2)|*γz<sup>D</sup>* / *sz*|-2*<sup>n</sup>*+5/3

<sup>6</sup> )(2*Dγz*)

5 <sup>3</sup> { *<sup>Γ</sup>*( <sup>7</sup>

<sup>3</sup> )*Γ*(*<sup>n</sup>* - <sup>5</sup> 6 )

<sup>6</sup> )*Γ*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> 6 ) -

<sup>6</sup> , *<sup>n</sup>* - <sup>5</sup>

<sup>2</sup> , *n* + 3 2 ; 10 <sup>3</sup> , <sup>10</sup> <sup>3</sup> ;| *sz <sup>γ</sup>z<sup>D</sup>* <sup>|</sup><sup>2</sup> )}

<sup>6</sup> ; - <sup>4</sup>

Further if |*sz*|≪*γzD*, the second-order asymptotic expansions of parameter | *sz*

<sup>3</sup> , 1;| *sz*

*<sup>π</sup> <sup>Γ</sup>*( <sup>17</sup>


expanding to second-order, the results are

The results are determined by the sizes of *sz* and *γzD*. If |*sz*|≥*γzD*, then

*Γ*(7 / 3) / *Γ*(17 / 6) *<sup>π</sup> <sup>Γ</sup>*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> / 6) -

> <sup>3</sup> ) / *<sup>Γ</sup>*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> 6 )

<sup>2</sup>4/3 *<sup>π</sup> <sup>Γ</sup>*( <sup>17</sup>

When the filter function is equal to

2

**Appendix**

vector, i.e.,

*Fn*,*m*(*κ*, *z*)=2*Nn*

*In*(*z*)=(*<sup>n</sup>* <sup>+</sup> 1)*Γ*(*<sup>n</sup>* - <sup>5</sup>

22*n*+5/3

*In*(*z*)=(1 <sup>+</sup> *<sup>n</sup>*)*Γ*(*<sup>n</sup>* - <sup>5</sup>

*In*(*z*)=21/3(*<sup>n</sup>* <sup>+</sup> 1)(*γzD*)5/3{ *<sup>Γ</sup>*( <sup>7</sup>

if |*sz*|<*γzD*, then

<sup>3</sup> )*Γ*(*<sup>n</sup>* - <sup>5</sup> 6 )

<sup>6</sup> )*Γ*(*<sup>n</sup>* <sup>+</sup> <sup>23</sup> <sup>6</sup> ) <sup>3</sup> *F*2 (- 11 <sup>6</sup> , - *<sup>n</sup>* - <sup>17</sup>



*<sup>π</sup> <sup>Γ</sup>*( <sup>17</sup>

*In*(*z*)=*∫* 0

**Figure 8.** The anisoplanatism of the defocus component

#### **6. Summary**

Using transverse spectral filtering techniques we reconsider the anisoplanatism of general AO systems. A general but simple formula was given to find the anisoplanatic variance of the turbulence-induced phase and its arbitrary Zernike components under the general ge‐ ometry of AO systems. This general geometry can describe most kinds of anisoplanatism appearing in currently running AO systems, including angular anisoplanatism, focal aniso‐ planatism and that induced by distributed sources or separated apertures, and so on. Under some special geometry, close-form solutions can be obtained and are consistent with classic results, which prove the effectiveness and universality of the general formula constructed in this chapter. We also give some numerical results of hybrid anisoplanatism under some more complex geometry.
