**4. Some special cases**

In the previous section the transverse anisoplanatic spectral filter functions for the general geometry of adaptive optical systems have been established. In this section we consider some special geometric cases, where asymptotic solutions of integrals can be obtained.

#### **4.1. The anisoplanatism induced by separated apertures and its related characteristic distances**

We first consider a simple case, where only the anisoplanatism induced by two separated apertures exists and the others are ignored. Let *τ* =0, *θ* =0, *Gs* (*κ*, *z*)=1, and *L* =*H* = + ∞ (i.e., *γ<sup>z</sup>* =*α<sup>z</sup>* =1), and taking into account the limitation *lim x*→0 *N*0 (*x*)=1, then Eq. (18) and Eq. (23) reduce to

$$F\_{\phi}(\kappa, z) = 2[1 - f\_0(\kappa d)]\tag{25}$$

$$F\_{n,m}(\kappa, z) = 2C\_m N\_n^2(\kappa) \mathbf{[1} \cdot J\_0(\kappa d)] \tag{26}$$

The anisoplanatic phase variance is easily obtained. Substituting Eq. (25) into Eq. (6), and us‐ ing the Kolmogrov spectrum, i.e., *g*(*κ*)=1, the integral is equal to

$$
\sigma\_{\phi}^{2} = \{d \mid d\_{0}\}^{5/3} \tag{27}
$$

Here we have calibrated the variance with a new characteristic distance *d*0, defined as

$$d\_0 = 0.526 \, k\_0^{-6/5} \mu\_0^{-3/5} \tag{28}$$

This is about *1/3* of the atmospheric coherence length *r*<sup>0</sup> =(0.423*k*<sup>0</sup> 2 *<sup>μ</sup>*0)-3/5, Where *μm* represents the mth (full) turbulence moments. From Eq. (27), we find that the anisoplanatic variance induced by separated apertures meets the *5/3* power scaling law with the distance of sepa‐ rated apertures.

For AO systems, the piston phase variance is not meaningful and can be removed from the total variance. Their difference, i.e., the piston-removed phase variance, cannot be expressed analytically for arbitrary distances, while for very small and very large distance their asymptotic solutions can be found. We first calculate in these limitations the wave vector in‐ tegral of the piston-removed anisoplanatic phase filter function *I*ϕeff,*aniso* = *I<sup>ϕ</sup>* - *I*0, which can be found easily from the Eq. (79) and (80) in Appendix with *n* =0.

When *d* ≫*D*, expanding *I*ϕeff,*aniso* to second order of (*D* / *d*), the result is

$$I\_{\text{>eff,axis}} \sim \left(\frac{D}{2}\right)^{5/3} \left[ \frac{4 \,\Gamma(\text{-}5/6) \Gamma(\text{-}3)}{\sqrt{\pi} \,\, \Gamma(17/6) \Gamma(23/6)} - \frac{\Gamma(1/6)}{4 \,\Gamma(5/6)} \left(\frac{D}{d}\right)^{1/3} \right] \tag{29}$$

While *d* ≪*D*, expanding it to fourth-order of (*d* / *D*), the result is

Where *Cm* is a constant factor related to the azimuthal order *m*. If *m*=0, then *Cm* =1; other‐

2

In the previous section the transverse anisoplanatic spectral filter functions for the general geometry of adaptive optical systems have been established. In this section we consider some special geometric cases, where asymptotic solutions of integrals can be obtained.

We first consider a simple case, where only the anisoplanatism induced by two separated

2(*κ*) 1 - *<sup>J</sup>*<sup>0</sup>

The anisoplanatic phase variance is easily obtained. Substituting Eq. (25) into Eq. (6), and us‐


the mth (full) turbulence moments. From Eq. (27), we find that the anisoplanatic variance induced by separated apertures meets the *5/3* power scaling law with the distance of sepa‐

For AO systems, the piston phase variance is not meaningful and can be removed from the total variance. Their difference, i.e., the piston-removed phase variance, cannot be expressed

*x*→0 *N*0

**4.1. The anisoplanatism induced by separated apertures and its related characteristic**

*Fϕ*(*κ*, *z*)=2 1 - *J*<sup>0</sup>

*Fn*,*m*(*κ*, *z*)=2*CmNn*

*σϕ*

*d*<sup>0</sup> =0.526 *k*<sup>0</sup>

Here we have calibrated the variance with a new characteristic distance *d*0, defined as

ing the Kolmogrov spectrum, i.e., *g*(*κ*)=1, the integral is equal to

This is about *1/3* of the atmospheric coherence length *r*<sup>0</sup> =(0.423*k*<sup>0</sup>

apertures exists and the others are ignored. Let *τ* =0, *θ* =0, *Gs*

(i.e., *γ<sup>z</sup>* =*α<sup>z</sup>* =1), and taking into account the limitation *lim*

(*γzκ*) (24)

(*κ*, *z*)=1, and *L* =*H* = + ∞

(*x*)=1, then Eq. (18) and Eq.

(*κd*) (25)


*<sup>μ</sup>*0)-3/5, Where *μm* represents

<sup>2</sup> =(*<sup>d</sup>* / *<sup>d</sup>*0)5/3 (27)

2

(*κd*) (26)

*Fn*,*m*(*κ*, *z*)=*CmNn*

Similarly for the case *z* ≥*H* , the corresponding result is

wise *Cm* =2.

198 Adaptive Optics Progress

**distances**

(23) reduce to

rated apertures.

**4. Some special cases**

$$I\_{\text{Qeff}, \text{amiso}} \sim -\frac{\Gamma(\text{-}5/6)}{\Gamma(11/6)} \left(\frac{D}{2}\right) \frac{5/3}{2} \binom{d}{\text{D}} \frac{5/3}{\text{D}} + \frac{\Gamma(7/3)\Gamma(\text{-}5/6)}{\sqrt{\pi}\,\Gamma(17/6)\Gamma(23/6)} \left(\frac{D}{2}\right) \frac{5/3}{\text{D}} \binom{d}{\text{D}} \frac{27}{41472} \left(\frac{d}{\text{D}}\right)^2 \sim \frac{935}{144} \,\text{L} \tag{30}$$

On the other hand, the wave vector integral of the piston-removed phase filter function for a single wave beam is easy to find and can be expressed as:

$$I\_{\text{Qeff,single}} = -\frac{2\,\Gamma(-5/6)\Gamma(7/3)}{\sqrt{\pi}\,\Gamma(17/6)\Gamma(23/6)} \left(\frac{D}{2}\right)^{5/3} \tag{31}$$

From the above equations we find that in the limitation of *d* ≫*D* the piston-removed aniso‐ planatic phase variance tends to be twice that of the piston-removed phase variance of a sin‐ gle wave. This is predictable, because when the separated distance of apertures is large enough, the correlation of waves from two separated aperture is gradually lost, and these beams are statistically independent of each other. We also find in the limitation of *d* ≪*D* the piston-removed anisoplanatic phase variance remains the 5/3 power scaling law with the separated distance, which is same as that for the total phase in Eq. (27).

There are many ways to define a related characteristic distance. For an AO system, if the pis‐ ton-removed anisoplanatic phase variance is greater than the same quantity for a single wave, that is to say

$$
\sigma^2\_{\text{\\$\text{\\$}\text{\\$}\text{\\$},\text{aniso}} \mid\_{d \tag{3}} > \sigma^2\_{\text{\\$}\text{\\$}\text{\\$},\text{single}} \tag{32}
$$

Then the compensation is ineffective and the AO system is not needed. We can define the uncorrected distance *dunc* of two separated apertures as the smallest distance satisfied above inequality. Using Eq. (30), Eq. (31) and Eq. (32), an approximation of this characteristic dis‐ tance can be given by *dunc* =0.828*D*.

On the other hand, to achieve a better performance, the residual error of corrected wave must be small enough. Similar to the isoplanatic angle, we can define the isoplanatic dis‐ tance as the separated distance of apertures at which the residual error is an exact unit. From the scaling law of Eq. (27), this distance is same as *d*0, i.e., *diso* =*d*0.

The above two characteristic distances (*dunc* and *diso*) give different restrictions to an aper‐ tures-separated AO system. Other characteristic distances can also be defined. For example, for such an AO system, we can define the effective corrected distance (*d*eff) as the separated distance of apertures at which the AO system can work effectively. Obviously this distance can be determined by the smaller of the above two characteristic distances, namely,

$$d\_{\rm eff} = \text{Min}\{d\_{\rm iso}, d\_{\rm unc}\} \tag{33}$$

From Eq. (34) and Eq. (37), and another inequality similar to Eq. (32), the uncorrected dis‐

Similarly, the effective distance of the Zernike mode *Z(m,n)* can be defined as (at *n* ≥1):

When the separated distance of the two apertures is smaller than this characteristic distance, the Zernike mode *Z(m,n)* of turbulence-induced phase can be compensated effectively by the AO system. In Eq. (40), the Minimum operator is evaluated throughout all the field of *m*,

In Figure 3, we show the typical values of the characteristic distances *dn*;eff defined above for the separated-apertures-induced anisoplanatism with *D=1.2m*. As a comparison with the total phase, the value of piston-removed quantity *d*0 is also showed in the same figures at n = 0.

In Figure 3(a), the relationship among *dn*;eff and the other two characteristic distances (for *dn*,*m*;*iso* and *dn*,*m*;*unc*, their values also select the minimum in all the ms) are showed for *λ=532nm*. From this figure, we find that the isoplanatic distance is monotonous - increas‐ ing with the radial order of Zernike mode - while the uncorrected distance is decreasing with it. Therefore, the effective distance is determined by the isoplanatic distance when the radial order is small (such as for the tip-tilt, defocus, et al) and by the uncorrected dis‐ tance when the radial order is large. We also find that the effective distances for small ns are usually greater than those for the (piston-removed) total phase, so when only a few low-level Zernike modes need to be compensated for, apertures with greater separated

Other sub-figures in Figure 3 show the effective distances for different compensational or‐ ders at different turbulent intensities and wavelengths. In Figure 3(b), four different turbu‐ lent intensities (*r*0*=3cm, 6cm, 9cm* and *12cm* at reference wavelength of *λ=500nm*) are compared. In Figure 3(c), the effective distances for two different turbulence intensities (*r*<sup>0</sup> *=5cm* and *10cm*) and two different wavelengths (*λ=532nm* and *1064nm*) are compared. We can find that the effective distances are smaller at stronger turbulences or smaller wave‐

In Figure 3(d), the relationships between the effective distances and turbulence intensities are showed for four different compensational orders (*n=1, 2, 3*, and *5*) at *λ=532nm*. This shows that the effective (or uncorrected) distances are not related to the turbulence intensi‐

ties for lager compensational orders, such as that for n=5.

*dn*;eff<sup>=</sup>*Minm*

*dn*,*m*;*unc* =12*D* / 11(6*n* - 5)(6*n* + 17) (39)

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(*dn*,*m*;*iso*, *dn*,*m*;*unc*) (40)

tance of the Zernike mode *Z(m,n)* can be defined as:

so the result is no longer dependent on *m*.

distance can be used.

lengths.

In general, the inequality *diso* <*dunc* is always satisfied, so the effective corrected distance is *d*eff=*diso* =*d*0.

Similar to the above analysis and definitions for total phase, anisoplanatic variances and re‐ lated characteristic distances can be determined for arbitrary Zernike modes. The final result is complex and can be expressed with generalized hypergeometric functions (Andrews 1998). In order to obtain a simpler close solution, we consider the limit case of very large or very small separating distance.

From Eq. (80), in the limitation *d* ≪*D*, the integral is approximately equal to

$$I\_{n,m}(\mathbf{z}) = C\_m \frac{11 \, \Gamma(7/3) \Gamma(n+1/6) (1+n)}{2^{83} \sqrt{\pi} \, \Gamma(17/6) \Gamma(17/6 \, \*, n)} \left(\frac{d}{D}\right)^2 D^{5/3} \tag{34}$$

Furthermore, performing the integration at the propagating path, the asymptotic value of the anisoplanatic phase variance for Zernike mode *Z(m,n)* is obtained as follows:

$$
\sigma\_{n,m}^2 = 0.879 \text{C}\_m k\_0^2 \frac{(1+n)\Gamma(n+1/6)}{\Gamma(n+17/6)\Gamma^{1/3}} \mu\_0 d^2 \tag{35}
$$

If we defined the isoplanatic distance of the Zernike mode *Z(m,n) dn*,*m*;*iso* as the distance sat‐ isfied the condition *σn*,*<sup>m</sup>* <sup>2</sup> =1, then the variance can be calibrated as:

$$
\sigma\_{n,m}^2 = \langle d \mid d\_{n,m;iso} \rangle^2 \tag{36}
$$

This characteristic distance can be determined as follows:

$$d\_{n,m;iso} = \frac{1.067}{k\_0} \sqrt{\frac{\Gamma(n+17/6) \, D^{1/3}}{\Gamma\_m \mu\_0 (n+1) \Gamma(n+1/6)}}\tag{37}$$

For a single beam, the expression corresponding to Eq. (34) is (*n* ≥1)

$$I\_{n,m;single}(\mathbf{z}) = \frac{\Gamma(7/3)\Gamma(n \cdot 5/6)\Gamma(1+n)}{2^{2/3}\sqrt{\pi}\,\Gamma(17/6)\Gamma(n \cdot 23/6)}\,\mathrm{D}^{5/3} \tag{38}$$

From Eq. (34) and Eq. (37), and another inequality similar to Eq. (32), the uncorrected dis‐ tance of the Zernike mode *Z(m,n)* can be defined as:

The above two characteristic distances (*dunc* and *diso*) give different restrictions to an aper‐ tures-separated AO system. Other characteristic distances can also be defined. For example, for such an AO system, we can define the effective corrected distance (*d*eff) as the separated distance of apertures at which the AO system can work effectively. Obviously this distance

In general, the inequality *diso* <*dunc* is always satisfied, so the effective corrected distance is

Similar to the above analysis and definitions for total phase, anisoplanatic variances and re‐ lated characteristic distances can be determined for arbitrary Zernike modes. The final result is complex and can be expressed with generalized hypergeometric functions (Andrews 1998). In order to obtain a simpler close solution, we consider the limit case of very large or

*d*eff=*Min*{*diso*, *dunc*} (33)

can be determined by the smaller of the above two characteristic distances, namely,

From Eq. (80), in the limitation *d* ≪*D*, the integral is approximately equal to

the anisoplanatic phase variance for Zernike mode *Z(m,n)* is obtained as follows:

11 *Γ*(7/3)*Γ*(*n* +1/6)(1 + *n*) 28/3 *π Γ*(17 / 6)*Γ*(17 / 6 + *n*)

Furthermore, performing the integration at the propagating path, the asymptotic value of

<sup>2</sup> (1 + *<sup>n</sup>*)*Γ*(*<sup>n</sup>* +1/6)

If we defined the isoplanatic distance of the Zernike mode *Z(m,n) dn*,*m*;*iso* as the distance sat‐

*Γ*(*n* + 17 / 6) *D* 1/3

<sup>2</sup> =1, then the variance can be calibrated as:

*Cmμ*<sup>0</sup>

*In*,*m*;*single*(*z*)= *<sup>Γ</sup>*(7/3)*Γ*(*<sup>n</sup>* -5/6)(1 + *<sup>n</sup>*)*<sup>C</sup> <sup>m</sup>*

( *d D* )2

*D* 5/3 (34)

*<sup>Γ</sup>*(*<sup>n</sup>* + 17 / 6)*<sup>D</sup>* 1/3 *<sup>μ</sup>*0*<sup>d</sup>* <sup>2</sup> (35)

<sup>2</sup> =(*<sup>d</sup>* / *dn*,*m*;*iso*)<sup>2</sup> (36)

<sup>2</sup>2/3 *<sup>π</sup> <sup>Γ</sup>*(17 / 6)*Γ*(*<sup>n</sup>* + 23 / 6) *<sup>D</sup>* 5/3 (38)

(*<sup>n</sup>* + 1)*Γ*(*<sup>n</sup>* +1/6) (37)

*In*,*m*(*z*)=*Cm*

*σn*,*<sup>m</sup>*

This characteristic distance can be determined as follows:

<sup>2</sup> = 0.879*Cmk*<sup>0</sup>

*σn*,*<sup>m</sup>*

*dn*,*m*;*iso* <sup>=</sup>1.067 *k*0

For a single beam, the expression corresponding to Eq. (34) is (*n* ≥1)

*d*eff=*diso* =*d*0.

200 Adaptive Optics Progress

very small separating distance.

isfied the condition *σn*,*<sup>m</sup>*

$$d\_{n,m;unc} = 12D \left\langle \sqrt{11(6n-5)(6n+17)} \right\rangle \tag{39}$$

Similarly, the effective distance of the Zernike mode *Z(m,n)* can be defined as (at *n* ≥1):

$$d\_{n, \text{eff}} = \underset{m}{\text{Min}} \; \left( d\_{n, m; \text{iso}}, d\_{n, m; \text{unc}} \right) \tag{40}$$

When the separated distance of the two apertures is smaller than this characteristic distance, the Zernike mode *Z(m,n)* of turbulence-induced phase can be compensated effectively by the AO system. In Eq. (40), the Minimum operator is evaluated throughout all the field of *m*, so the result is no longer dependent on *m*.

In Figure 3, we show the typical values of the characteristic distances *dn*;eff defined above for the separated-apertures-induced anisoplanatism with *D=1.2m*. As a comparison with the total phase, the value of piston-removed quantity *d*0 is also showed in the same figures at n = 0.

In Figure 3(a), the relationship among *dn*;eff and the other two characteristic distances (for *dn*,*m*;*iso* and *dn*,*m*;*unc*, their values also select the minimum in all the ms) are showed for *λ=532nm*. From this figure, we find that the isoplanatic distance is monotonous - increas‐ ing with the radial order of Zernike mode - while the uncorrected distance is decreasing with it. Therefore, the effective distance is determined by the isoplanatic distance when the radial order is small (such as for the tip-tilt, defocus, et al) and by the uncorrected dis‐ tance when the radial order is large. We also find that the effective distances for small ns are usually greater than those for the (piston-removed) total phase, so when only a few low-level Zernike modes need to be compensated for, apertures with greater separated distance can be used.

Other sub-figures in Figure 3 show the effective distances for different compensational or‐ ders at different turbulent intensities and wavelengths. In Figure 3(b), four different turbu‐ lent intensities (*r*0*=3cm, 6cm, 9cm* and *12cm* at reference wavelength of *λ=500nm*) are compared. In Figure 3(c), the effective distances for two different turbulence intensities (*r*<sup>0</sup> *=5cm* and *10cm*) and two different wavelengths (*λ=532nm* and *1064nm*) are compared. We can find that the effective distances are smaller at stronger turbulences or smaller wave‐ lengths.

In Figure 3(d), the relationships between the effective distances and turbulence intensities are showed for four different compensational orders (*n=1, 2, 3*, and *5*) at *λ=532nm*. This shows that the effective (or uncorrected) distances are not related to the turbulence intensi‐ ties for lager compensational orders, such as that for n=5.

**Figure 3.** The characteristic distances for the anisoplanatism of separated apertures. (a) The relationship among three characteristic distances, λ=532nm; (b) The effective distances at for four different turbulent intensities, λ=532nm; (c) The effective compensational distances at different turbulent intensities and wavelengths; (d) The relationship be‐ tween the effective distances and turbulence intensities for four different compensational orders, λ=532nm

#### **4.2. The anglular anisoplanatism and related characteristic angles**

Now we consider the geometry where only angular anisoplanatism exits. Let *d* =0, *τ* =0, *γ<sup>z</sup>* =*α<sup>z</sup>* =1, and *Gs* (*κ*, *z*)=1, then Eq. (18) and Eq. (23) reduce to

$$F\_{\phi}(\kappa, z) = 2\mathbf{\tilde{1}} \mathbf{1} - J\_0(\kappa \Theta z) \mathbf{\tilde{1}} \tag{41}$$

Using Eq. (42), the angular anisoplanatism of Zernike modes can also be calculated. The re‐ sults can be expressed with the generalized hypergeometric functions, and in some limit

We consider the limitation of *θz*≪*D*. Using Eq. (80) in Appendix, the angular anisoplana‐ tism of Zernike mode *Z(m,n)* can be expanded to the turbulence second-order structure con‐

*Γ*(*n* + 17 / 6) *D* 1/3

can be defined as the isoplanatic angle for Zernike mode *Z(m,n)*, and it is the size of the off‐ set-axis angle between the beacon and the target when the angular anisoplanatism of Zer‐

When *n* =1 and *m*=1, the tip-tilt isoplanatic angle (also called isokinetic angle) is obtained.

2

Similar to anisoplanatism of separated apertures, other characteristic angles can be defined

In Figure 4, the typical values of the characteristic angles *θn*;eff defined above are showed at *D=1.2m*. In Figure 4(a), we compare the values for two different turbulent intensities (*r*0*=5cm, 10cm*) and two different wavelengths (*λ=532nm*, and *1064nm*). We can also find that the ef‐ fective offset angles *θn*;eff for small ns are usually greater than those for the (piston-removed) total phase, the same as the characteristic quantities *dn*;eff. In fact, this is one of main reasons that the use of LGS can partially solve the so-called "beacon difficulty", because a NGS may be find to correct the lower order modes of the turbulence-induced phase in a field far wider

*θn*,*m*;*unc* =12 *D μ*<sup>0</sup> / *μ*<sup>2</sup> / 11(6*n* - 5)(6*n* + 17) (46)

<sup>2</sup> =(*<sup>θ</sup>* / *<sup>θ</sup>n*,*m*;*iso*)<sup>2</sup> (43)

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*Cm*(*<sup>n</sup>* + 1)*Γ*(*<sup>n</sup>* +1/6) (44)

*<sup>μ</sup>*2*<sup>D</sup>* -1/3)-1/2 (45)

(*θn*,*m*;*iso*, *θn*,*m*;*unc*) (47)

conditions, a more compact expression can be obtained.

*σn*,*<sup>m</sup>*

*<sup>θ</sup>n*,*m*;*iso* <sup>=</sup> 1.06665 *k*0*μ*<sup>2</sup> 1/2

*θTA* =*θ*1,1;*iso* =1.224(*k*<sup>0</sup>

and calculated. The uncorrected offset angle of *Z(m,n)* can be expressed as:

and the effective offset angle of the n-order Zernike mode can be determined by

*θn*;eff=*min m*

This is consistent with the results in other research (Sasiela and Shelton 1993).

stant moments and can be expressed as (*n* ≥1)

.

This characteristic angle can be expressed as:

where the characteristic angle

nike mode is unit *rad*<sup>2</sup>

$$F\_{n,m}(\kappa, z) = 2C\_m N\_n^2(\kappa) \mathbf{[1 - J\_0(\kappa \Theta z)]} \tag{42}$$

Substituting Eq. (41) into Eq. (6), and using the Kolmogrov spectrum, the result is *σϕ* <sup>2</sup> =(*<sup>θ</sup>* / *<sup>θ</sup>*0)5/3, here *θ*0 is the well-known isoplanatic angle defined as (Fried 1982) *θ*<sup>0</sup> =(2.914*k*<sup>0</sup> 2 *<sup>μ</sup>*5/3)-3/5.

Similarly, in the limitation of very small offset angle, i.e., *θz*≪*D*, the effective corrected off‐ set angle between beacon and target can be defined and determined by *θ*eff=*θ*0.

Using Eq. (42), the angular anisoplanatism of Zernike modes can also be calculated. The re‐ sults can be expressed with the generalized hypergeometric functions, and in some limit conditions, a more compact expression can be obtained.

We consider the limitation of *θz*≪*D*. Using Eq. (80) in Appendix, the angular anisoplana‐ tism of Zernike mode *Z(m,n)* can be expanded to the turbulence second-order structure con‐ stant moments and can be expressed as (*n* ≥1)

$$
\sigma\_{n,m}^2 = \{ \Theta \mid \Theta\_{n,m;\text{iso}} \}^2 \tag{43}
$$

where the characteristic angle

**Figure 3.** The characteristic distances for the anisoplanatism of separated apertures. (a) The relationship among three characteristic distances, λ=532nm; (b) The effective distances at for four different turbulent intensities, λ=532nm; (c) The effective compensational distances at different turbulent intensities and wavelengths; (d) The relationship be‐

Now we consider the geometry where only angular anisoplanatism exits. Let *d* =0, *τ* =0,

2(*κ*) 1 - *<sup>J</sup>*<sup>0</sup>

Substituting Eq. (41) into Eq. (6), and using the Kolmogrov spectrum, the result is

<sup>2</sup> =(*<sup>θ</sup>* / *<sup>θ</sup>*0)5/3, here *θ*0 is the well-known isoplanatic angle defined as (Fried 1982)

Similarly, in the limitation of very small offset angle, i.e., *θz*≪*D*, the effective corrected off‐

(*κθz*) (41)

(*κθz*) (42)

tween the effective distances and turbulence intensities for four different compensational orders, λ=532nm

(*κ*, *z*)=1, then Eq. (18) and Eq. (23) reduce to

*Fϕ*(*κ*, *z*)=2 1 - *J*<sup>0</sup>

set angle between beacon and target can be defined and determined by *θ*eff=*θ*0.

*Fn*,*m*(*κ*, *z*)=2*CmNn*

**4.2. The anglular anisoplanatism and related characteristic angles**

*γ<sup>z</sup>* =*α<sup>z</sup>* =1, and *Gs*

202 Adaptive Optics Progress

*σϕ*

*θ*<sup>0</sup> =(2.914*k*<sup>0</sup>

2 *<sup>μ</sup>*5/3)-3/5.

$$
\Theta\_{n,m;\text{iso}} = \frac{1.06665}{k\_0 \mu\_2^{1/2}} \sqrt{\frac{\Gamma(n+17/6) \text{ D}^{1/3}}{\Gamma\_m (n+1) \Gamma(n+1/6)}}\tag{44}
$$

can be defined as the isoplanatic angle for Zernike mode *Z(m,n)*, and it is the size of the off‐ set-axis angle between the beacon and the target when the angular anisoplanatism of Zer‐ nike mode is unit *rad*<sup>2</sup> .

When *n* =1 and *m*=1, the tip-tilt isoplanatic angle (also called isokinetic angle) is obtained. This characteristic angle can be expressed as:

$$
\Theta\_{\rm TA} = \Theta\_{\rm 1, 1; \rm iso} = 1.224 \{ k\_0^2 \mu\_2 D^{-1/3} \}^{-1/2} \tag{45}
$$

This is consistent with the results in other research (Sasiela and Shelton 1993).

Similar to anisoplanatism of separated apertures, other characteristic angles can be defined and calculated. The uncorrected offset angle of *Z(m,n)* can be expressed as:

$$\Theta\_{n,m;unc} = 12 \text{ D} \sqrt{\mu\_0/\mu\_2} \left| \sqrt{11(6n-5)(6n+17)} \right. \tag{46}$$

and the effective offset angle of the n-order Zernike mode can be determined by

$$
\Theta\_{n, \text{eff}} = \min\_{m} \left( \Theta\_{n, m; \text{iso}}, \; \Theta\_{n, m; \text{unc}} \right) \tag{47}
$$

In Figure 4, the typical values of the characteristic angles *θn*;eff defined above are showed at *D=1.2m*. In Figure 4(a), we compare the values for two different turbulent intensities (*r*0*=5cm, 10cm*) and two different wavelengths (*λ=532nm*, and *1064nm*). We can also find that the ef‐ fective offset angles *θn*;eff for small ns are usually greater than those for the (piston-removed) total phase, the same as the characteristic quantities *dn*;eff. In fact, this is one of main reasons that the use of LGS can partially solve the so-called "beacon difficulty", because a NGS may be find to correct the lower order modes of the turbulence-induced phase in a field far wider than that limited by the isoplanatic angle *θ*0. Unlike *dn*;eff, the effective offset angle *θn*;eff is not only dependent on aperture diameter *D*, but also turbulence intensity. Therefore, for higher-order Zernike modes, the effectively offset angle is also dependent on the turbulence intensity. In Figure 4(b), the relationships between effective offset angles and turbulence in‐ tensities are showed for four different compensational orders (*n=1, 2, 3,* and *5*) at *λ=532nm*.

*f* <sup>0</sup> ≈0.134 / *τ*<sup>0</sup> (50)

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

*<sup>D</sup>* -1/3*ν*2*<sup>τ</sup>* <sup>2</sup> (51)

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

205

<sup>2</sup> =1 to rescale, then the variance can be ex‐

*Cm*(*<sup>n</sup>* + 1)*Γ*(*<sup>n</sup>* +1/6) (53)

*<sup>Γ</sup>*(*<sup>n</sup>* + 17 / 6) *<sup>D</sup>* 1/3 (54)

(*τn*,*m*;*iso*, *τn*,*m*;*unc*) (55)

*τn*,*m*;*unc* =12 *D μ*<sup>0</sup> / *ν*<sup>2</sup> / 11(6*n* - 5)(6*n* + 17) (56)

<sup>2</sup> =(*<sup>τ</sup>* / *<sup>τ</sup>n*,*m*;*iso*)<sup>2</sup> (52)

Similarly, in the limitation *vzτ*≪*D*, the effective corrected time can be defined and deter‐ mined by *τ*eff=*τ*0. For arbitrary Zernike mode of phase, from Eq. (80), when we consider the

*<sup>Γ</sup>*(*<sup>n</sup>* + 17 / 6) *Cmk*<sup>0</sup>

2

*Γ*(*n* + 17 / 6) *D* 1/3

1/2 *Cm*(*<sup>n</sup>* + 1)*Γ*(*<sup>n</sup>* +1/6)

Similar to Greenwood frequency, we can apply Eq. (50) to define a characteristic frequency

This is the characteristic frequency using a single-poles filter to compensate for the Zernike

Further, the effective correction time of arbitrary n-order Zernike model of phase can be de‐

When using an AO system with a time delay exceeding this characteristic time to compen‐

second order approximation, the anisoplanatic variance is equal to

<sup>2</sup> = 0.879 (1 + *<sup>n</sup>*)*Γ*(*<sup>n</sup>* +1/6)

*σn*,*<sup>m</sup>*

*<sup>τ</sup>n*,*m*;*iso* <sup>=</sup> 1.06665 *k*0*ν*<sup>2</sup> 1/2

*f <sup>n</sup>*,*m*;*iso* =0.1256*k*0*ν*<sup>2</sup>

*τn*;eff=*min m*

sate for the n-order Zernike model of phase, the compensation is ineffective.

*σn*,*<sup>m</sup>*

Using the isoplanatic time *τn*,*m*;*iso* satisfied *σn*,*<sup>m</sup>*

related to the isoplanatic time in Eq. (53) as follows:

mode Z(m,n) of the turbulence-induced phase.

where the uncorrected time can be expressed as

pressed as:

fined as:

and its expression is

as is first noted by Fried (Fried 1990).

**Figure 4.** The characteristic angles of the angular anisoplanatism for separated beacon and target. (a) the effective offset angles at different turbulent intensities and wavelengths; (b) the relationship between effective offset angles and turbulence intensities for four different compensational orders at λ=532nm

#### **4.3. The time-delayed anisoplanatism and related characteristic quantities**

When *d* =0, *θ* =0, *γ<sup>z</sup>* =*α<sup>z</sup>* =1, and *Gs* (*κ*, *z*)=1, then there is only time-delayed anisoplanatism. Now Eq. (18) and Eq. (23) reduce to

$$F\_{\phi}(\kappa, z) = 2\mathbf{\tilde{1}} \mathbf{1} \cdot J\_0(\kappa \ \upsilon\_z \mathbf{\tau}) \mathbf{\tilde{1}} \tag{48}$$

$$F\_{n,m}(\kappa, z) = 2C\_m N\_n^2(\kappa) \mathbf{\tilde{l}} \mathbf{1} \cdot J\_0(\kappa \ \upsilon\_z \tau) \mathbf{\tilde{l}} \tag{49}$$

Using Eq. (48) and *g*(*κ*)=1 to perform the integration in Eq. (6), the total phase anisoplana‐ tism variance can be expressed as *σϕ* <sup>2</sup> =(*<sup>τ</sup>* / *<sup>τ</sup>*0)5/3, where τ0 is normally-defined atmospheric coherence time and equal to *τ*<sup>0</sup> =(2.913*k*<sup>0</sup> 2 *<sup>ν</sup>*5/3)-3/5, and *νn* is the nwth velocity moments of at‐ mospheric turbulence defined by *ν<sup>n</sup>* =*∫* 0 *<sup>L</sup> dzCn* 2(*z*)*<sup>v</sup> <sup>m</sup>*(*z*). This characteristic quantity *τ*<sup>0</sup> is relat‐ ed to the Greenwood frequency. For a single-poles filter (controller), the variance of compensated phase can be scaled as *σϕ* <sup>2</sup> =( *<sup>f</sup>* <sup>0</sup> / *<sup>f</sup>* <sup>3</sup>*db*)5/3, where f is the effective control band‐ width of AO system and *f* 0 is the Greenwood frequency, defined by *f* <sup>0</sup> =(0.103*k*<sup>0</sup> 2 *<sup>ν</sup>*5/3)3/5. We can easily find there is a simple relationship between these two characteristic quantities:

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems http://dx.doi.org/10.5772/54602 205

$$f\_0 \approx 0.134 / \tau\_0 \tag{50}$$

as is first noted by Fried (Fried 1990).

than that limited by the isoplanatic angle *θ*0. Unlike *dn*;eff, the effective offset angle *θn*;eff is not only dependent on aperture diameter *D*, but also turbulence intensity. Therefore, for higher-order Zernike modes, the effectively offset angle is also dependent on the turbulence intensity. In Figure 4(b), the relationships between effective offset angles and turbulence in‐ tensities are showed for four different compensational orders (*n=1, 2, 3,* and *5*) at *λ=532nm*.

**Figure 4.** The characteristic angles of the angular anisoplanatism for separated beacon and target. (a) the effective offset angles at different turbulent intensities and wavelengths; (b) the relationship between effective offset angles

Using Eq. (48) and *g*(*κ*)=1 to perform the integration in Eq. (6), the total phase anisoplana‐

ed to the Greenwood frequency. For a single-poles filter (controller), the variance of

2

width of AO system and *f* 0 is the Greenwood frequency, defined by *f* <sup>0</sup> =(0.103*k*<sup>0</sup>

can easily find there is a simple relationship between these two characteristic quantities:

0 *<sup>L</sup> dzCn*

(*κ*, *z*)=1, then there is only time-delayed anisoplanatism.

*Fϕ*(*κ*, *z*)=2 1 - *J*0(*κ vzτ*) (48)

2(*κ*) 1 - *<sup>J</sup>*0(*<sup>κ</sup> vzτ*) (49)

<sup>2</sup> =(*<sup>τ</sup>* / *<sup>τ</sup>*0)5/3, where τ0 is normally-defined atmospheric

*<sup>ν</sup>*5/3)-3/5, and *νn* is the nwth velocity moments of at‐

<sup>2</sup> =( *<sup>f</sup>* <sup>0</sup> / *<sup>f</sup>* <sup>3</sup>*db*)5/3, where f is the effective control band‐

2(*z*)*<sup>v</sup> <sup>m</sup>*(*z*). This characteristic quantity *τ*<sup>0</sup> is relat‐

2

*<sup>ν</sup>*5/3)3/5. We

and turbulence intensities for four different compensational orders at λ=532nm

When *d* =0, *θ* =0, *γ<sup>z</sup>* =*α<sup>z</sup>* =1, and *Gs*

204 Adaptive Optics Progress

Now Eq. (18) and Eq. (23) reduce to

tism variance can be expressed as *σϕ*

mospheric turbulence defined by *ν<sup>n</sup>* =*∫*

compensated phase can be scaled as *σϕ*

coherence time and equal to *τ*<sup>0</sup> =(2.913*k*<sup>0</sup>

**4.3. The time-delayed anisoplanatism and related characteristic quantities**

*Fn*,*m*(*κ*, *z*)=2*CmNn*

Similarly, in the limitation *vzτ*≪*D*, the effective corrected time can be defined and deter‐ mined by *τ*eff=*τ*0. For arbitrary Zernike mode of phase, from Eq. (80), when we consider the second order approximation, the anisoplanatic variance is equal to

$$
\sigma\_{n,m}^2 = 0.879 \frac{(1+n)\Gamma(n+1/6)}{\Gamma\{n+17/6\}} \mathcal{C}\_m k\_0^2 D^{-1/3} \nu\_2 \tau^2 \tag{51}
$$

Using the isoplanatic time *τn*,*m*;*iso* satisfied *σn*,*<sup>m</sup>* <sup>2</sup> =1 to rescale, then the variance can be ex‐ pressed as:

$$
\sigma\_{n,m}^2 = (\tau \mid \tau\_{n,m;iso})^2 \tag{52}
$$

and its expression is

$$\tau\_{n,m;iso} = \frac{1.06665}{k\_0 \nu\_2^{1/2}} \sqrt{\frac{\Gamma(n+17/6) \, D^{1/3}}{\Gamma\_m(n+1)\Gamma(n+1/6)}}\tag{53}$$

Similar to Greenwood frequency, we can apply Eq. (50) to define a characteristic frequency related to the isoplanatic time in Eq. (53) as follows:

$$f\_{n,m;\text{iso}} = 0.1256 k\_0 \nu\_2^{1/2} \sqrt{\frac{C\_n(n+1)\Gamma(n+1/6)}{\Gamma(n+17/6)\ \, D}}\tag{54}$$

This is the characteristic frequency using a single-poles filter to compensate for the Zernike mode Z(m,n) of the turbulence-induced phase.

Further, the effective correction time of arbitrary n-order Zernike model of phase can be de‐ fined as:

$$\pi\_{n; \text{eff}} = \min\_{m} \left( \pi\_{n, m; \text{iso}}, \pi\_{n, m; \text{unc}} \right) \tag{55}$$

where the uncorrected time can be expressed as

$$
\pi\_{n,m;unc} = 12 \text{ D} \sqrt{\mu\_0/\nu\_2} \left| \sqrt{11(6n-5)(6n+17)} \right. \tag{56}
$$

When using an AO system with a time delay exceeding this characteristic time to compen‐ sate for the n-order Zernike model of phase, the compensation is ineffective.

The characteristic quantities *τn*;eff are similar to *θn*;eff and *dn*;eff. In Figure 5, we show some typical values of the characteristic angles *τn*;eff.

Substituting Eq. (57) into Eq. (6), the anisoplanatic variance for total phase is given by

Similarly, using Eq. (58) the anisoplanatic variance of Zernike mode *Z(m,n)* can also be cal‐ culated. In order to obtain a more simple close solution, we consider the limit case of a very high altitude beacon, i.e., *H* ≫*z*. From Eq. (81), when the second-order small quantities are

By this expression, the first two components, i.e., the anisoplanatic variances of the piston

When analyzing a LGS AO system with a telescope aperture of diameter D, it is useful to

measure of effective diameter of the LGS AO system (Tyler 1994) (i.e., a telescope with a di‐ ameter equal to *de* will have *1 rad* of rms wave-front error). Considering the fact that for a LGS system piston is meaningless and tip-tilt is non-detectable (Rigaut and Gendron 1992;

Esposito, Ragazzoni et al. 2000), then an approximated value of *de* can be obtained by

*Fn*,*m*(*κ*, *z*)=*CmNn*

Performing the integration Eq. (6), the corresponding variance is obtained as

<sup>2</sup> =0.0569*k*<sup>0</sup>

*σ*eff,*up*


We can further consider the effect of turbulence above the beacon. From Eq. (17) and Eq. (24), the filter functions for the total phase and its Zernike mode *Z(m,n)* above the beacon are

Therefore the anisoplanatic filter function of the partial phase in which the components of

2 *μ*0 + -

*F*eff,*up*(*κ*, *z*)= *Fϕ*(*κ*, *z*) - *F*0,0(*κ*, *z*) - *F*1,1(*κ*, *z*) (63)

is the mth lower turbulence moment, defined by

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

<sup>2</sup> <sup>=</sup>*σ*0,0

<sup>2</sup> =(*<sup>D</sup>* / *de*)5/3, where the characteristic quantity *de* is a

*Fϕ*(*κ*, *z*)=1 (61)

<sup>2</sup> =0.0834*k*<sup>0</sup>

*D* 5/3 / *H* <sup>2</sup> (59)

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

2 *<sup>D</sup>* 5/3*μ*<sup>2</sup> - / *H* <sup>2</sup>

/ *H* <sup>2</sup> }-3/5 (60)

2(*κ*) (62)

*D* 5/3 (64)

and

207

2 *μ*2 -

*σϕ* <sup>2</sup> =0.5*k*<sup>0</sup> 2 *μ*5/3

*μm* - =*∫* 0 *<sup>H</sup> Cn*

*σ*T <sup>2</sup> <sup>=</sup>*σ*1,1 - (*<sup>D</sup>* / *<sup>H</sup>* )5/3, here *μm*

*σn*,*<sup>m</sup>* <sup>2</sup> <sup>=</sup> 3.317 6237

2 *<sup>D</sup>* 5/3*μ*<sup>2</sup> - / *H* <sup>2</sup> .

express the anisoplanatic variance by *σϕ*

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

the piston and tip-tilt are removed can be expressed as:

<sup>2</sup> 0.5*μ*5/3

2(*z*)*z mdz*.

<sup>2</sup> =0.3549*k*<sup>0</sup>


retained, the anisoplanatic variance for *Z(m,n)* can be approximated by

and tip-tilt, can be obtained immediately as follows: *σ*<sup>P</sup>

(1 + *n*)*Γ* 2(-8 / 3) 108*n*(*n* + 2) - 55 *<sup>Γ</sup>*(-10 / 3)*Γ*(*<sup>n</sup>* + 23 / 6) / *<sup>Γ</sup>*(*<sup>n</sup>* -5/6) *Cmk*<sup>0</sup>

**Figure 5.** The characteristic times for the time-delay anisoplanatism. (a) The effective times at different turbulent in‐ tensities and wavelengths; (b) The relationship between effective times and turbulence intensities for different com‐ pensational orders at λ=532nm.

When *n = 1* and *m = 1*, the isoplanatic times or the characteristic frequencys for the tip-tilt component of the turbulence-induced phase are obtained as: *τ*1,1;*iso* =(0.668*k*<sup>0</sup> 2 *<sup>ν</sup>*2*<sup>D</sup>* -1/3)-1/2 or *<sup>f</sup>* 1,1;*iso* =0.4864 *<sup>λ</sup>* -1*ν*<sup>2</sup> 1/2 *D* -1/6. It should be noted that these results are slightly different with others. In many studies, the tilt anisoplanatic variances are calibrated as: *σα* <sup>2</sup> <sup>=</sup> <sup>1</sup> 5 ( *λ D* )2 ( *τst τ*0*t* ) 2 or *σα* <sup>2</sup> =( *<sup>λ</sup> D* )2 ( *<sup>f</sup> <sup>T</sup> f* <sup>3</sup>*db* ) 2 , Where the characteristics time (Parenti and Sasiela 1994) and frequency (Ty‐ ler 1994) are defined by *τ*0*<sup>t</sup>* =(0.512*k*<sup>0</sup> 2 *ν*-1/3 8/15*ν*14/3 7/15*<sup>D</sup>* -1/3)-1/2 or *<sup>f</sup> <sup>T</sup>* =0.368 *<sup>λ</sup>* -1*ν*<sup>2</sup> 1/2 *D* -1/6. These re‐ sults are slightly different from ours because different methods of series expanding are used. However, the differences are minor and our expressions have simpler forms and are more convenient to use.

#### **4.4. The focal anisoplanatism**

If the altitudes of beacon and target are different, then focal anisoplanatism appears. When other anisoplanatic effects are neglect (i.e., *θ* =0, *d* =0, *τ* =0, *L* = + ∞, *Gs* (*κ*, *z*)=1), the aniso‐ planatic filter function below the beacon are simplified to

$$\,\_2F\_{\phi}(\kappa,\,\,z) = 2\left[1 \cdot \frac{H}{\kappa Dz} J\_1\left(\frac{\kappa Dz}{2H}\right)\right] \tag{57}$$

$$F\_{n,m}(\kappa, z) = \mathbb{C}\_m \mathsf{f} N\_n(\kappa) \text{ - } N\_n(\alpha, \kappa) \mathsf{I}^2 \tag{58}$$

Substituting Eq. (57) into Eq. (6), the anisoplanatic variance for total phase is given by *σϕ* <sup>2</sup> =0.5*k*<sup>0</sup> 2 *μ*5/3 - (*<sup>D</sup>* / *<sup>H</sup>* )5/3, here *μm* is the mth lower turbulence moment, defined by *μm* - =*∫* 0 *<sup>H</sup> Cn* 2(*z*)*z mdz*.

The characteristic quantities *τn*;eff are similar to *θn*;eff and *dn*;eff. In Figure 5, we show some

**Figure 5.** The characteristic times for the time-delay anisoplanatism. (a) The effective times at different turbulent in‐ tensities and wavelengths; (b) The relationship between effective times and turbulence intensities for different com‐

When *n = 1* and *m = 1*, the isoplanatic times or the characteristic frequencys for the tip-tilt

sults are slightly different from ours because different methods of series expanding are used. However, the differences are minor and our expressions have simpler forms and are

If the altitudes of beacon and target are different, then focal anisoplanatism appears. When

*<sup>κ</sup>Dz J*<sup>1</sup>

( *<sup>κ</sup>Dz*

*Fn*,*m*(*κ*, *z*)=*Cm Nn*(*κ*) - *Nn*(*αzκ*) <sup>2</sup> (58)

1/2 *D* -1/6. It should be noted that these results are slightly different with

, Where the characteristics time (Parenti and Sasiela 1994) and frequency (Ty‐

7/15*<sup>D</sup>* -1/3)-1/2 or *<sup>f</sup> <sup>T</sup>* =0.368 *<sup>λ</sup>* -1*ν*<sup>2</sup>

2

<sup>2</sup> <sup>=</sup> <sup>1</sup> 5 ( *λ D* )2 ( *τst τ*0*t* ) 2 or

*<sup>ν</sup>*2*<sup>D</sup>* -1/3)-1/2 or

1/2 *D* -1/6. These re‐

(*κ*, *z*)=1), the aniso‐

<sup>2</sup>*<sup>H</sup>* ) (57)

component of the turbulence-induced phase are obtained as: *τ*1,1;*iso* =(0.668*k*<sup>0</sup>

others. In many studies, the tilt anisoplanatic variances are calibrated as: *σα*

2 *ν*-1/3 8/15*ν*14/3

other anisoplanatic effects are neglect (i.e., *θ* =0, *d* =0, *τ* =0, *L* = + ∞, *Gs*

*<sup>F</sup>ϕ*(*κ*, *<sup>z</sup>*)=2 1 - *<sup>H</sup>*

planatic filter function below the beacon are simplified to

typical values of the characteristic angles *τn*;eff.

pensational orders at λ=532nm.

206 Adaptive Optics Progress

*<sup>f</sup>* 1,1;*iso* =0.4864 *<sup>λ</sup>* -1*ν*<sup>2</sup>

more convenient to use.

**4.4. The focal anisoplanatism**

ler 1994) are defined by *τ*0*<sup>t</sup>* =(0.512*k*<sup>0</sup>

*σα* <sup>2</sup> =( *<sup>λ</sup> D* )2 ( *<sup>f</sup> <sup>T</sup> f* <sup>3</sup>*db* ) 2 Similarly, using Eq. (58) the anisoplanatic variance of Zernike mode *Z(m,n)* can also be cal‐ culated. In order to obtain a more simple close solution, we consider the limit case of a very high altitude beacon, i.e., *H* ≫*z*. From Eq. (81), when the second-order small quantities are retained, the anisoplanatic variance for *Z(m,n)* can be approximated by

$$\sigma\_{n,m}^{2} = \frac{3.317}{6237} \frac{(1+n)\Gamma^{2}(-8/3)\mathsf{I}\{108n\langle \mathfrak{u} + 2 \rangle - 55\}}{\Gamma(-10/3)\Gamma(n + 23/6)\langle \mathfrak{I}\{n - 5/6\})} \mathbb{C}\_{m} k\_{0}^{2} \mu\_{2}^{-} \mathsf{D}^{5/3} \left/ H^{2} \right. \tag{59}$$

By this expression, the first two components, i.e., the anisoplanatic variances of the piston and tip-tilt, can be obtained immediately as follows: *σ*<sup>P</sup> <sup>2</sup> <sup>=</sup>*σ*0,0 <sup>2</sup> =0.0834*k*<sup>0</sup> 2 *<sup>D</sup>* 5/3*μ*<sup>2</sup> - / *H* <sup>2</sup> and *σ*T <sup>2</sup> <sup>=</sup>*σ*1,1 <sup>2</sup> =0.3549*k*<sup>0</sup> 2 *<sup>D</sup>* 5/3*μ*<sup>2</sup> - / *H* <sup>2</sup> .

When analyzing a LGS AO system with a telescope aperture of diameter D, it is useful to express the anisoplanatic variance by *σϕ* <sup>2</sup> =(*<sup>D</sup>* / *de*)5/3, where the characteristic quantity *de* is a measure of effective diameter of the LGS AO system (Tyler 1994) (i.e., a telescope with a di‐ ameter equal to *de* will have *1 rad* of rms wave-front error). Considering the fact that for a LGS system piston is meaningless and tip-tilt is non-detectable (Rigaut and Gendron 1992; Esposito, Ragazzoni et al. 2000), then an approximated value of *de* can be obtained by

$$d\_e = \left| k\_0^2 \text{[0.5} \mu\_{5/3}^- \right| H^{\ 5/3} \text{ - 0.4383} \mu\_2^- \left| H^{\ 2} \right| \text{]}^{\ \cdot 3\%} \tag{60}$$

We can further consider the effect of turbulence above the beacon. From Eq. (17) and Eq. (24), the filter functions for the total phase and its Zernike mode *Z(m,n)* above the beacon are

$$F\_{\phi}(\kappa, z) = 1\tag{61}$$

$$F\_{n,m}(\kappa, z) = C\_m N\_n^2(\kappa) \tag{62}$$

Therefore the anisoplanatic filter function of the partial phase in which the components of the piston and tip-tilt are removed can be expressed as:

$$F\_{\rm eff,up}(\kappa, z) = F\_{\phi}(\kappa, z) - F\_{0,0}(\kappa, z) - F\_{1,1}(\kappa, z) \tag{63}$$

Performing the integration Eq. (6), the corresponding variance is obtained as

$$
\sigma\_{\rm eff,up}^2 = 0.0569 k\_0^2 \mu\_0^\* D^{5/3} \tag{64}
$$

Where *μ*<sup>0</sup> + is the mth upper turbulence moment, defined by *μm* <sup>+</sup> =*∫ H* <sup>∞</sup>*Cn* 2(*z*)*z mdz*. So when con‐ sider the effect of turbulence above the beacon, the effective diameter can be expressed ap‐ proximately as

$$d\_e = \left| k\_0 \right\| 0.0569 \mu\_0^+ + 0.5 \mu\_{5/3}^- \left| H^{\ 5/3} \text{ - } 0.4383 \mu\_2^+ \left| H^{\ 2} \right\| \right\|^{\cdot 3/5} \tag{65}$$

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

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

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‐ duced to

$$F\_{\phi}(\kappa, z) = \left[1 \cdot G\_{s}(\kappa, z)\right]^{2} \tag{66}$$

*Cn*

will be use. Where *κo* and *κ<sup>i</sup>*

this paper.

hybrid. Let *Gs*

are reduced to

2(*z*)= *<sup>A</sup> exp*(- *<sup>z</sup>*

time-delay, we use the Bufton wind model

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

<sup>10</sup><sup>2</sup> ) +

2.7 <sup>10</sup><sup>16</sup> *exp*(- *<sup>z</sup>*

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

(*κ*, *z*)=1, *L* =*H* (=3.8×10<sup>8</sup>

*Fn*,*m*(*κ*, *z*)=2*CmNn*

<sup>1500</sup> ) +

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‐

and the inner scale of the atmospheric turbulence, respectively. To consider the effect of

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

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

angle of apertures and sources as d and *θ*, then the anisoplanatic filter function in altitude z

2

lence strength at the ground. At the same time, the modified von Karman spectrum

*g*(*κ*)= 1 + (*κ<sup>o</sup>* / *κ*)<sup>2</sup> -11/6*exp* -(*κ* / *κ<sup>i</sup>*

5.94 <sup>10</sup><sup>3</sup> ( *<sup>w</sup>* 27 )2 ( *z* <sup>10</sup><sup>5</sup> )<sup>10</sup>

*exp*(- *<sup>z</sup>*

are the space wave numbers corresponding to the outer scale

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

*vz* <sup>=</sup>*vg* + 30 *exp* -(*<sup>z</sup>* - 9400)2 / <sup>4800</sup><sup>2</sup> (70)

*m*), and denote respectively the offset distance and

(*γzκ*) 1 - *J*0(*szκ*) (72)

*Fϕ*(*κ*, *z*)=2 1 - *J*0(*szκ*) (71)

<sup>10</sup><sup>3</sup> ) (68)

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

209

)<sup>2</sup> (69)

$$F\_{n,m}(\kappa, z) = C\_m N\_n^2(\kappa) \mathbf{[1 - G\_s(\kappa, z)]^2} \tag{67}$$

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

For the total phase, the integration can easily be obtained. The result is *σϕ* <sup>2</sup> =0.5327 *θ<sup>r</sup>* 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‐ mospheric isoplanatic angle. Obviously, the result is similar to the classic *5/3* power scaling law for angular anisoplanatism.

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‐ ton and tip-tilt) as follows: *σ<sup>P</sup>* <sup>2</sup> <sup>=</sup>*σ*0,0 <sup>2</sup> =0.5327*θ<sup>r</sup>* 5/3 *<sup>μ</sup>*5/3 - 0.4369 *<sup>D</sup>* 5/3*μ*0 and *σT* <sup>2</sup> <sup>=</sup>*σ*1,1 <sup>2</sup> =0.3799 *<sup>D</sup>* 5/3*μ*0.
