## 1. Introduction

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

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

Conventionally, all kinds of anisoplanatic effects are studied individually, assuming that they are statistically uncorrelated, and the total effects are obtained by summing them all together when necessary (Gavel, Morris et al. 1994). This conventional approach has a rich history dating back to the earliest days of AO technology and has obtained many good results. But this approach is very limited, because for actual applications of AO systems, many kinds of anisoplanatic effects exist simultaneously and are dependent on each other (Tyler 1994). It is increasingly obvious that these methods are inadequate to treat the diverse nature of new AO applications and the concept of anisoplanatism, and the associated analysis methods must be expanded to treat these new systems so their performance may be properly assessed.

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

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

## 2. General analysis geometry

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

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

where

Under some hypotheses, these expressions can be further simplified. We suppose two apertures are at the same altitudes and select the centre of the sensing aperture as the origin of coordinates. We express the positions of target and beacon with the zenith angle and altitude as* ( θ → t , L)* and* (**, H)*, respectively. We notice that in studying anisoplanatic effects, the offsets angular is very small in general (Welsh and Gardner 1991), i.e.,

where

Further, if we consider delayed-time (

where

The above is the most general geometric relationship of AO systems. Depending on the conditions of application, more simple geometry can often be used to consider the anisoplanatism of AO systems. Some examples are showed in Figure 2. When the target is sufficiently bright, wave-front perturbation can be measured by directly observing the target. Thus an ideal compensation can be obtained and no anisoplanatism exists. This case is showed in Figure 2(a). In general, the target we are interested in is too dim to provide wave-front sensing, another bright beacon in the vicinity of the target must be used, as depicted in Figure 2(b). In this case, the so-called angular anisoplanatism exits (Fried 1982). In more general cases, a naturally existed object (NGS) cannot be find appropriately, 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) appears 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 distributed 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, Roggemann et al. 1998). Figure 2(f) illustrates a hybrid case, in which many anisoplanatic effects coexist at the same time.

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

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

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 turbulence-induced phase-related quantity for the propagating waves, when diffraction is ignored, can be written as:

where *L* is the propagation distance and * λ* by

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

where

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

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

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

While when

In above two equations,

Below we list some explicit expressions of complex filter functions.

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

Here

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

For the global phase, the complex filter function is

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

For

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

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

While when

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

When *x, y* component of Zernike mode, we can write their anisoplanatic filter functions as follows:

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

then we can obtain

Where

Similarly for the case

## 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

The anisoplanatic phase variance is easily obtained. Substituting Eq. (25) into Eq. (6), and using the Kolmogrov spectrum, i.e.,

Here we have calibrated the variance with a new characteristic distance

This is about *1/3* of the atmospheric coherence length *5/3* power scaling law with the distance of separated 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 integral of the piston-removed anisoplanatic phase filter function

When

While

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:

From the above equations we find that in the limitation of

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

Then the compensation is ineffective and the AO system is not needed. We can define the uncorrected distance

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 distance 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

The above two characteristic distances (

In general, the inequality

Similar to the above analysis and definitions for total phase, anisoplanatic variances and related 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

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:

If we defined the isoplanatic distance of the Zernike mode *Z(m,n)*

This characteristic distance can be determined as follows:

For a single beam, the expression corresponding to Eq. (34) is (

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

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

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 *D=1.2m*. As a comparison with the total phase, the value of piston-removed quantity

In Figure 3(a), the relationship among *λ=532nm*. From this figure, we find that the isoplanatic distance is monotonous - increasing 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 distance 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 orders at different turbulent intensities and wavelengths. In Figure 3(b), four different turbulent intensities (*=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 (*=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 wavelengths.

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

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

Now we consider the geometry where only angular anisoplanatism exits. Let

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

Similarly, in the limitation of very small offset angle, i.e.,

Using Eq. (42), the angular anisoplanatism of Zernike modes can also be calculated. The results 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 *Z(m,n)* can be expanded to the turbulence second-order structure constant moments and can be expressed as (

where the characteristic angle

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

When

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:

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

In Figure 4, the typical values of the characteristic angles *D=1.2m*. In Figure 4(a), we compare the values for two different turbulent intensities (*=5cm, 10cm*) and two different wavelengths (*λ=532nm*, and *1064nm*). We can also find that the effective offset angles *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 intensities are showed for four different compensational orders (*n=1, 2, 3,* and* 5*) at *λ=532nm*.

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

When

Using Eq. (48) and

as is first noted by Fried (Fried 1990).

Similarly, in the limitation

Using the isoplanatic time

and its expression is

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

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 defined as:

where the uncorrected time can be expressed as

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

The characteristic quantities

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:

### 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.,

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 calculated. In order to obtain a more simple close solution, we consider the limit case of a very high altitude beacon, i.e., *Z(m,n)* can be approximated by

By this expression, the first two components, i.e., the anisoplanatic variances of the piston and tip-tilt, can be obtained immediately as follows:

When analyzing a LGS AO system with a telescope aperture of diameter D, it is useful to express the anisoplanatic variance by *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

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

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

Where

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

Substituting above two equations into Eq. (6) and performing the integration, the anisoplanatic 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

For the total phase, the integration can easily be obtained. The result is *5/3* power scaling law for angular anisoplanatism.

For Zernike component *Z(m,n)*, we consider the limit case of very big

## 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 obtained - only numeric results.

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

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

will be use. Where

Where

### 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 successful LLR system (Dickey, Bender et al. 1994). One of the main reasons is that the quality of the outgoing laser beams deteriorates sharply due to the effect of atmospheric turbulence, including the wandering, expansion, and scintillation. To mitigate these effects 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 measured 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

where

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

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 variances usually monotonously increase with the separated distance. We can also see that increasing 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

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 separated apertures) exist at the same time and the corresponding parameters are selected as

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

A laser beacon is insensitive to full-aperture tilt because the beam wanders on both the upward 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 other Zernike modes except tip-tilt can be corrected by LGS subsystems, but the corrected performance 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 anisoplanatism of the defocus component is very important (Esposito, Riccardi et al. 1996; Neyman 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 defocus 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 statistically independent of each other. Then the anisoplanatic filter functions for the NGS subsystem are reduced to

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

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

In above equations,

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

In this equation, the first two items in parentheses are the contribution of the NGS subsystem, 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 variance of the total phase, the piston, the tip-tilt and the defocus mode, sequentially. As a comparison, 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:

Obviously, for this special LGS AO system, the contribution of the defocus mode to the effective anisoplanatic variance comes from the NGS system, i.e.,

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 (

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 component 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 altitude *H=15km* and *H=90km* respectively. It is easy to see that the variances generally increase 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 increases. 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* *0* to *15km* Rayleigh LGS, as showed in Figure 7(b).

In Figure 7(c) and (d), the components of anisoplanatic variance below and above the beacon, are given respectively. The values for the total phase and its first three components (piston, 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 anisoplanatic 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 sub-system and not from the LGS subsystems as usual LGS AO systems. In Figure 8, we compare 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 altitude of LGS is showed in the left longitudinal coordinates. Similarly, the dotted line describes 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*

## 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 geometry of AO systems. This general geometry can describe most kinds of anisoplanatism appearing in currently running AO systems, including angular anisoplanatism, focal anisoplanatism 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.

## Appendix

Here we give some expressions describing the integrations of the anisoplanatic filter function *Z(m,n)* with respect to the radial component of the wave vector, i.e.,

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

When the filter function is equal to

The results are determined by the sizes of

Further, if

if

Further if

### Expressions used in section 4.4.

When the filter function is equal to

The integrations can be expressed by

When the beacon is high enough (

### Expressions used in section 4.5.

When the filter function is equal to

For the Gaussian source Eq. (12), the results can be expressed as:

If the widths of the distributed source are very large (