2.1 Atmospheric turbulence

The task of an AO system is to correct wavefront aberrations introduced by the turbulent atmosphere above the telescope. Turbulent motions in the atmosphere mix eddies with different temperatures and thus different densities. As a consequence light propagating along different paths through a turbulent medium experiences a different refractive index. The result is an optical path difference, which in turn leads to deformations of the incoming wavefronts.

A good understanding of the atmospheric properties above the telescope site and the resulting wavefront aberrations is crucial for the design and operation of an AO system. The importance of site survey and site characterization efforts cannot be overemphasized in this context. Key atmospheric parameters that determine the design and performance of an AO system include the Fried parameter r0, the Greenwood frequency fc, and the atmospheric turbulence profile 2 Cn (h). 2 Cn is the refractive index (n = refractive index) structure constant, which will be defined and discussed below. By using turbulence theory, these properties can be used to design AO systems and predict their performance. The most widely used model to describe atmospheric turbulence is the Kolmogorov model (Kolmogorov, 1941Jump To The Next Citation Point, 1991Jump To The Next Citation Point). Energy is introduced into the system by wind flows at a large scale (the outer scale) and cascades down to ever smaller scales until, finally, energy is dissipated at molecular scales (the inner scale). The inertial range is bound by the outer and the inner scale and marks the regime where the turbulent power of the temperature fluctuations ΦT as a function of spatial wave number κ = 2π ∕l can be expressed by a power law:

ΦT (κ ) ∝ κ−5∕3. (1 )

Similar power laws can be derived for other quantities such as the refractive index power spectral density in one dimension (Hardy, 1998Jump To The Next Citation Point; Tatarskii, 1967):

( ) ( ) Γ 5 sin π ΦN (κ) = ---3---5∕33--C2n κ−5∕3 = 0.0365 C2nκ−5∕3. (2 ) (2π)

The equivalent in three dimensions leads to the well known − 11∕3 κ power law (Roggemann and Welsh, 1996Jump To The Next Citation Point; Quirrenbach, 2002Jump To The Next Citation Point):

( ) ( ) Γ 8 sin π 3D Φn(κ) = ---3--2---3-C2nκ− 11∕3 4π 2 −11∕3 = 0.033 Cn κ . (3 )

The refractive index structure constant 2 Cn is a measure for the strength of the turbulence and is related to the temperature structure constant 2 CT by: Cn = δn∕δT CT.

At this point it is convenient to discuss the concept of the structure function, which was introduced by Kolmogorov to describe non stationary random processes such as turbulence. In general, structure functions are very useful in assessing the impact of turbulence on the image quality provided by an imaging system, independent of whether the aberrations are caused by atmospheric turbulence or optical imperfections. Today, structure functions are used to, for example, specify optical polishing tolerances.

In the context of atmospheric turbulence the refractive index structure function is assumed to be isotropic and is defined as:

2 Dn (ρ) = ⟨|n (⃗r) − n (⃗r + ⃗ρ )| ⟩ = C2nρ2∕3. (4 )
The angled brackets ⟨...⟩ indicate an ensemble average and ⃗ρ defines a spatial separation, for example in the pupil plane of a telescope. The refractive index structure function can be related to the phase structure function, which is essential in determining the performance of an imaging system in the presence of turbulence. The phase structure function produced by a layer of thickness δh is given by Hardy (1998Jump To The Next Citation Point), Roggemann and Welsh (1996), and Quirrenbach (2002Jump To The Next Citation Point):
D (ρ,h) = 2.914k2 δhC2 (h )ρ5∕3. (5 ) φ ni
The wavenumber k = 2πλ, where λ is the wavelength. It should be noted that the phase shift introduced can be related to the refractive index fluctuations along the optical path by ∫ Φ = n(h)dh.

Hence, if multiple turbulence layers (or a continuum) are present, Equation (5View Equation) has to be integrated along the line-of-sight. The zenith angle γ is introduced to account for changes in the length of the line-of-sight travel path with observing angle:

∫ D φ(ρ) = 2.914 k2(secγ )ρ5∕3 C2n(h)dh. (6 )

The Fried parameter r0 is a measure for the strength of the turbulence and is defined as (Fried, 1966a,b):

[ ∫ ]−3∕5 2 2 r0 ≡ 0.423 k (secγ) C n(h)dh . (7 )
The Fried parameter gives the diameter of a patch in the aperture plane over which the wavefront can be regarded as flat. More precisely, flat in this case means the wavefront variance is less than 1 rad2. In that sense the Fried parameter can be interpreted as the smallest AO relevant scale of turbulence. Of course, the turbulent spectrum contains much smaller spatial scales. The Fried parameter is often used to quantify the seeing quality at an astronomical site. The statistical distribution as well as the temporal evolution of the Fried parameter ultimately determines the performance of an AO system at the site. The value of the Fried parameter depends on wavelength according to r0 ∝ λ6∕5. This means r 0 has to be specified at a certain wavelength, typically 500 nm. The Fried parameter r0 is larger for longer wavelengths, which means the seeing is significantly better at infrared wavelengths. Hence, correcting the seeing with AO becomes easier at longer wavelengths.

The phase structure function can be expressed in terms of the Fried parameter resulting in a much simpler form of D φ(ρ):

( ρ )5 ∕3 Dlong(ρ) = 6.88 -- . (8 ) r0

This phase structure function describes the performance of imaging systems in the presence of turbulence. Equation 8View Equation is often referred to as the uncorrected, long exposure phase structure function. This simple form for the structure function is valid only in a statistical sense, i.e., when averaged over a sufficient amount of independent realizations of the atmospheric turbulence. Marino (2007Jump To The Next Citation Point) showed that, for daytime observations, exposures of only a few seconds can be regarded as long exposures. Solar astronomers tend to use such long exposure times when performing precision polarimetric measurements. High resolution spectroscopy sometimes requires exposure times of similar length. However, broad-band solar imaging is typically performed with very short exposures on the order of a few milliseconds with the intention to freeze the seeing. Image motion (tip and tilt) can essentially be removed in this way and the images contain speckle, i.e., structure with width of the diffraction limit λ- D. The fact that diffraction limited information is contained in the short exposure images can then be used to restore the true image by applying post-facto image reconstruction techniques. Löfdahl et al. (2007Jump To The Next Citation Point) give a review of various post-facto reconstruction in solar astronomy.

An analytical expression for the short exposure structure function can also be given (see, e.g., Hardy, 1998Jump To The Next Citation Point, p. 92):

( )5∕3( ( ) ) D (ρ) = 6.88 ρ- 1 − -ρ 1∕3 . (9 ) short r0 D

The negative impact of atmospheric turbulence on imaging systems has been described in detail in textbooks (e.g., Hardy, 1998Jump To The Next Citation Point, Section 3). It is the de-correlation of the phase over distances larger than r0 that results in blurred images where the typical width of the blurred long exposure image is λ- r0, i.e., the resolution is seeing limited.

The long exposure Optical Transfer Function (OTF) is defined as the ensemble average of the instantaneous OTFs over the entire exposure:

OTF (⃗ρ∕λ) = ⟨OTF (⃗ρ∕λ,t)⟩ = 1 ∫ = -- P (⃗x)P ∗(⃗x + ⃗ρ)⟨exp [iφ (⃗x,t) − iφ (⃗x + ⃗ρ,t)]⟩d⃗x, (10 ) S
where P(⃗x ) is the pupil function and S, the surface area of the pupil, normalizes the energy contained of the PSF to unity. Using the definition of the structure function the long exposure OTF can be simplified to:
∫ [ ] 1 ∗ 1 OTF (ρ⃗∕λ) = S- P (⃗x)P (⃗x + ⃗ρ) exp − 2D φ(⃗x,⃗ρ) d⃗x, (11 )
which leads directly to:
[ ( ) ] -ρ 5∕3 OTFatm (⃗ρ∕λ) = exp − 3.44 r . (12 ) 0

The OTF and the PSF are related through a Fourier transform. Figure 4View Image shows the OTFs and PSFs of the seeing limited long exposure (Equation 12View Equation), the aberration free, diffraction limited telescope and a typical AO corrected case. The seeing limited OTF does not transfer high spatial frequency information. The AO system is able to retain high spatial resolution information potentially up to the diffraction limit. However, the amplitudes are attenuated in particular, for high spatial frequencies where, depending on the particular observation, noise may begin to dominate before the theoretical diffraction limit.

View Image

Figure 4: Left: long exposure Point Spread Function (PSF) of the turbulent atmosphere (red,dashed), the perfect, diffraction limited telescope (black, solid) and a typical partially AO corrected PSF (blue, dotted). The corresponding modulation transfer functions (MTF) are shown on the right.

The AO corrected PSF consists of two parts. A diffraction limited core and a seeing limited halo. The width of the core is λD-, while the width of the halo is λr0-. The Strehl, which is defined as the ratio of the peak intensity of the observed PSF compared to the peak intensity of the ideal telescope PSF, of the AO corrected PSF shown in Figure 4View Image is S = 0.6. The Strehl ratio is also a measure for the energy contained in the core vs. energy in the halo. The AO corrected long exposure phase structure function will be revisited in detail in Section 6.3.

The time τ0 after which wavefront aberrations change significantly is another important parameter that determines and limits the performance of solar AO and also defines the meaning of a short or long exposure. Using the Taylor hypothesis of frozen in turbulence one can easily derive an estimate for τ 0. The assumption is that a turbulence screen is carried across the telescope aperture by wind at time scales much faster than the intrinsic evolution of the turbulence. This assumption has been experimentally verified (Poyneer et al., 2009) and can be used to implement predictive control of AO systems (e.g., Dessenne et al., 1998, 1999; Poyneer and Véran, 2008; Johnson et al., 2008) in order to improve performance. With this assumption the turbulence or seeing time constant τ 0 can be estimated by the simple equation:

τ0 ≡ r0∕v, (13 )
where v is the wind speed of the dominant turbulence layer. For visible wavelengths typical values for r ∼ 10 cm 0 and v ∼ 10 m∕s result in a τ ∼ 0 of 10 ms. The wavelengths dependence of τ 0 is the same as for 6 r0 ∼ λ 5. Thus the bandwidth requirements for AO can be relaxed in the infrared.
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