**3. Transverse spectral filtering method and general expressions of corrected (anisoplanatic) wave-front variance**

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‐

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‐

**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‐

fects coexist at the same time.

194 Adaptive Optics Progress

planatic variance under the most general geometry.

brid beacon - many anisoplanatic effects existing at the same time.

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‐ ed, can be written as:

*<sup>σ</sup>* <sup>2</sup> =2*πk*<sup>0</sup> 2 *∫* 0 *<sup>L</sup> dz*(*∫*Φ(*κ* <sup>→</sup> , *z*) *f* (*κ* <sup>→</sup> , *z*)*dκ* <sup>→</sup> ) (4)

where *L* is the propagation distance and k0 is the space wave number, which when related to wavelength *λ* by *k*<sup>0</sup> =2*π* / *λ*; Φ(*κ* <sup>→</sup> , *z*) is two-dimensional transverse power spectrum of fluctuated refractive-index at the plane vertical to the direction of wave propagation and *κ* <sup>→</sup> =(*κ*, *φ*); *f* (*κ* <sup>→</sup> , *z*) is the transverse spectral filter function related to this calculated quantity, whose explicit form can be determined by the corresponding physical processes.

For the atmospheric turbulence, the two-dimensional transverse power spectrum of fluctu‐ ated refractive-index can generally be written as:

$$\Phi(\vec{\kappa}, z) = 0.033 C\_n^2(z) g(\kappa) \kappa^{-11/3} \tag{5}$$

where *Cn* 2(*z*) is refractive-index structure parameter which is allowed to vary along the propagation path, and *g*(*κ*) is the normalized spectrum. If *g*(*κ*)=1, then the classic Kolmog‐ rov spectrum is obtained.

Now substitute Eq. (5) into Eq. (4), and sequentially perform the integration of wave vector *κ* <sup>→</sup> =(*κ*, *φ*) at the angular and radial components (Sasiela and Shelton 1993), then the variance reduces to

$$
\sigma^2 = 0.4147 \pi k\_0^2 \mathbf{j}\_0^L \text{ } dz \mathbf{C}\_n^2(z) \mathbf{I}\_F(z) \tag{6}
$$

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

$$I\_F(z) = \int\_0^\infty F(\kappa, z) g(\kappa) \kappa^{-8/3} d\kappa \tag{7}$$

$$F(\kappa, z) = \frac{1}{2\pi} \zeta\_0^{2\pi} f(\vec{\kappa}, z) d\varphi \tag{8}$$

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 anisoplanatic filter function can be created from some complex filter functions, describing the process related to the observed target and beacon respectively, by taking the absolute value squared of their difference.

Clearly, when *z* ≥*H* , the anisoplanatic filter function is

$$|f\left(\vec{\kappa}, z\right) = |G\left(\mathbf{y}\_z \vec{\kappa}\right)|^2\tag{9}$$

*Gn*,*<sup>m</sup> <sup>x</sup>* (*κ* → ) *Gn*,*<sup>m</sup> <sup>y</sup> κ*

In previous two equations, *D* is the diameter of aperture and

global phase and its Zernike modes can be established explicitly.

For the total phase, When *z* ≥*H* , from Eq. (9) and Eq. (13), it is

While when z<*H* , from Eq. (10) and Eq. (13), the result is

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

the case z<*H* , when *m*=0, it can be given by the expression

2 (*αzκ*)*Gs*

*<sup>x</sup>* (*κ*, *<sup>z</sup>*)= *Nn*

*<sup>x</sup>* (*κ*, *<sup>z</sup>*)= *Nn*



It is easy to find that if we define a new quantity as follows:

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

2 (*αzκ*) *Gs*

2

2

(*γzκ*) + *Nn*

(*γzκ*) + *Nn*

(*γzκ*) + *Nn*

*Fn*,*<sup>m</sup>*

*Fn*,*<sup>m</sup>*

*Fn*,0(*κ*, *z*)= *Nn*

functions as follows:

then we can obtain

*Fn*,*m*(*κ*, *z*) = *Cm Nn*

2

(*γzκ*) + *Nn*

2

*Nn*(*κ*

<sup>→</sup> } =*i <sup>m</sup>* 2(-1)

(*n*-*m*)/2*Nn*(*<sup>κ</sup>*

By the above complex filter functions, the expressions of anisoplanatic filter functions of

Similarly, the anisoplanatic filter functions for Zernike modes can also be established. For

When *m*≠0, for the *x, y* component of Zernike mode, we can write their anisoplanatic filter

2 (*αzκ*)*Gs*

> 2 (*αzκ*)*Gs*

*<sup>x</sup>* (*κ*, *<sup>z</sup>*) <sup>+</sup> *Fn*,*<sup>m</sup>*

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

<sup>→</sup> ){

(*κ*, *z*)*J*0(*szκ*) + *Gs*

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

2(*κ*, *z*)+

2(*κ*, *z*)

(*κ*, *<sup>z</sup>*) *<sup>J</sup>*0(*szκ*) - (-1)*mJ*2*m*(*szκ*) (20)

(*κ*, *<sup>z</sup>*) *<sup>J</sup>*0(*szκ*) <sup>+</sup> (-1)*mJ*2*m*(*szκ*) (21)

*<sup>y</sup>* (*κ*, *z*) (22)

(*κ*, *z*) *J*0(*sz κ*) (23)

*cos*(*mφ*)

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

<sup>→</sup> )=2 *n* + 1*Jn*+1(*κD* / 2) / (*κD* / 2) (16)

*Fϕ*(*κ*, *z*)=1 (17)

*sin*(*mφ*) (15)

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197

2(*κ*, *z*) (18)

(*κ*, *z*)*J*0(*szκ*) (19)

While when z<*H* , this item can be expressed as:

$$f(\vec{\kappa}, z) = |G(\vec{\gamma}, \vec{\kappa}) \exp(i \vec{\kappa} \bullet \vec{s}\_z) - G(a\_z \vec{\kappa}) G\_s(\vec{\kappa}, z)|^2 \tag{10}$$

In above two equations, *G*(*κ* <sup>→</sup> ) is the complex filter function corresponding to the wanted quantity, while *Gs* (*κ* <sup>→</sup> , *z*) is a complex function which can describe the characteristic of the beacon (such as distributed or point-like). When writing this equation, we have supposed that the main physical processes are linear and their complex filter function can be cascaded to form the total filter functions.

Below we list some explicit expressions of complex filter functions.

The transverse complex filter function for a uniform, circular source with angular diameter *θr*, can be expressed as:

$$\mathcal{G}\_s(\vec{\kappa}\_\prime \ z) = 2J\_1(\kappa \theta\_r z) / \langle \kappa \theta\_r z \rangle \tag{11}$$

Here *Jn*(∙ ) is the nth-order of Bessel function of the first kind; Similarly, the filter function for a Gaussian intensity distribution with 1 /*e* radius *θr*, has a complex filter function

$$G\_s(\vec{\kappa}, z) = \exp\left[ \cdot (\kappa \theta\_r z)^2 / 2 \right] \tag{12}$$

We also notice that for a point-like beacon, the filter function is simply 1.

For the global phase, the complex filter function is

$$G\_{\phi}(\vec{\kappa}, \vec{\rho}) = \exp(i \,\, \vec{\kappa} \bullet \vec{\rho}) \tag{13}$$

The expression of the complex filter function for Zernike mode *Z(m,n)* depends on its radial (*n*) and azimuthal (*m*) order. For *m*=0, it can be written as:

$$G\_{n,0}(\vec{\kappa}) = (-1)^{n/2} N\_n(\vec{\kappa}) \tag{14}$$

For *m*≠0, it is given by

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems http://dx.doi.org/10.5772/54602 197

$$
\begin{vmatrix} G\_{n,m}^{\ast,\ast}(\vec{\kappa}) \\ G\_{n,n}^{\ast,\ast} \end{vmatrix} = \mathbf{i}^m \sqrt{2} (-1)^{(n-m)/2} N\_n(\vec{\kappa}) \begin{vmatrix} \cos(m\varphi) \\ \sin(m\varphi) \end{vmatrix} \tag{15}
$$

In previous two equations, *D* is the diameter of aperture and

anisoplanatic filter function can be created from some complex filter functions, describing the process related to the observed target and beacon respectively, by taking the absolute

<sup>→</sup> )|<sup>2</sup> (9)

<sup>→</sup> , *z*)|2 (10)

value squared of their difference.

In above two equations, *G*(*κ*

to form the total filter functions.

*θr*, can be expressed as:

For *m*≠0, it is given by

quantity, while *Gs*

196 Adaptive Optics Progress

Clearly, when *z* ≥*H* , the anisoplanatic filter function is

<sup>→</sup> , *z*)=|*G*(*γzκ*

Below we list some explicit expressions of complex filter functions.

*Gs* (*κ*

> *Gs* (*κ*

For the global phase, the complex filter function is

(*n*) and azimuthal (*m*) order. For *m*=0, it can be written as:

We also notice that for a point-like beacon, the filter function is simply 1.

*Gϕ*(*κ* <sup>→</sup> , *ρ*

*Gn*,0(*κ*

While when z<*H* , this item can be expressed as:

*f* (*κ*

(*κ*

*f* (*κ*

<sup>→</sup> )*exp*(*iκ*

<sup>→</sup> , *z*)=|*G*(*γzκ*

<sup>→</sup> ∙*s* →

beacon (such as distributed or point-like). When writing this equation, we have supposed that the main physical processes are linear and their complex filter function can be cascaded

The transverse complex filter function for a uniform, circular source with angular diameter

Here *Jn*(∙ ) is the nth-order of Bessel function of the first kind; Similarly, the filter function

for a Gaussian intensity distribution with 1 /*e* radius *θr*, has a complex filter function

<sup>→</sup> )=*exp*(*i κ*

The expression of the complex filter function for Zernike mode *Z(m,n)* depends on its radial

<sup>→</sup> )=(-1)*<sup>n</sup>*/2*Nn*(*<sup>κ</sup>*

<sup>→</sup> ∙*ρ*

*<sup>z</sup>*) - *G*(*αzκ*

<sup>→</sup> )*Gs* (*κ*

<sup>→</sup> , *z*) is a complex function which can describe the characteristic of the

<sup>→</sup> ) is the complex filter function corresponding to the wanted

<sup>→</sup> , *z*)=2*J*1(*κθrz*) /(*κθrz*) (11)

<sup>→</sup> , *z*)=*exp* -(*κθrz*)<sup>2</sup> / 2 (12)

<sup>→</sup> ) (13)

<sup>→</sup> ) (14)

$$N\_n(\vec{\kappa}) = 2\sqrt{n+1}I\_{n+1}(\kappa D/2) / (\kappa D/2) \tag{16}$$

By the above complex filter functions, the expressions of anisoplanatic filter functions of global phase and its Zernike modes can be established explicitly.

For the total phase, When *z* ≥*H* , from Eq. (9) and Eq. (13), it is

$$F\_{\phi}(\kappa, z) = 1\tag{17}$$

While when z<*H* , from Eq. (10) and Eq. (13), the result is

$$F\_{\phi}(\kappa, z) = 1 - 2N\_0(\{\gamma\_z - \alpha\_z\} \kappa) G\_s(\kappa, z) I\_0(s\_z \kappa) + G\_s^2(\kappa, z) \tag{18}$$

Similarly, the anisoplanatic filter functions for Zernike modes can also be established. For the case z<*H* , when *m*=0, it can be given by the expression

$$F\_{n,0}(\kappa, z) = N\_n^2(\chi, \kappa) + N\_n^2(a\_z \kappa) \mathbb{G}\_s^2(\kappa, z) - 2N\_n(\chi, \kappa) N\_n(a\_z \kappa) \mathbb{G}\_s(\kappa, z) I\_0(s\_z \kappa) \tag{19}$$

When *m*≠0, for the *x, y* component of Zernike mode, we can write their anisoplanatic filter functions as follows:

$$\begin{aligned} \mathbf{F}\_{n,m}^{\times}(\kappa,\ z) &= \mathbf{N}\_n^2(\gamma\_z \kappa) + \mathbf{N}\_n^2(\alpha\_z \kappa) \mathbf{G}\_s^2(\kappa, z) + \\ -2\mathbf{N}\_n(\gamma\_z \kappa) \mathbf{N}\_n(\alpha\_z \kappa) \mathbf{G}\_s(\kappa, z) \mathbf{I}\_0(\kappa, z) \mathbf{I}\_0(\kappa\_z \kappa) - (-1)^m \mathbf{I}\_{2m}(\kappa\_z \kappa) \mathbf{I}\_1 \end{aligned} \tag{20}$$

$$\begin{aligned} \mathbf{F}\_{n,m}^{\times}(\kappa, z) &= \mathbf{N}\_n^2(\gamma\_z \kappa) + \mathbf{N}\_n^2(\alpha\_z \kappa) \mathbf{G}\_s^2(\kappa, z) \\ -2\mathbf{N}\_n(\gamma\_z \kappa) \mathbf{N}\_n(\alpha\_z \kappa) \mathbf{G}\_s(\kappa, z) \mathbf{I}\_0(\kappa\_z \kappa) + (-1)^m \mathbf{I}\_{2m}(\kappa\_z \kappa) \mathbf{I}\_1 \end{aligned} \tag{21}$$

It is easy to find that if we define a new quantity as follows:

$$F\_{n,m}(\kappa, z) = F\_{n,m}^{\;\;\;x}(\kappa, z) + F\_{n,m}^{\;\;y}(\kappa, z) \tag{22}$$

then we can obtain

$$F\_{n,m}(\kappa, z) = \mathbb{C}\_m \mathbb{E}\left[N\_n^2(\mathbb{y}, \kappa) + N\_n^2(\alpha, \kappa) \: \mathrm{G}\_s^2(\kappa, z) \cdot 2 \; N\_n(\mathbb{y}, \kappa) \; N\_n(\alpha, \kappa) \: \mathrm{G}\_s(\kappa, z) \; I\_0(\mathrm{s}, \kappa)\right] \tag{23}$$

Where *Cm* is a constant factor related to the azimuthal order *m*. If *m*=0, then *Cm* =1; other‐ wise *Cm* =2.

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

$$F\_{n,m}(\kappa, z) = \mathbb{C}\_m \mathbf{N}\_n^2(\mathbb{Y}\_z \kappa) \tag{24}$$

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

*<sup>π</sup> <sup>Γ</sup>*(17 / 6)*Γ*(23 / 6) - *<sup>Γ</sup>*(1/6)

*Γ*(7/3)*Γ*(-5 / 6) *π Γ*(17 / 6)*Γ*(23 / 6)

On the other hand, the wave vector integral of the piston-removed phase filter function for a

*π Γ*(17 / 6)*Γ*(23 / 6)

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

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

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‐

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

<sup>2</sup> <sup>|</sup>*d*≪*<sup>D</sup>* <sup>&</sup>gt;*σ*ϕeff,*single*

<sup>4</sup>*Γ*(5/6) ( *<sup>D</sup>*

A Unified Approach to Analysing the Anisoplanatism of Adaptive Optical Systems

*<sup>D</sup>* )<sup>2</sup> <sup>51425</sup> <sup>41472</sup> ( *<sup>d</sup> D* )2 - <sup>935</sup>

( *D* <sup>2</sup> )5/3( *<sup>d</sup>*

( *D*

*<sup>d</sup>* )1/3 (29)

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

199

<sup>2</sup> )5/3 (31)

<sup>2</sup> (32)

<sup>144</sup> (30)

be found easily from the Eq. (79) and (80) in Appendix with *n* =0.

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

*<sup>D</sup>* )5/3 +

<sup>2</sup> )5/3( *<sup>d</sup>*

single wave beam is easy to find and can be expressed as:

*<sup>I</sup>*ϕeff,*aniso* ~ ( *<sup>D</sup>*

*<sup>Γ</sup>*(11 / 6) ( *<sup>D</sup>*

*<sup>I</sup>*ϕeff,*aniso* ~- *<sup>Γ</sup>*(-5 / 6)

wave, that is to say

tance can be given by *dunc* =0.828*D*.

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

<sup>2</sup> )5/3 - <sup>4</sup>*Γ*(-5 / 6)*Γ*(7/3)

*<sup>I</sup>*ϕeff,*single* = - <sup>2</sup>*Γ*(-5 / 6)*Γ*(7/3)

separated distance, which is same as that for the total phase in Eq. (27).

*σ*ϕeff,*aniso*

the scaling law of Eq. (27), this distance is same as *d*0, i.e., *diso* =*d*0.
