AO theory begins with an understanding of turbulence and its structure. The theory of velocity fields of turbulence was first developed by Kolmogorov (1941) who proposed a simple scaling relation in which turbulent energy was added to the fluid at a large spatial scale (denoted asL0, the ‘outer scale’) and the energy was passed down to eddies at smaller and smaller spatial scales till it dissipated when the spatial scale was small enough for viscosity to become significant4. This smaller scale is known as the ‘inner scale’, denoted byl0 For the energy passage through spatial scales to be stable, the velocity fluctuationsV must depend only on the spatial scale l and the rate of energy input or dissipation ( per unit mass) at that scale. Dimensional analysis shows that for scales betweenl0 and L0 the following should be valid:
V ∝1/3l1/3. (1.1)
This equation defines the one-dimensional Kolmogorov turbulence power spectrum Φ(κ), where κ = 2π/l is the spatial wavenumber of the turbulence. Since the energy in the increment dκis proportional to V2,
Φ(κ)dκ∝V2 ∝κ−2/3,Φ(κ)∝κ−5/3. (1.2) The three-dimensional power spectrum, required for calculating the propagation of elec- tromagnetic radiation through the atmosphere was calculated by Tatarskii(1961) as,
Φ(κ)∝κ−11/3. (1.3)
The proportionality factor is determined by the strength of the turbulence. For measur- ing wavefront distortions, we measure the structure parameter of refractive index changes, denoted asCn. The final Kolmogorov three-dimensional power spectrum of refractive index variations is given by,
Φ(κ) = 0.033Cn2κ−11/3. (1.4)
This equation does not account for the behavior of the power spectrum towards the end
4Flows are characterized by their Reynolds number,Re=V l/ν whereV,landνare the characteristic velocity, spatial scale and kinematic viscosity respectively. Flows become non-turbulent when Re is less than a geometry-dependent critical value.
10-4 10-3 10-2 10-1 100 101 102 103 Spatial Scale (m)
10-36 10-33 10-30 10-27 10-24 10-21 10-18 10-15 10-12 10-9 10-6
Power Spectrum
von Karmann Kolmogorov
Figure 1.2. The spatial power spectra for isotropic turbulence is shown for the Kolmogorov model (red line) an the von Karmann turbulence (blue curve). The von Karmann spectrum is plotted forL0= 20 m and with an exponential drop-off atl0= 1 mm.
of its valid range, i.e. froml0 toL0. The outer scale is of tremendous importance to large telescopes (diameter & 8 m) where the aperture may be comparable in size to the outer scale. The von Karmann spectrum (Ishimaru,1978) accounts for this roll off as,
Φ(κ) = 0.033Cn2
(κ2+κ20)−11/6, (1.5)
κ0 is the wavenumber corresponding to the outer scaleL0.
At the lower spatial scales, the viscosity dissipates the turbulent eddies and the turbulent power drops. Tatarskii suggested an ad-hoc exponential term to account for the lowest spatial scales as,
Φ(κ) = 0.033Cn2
(κ2+κ20)(11/6)exp(−κ2
κ2m ), (1.6)
whereκm is the wavenumber corresponding to the smallest scale (. 10−3m). The highest wavenumber of interest in AO systems is that corresponding to the WFS subaperture size (∼10 cm) andκmκ, hence the exponential term evaluates to unity. Figure1.2shows the typical roll-off of the von Karmann model.
The strength of turbulence changes as a function of height (h) in the atmosphere. This distribution profile (Cn2(h)) significantly affects the correction method and the optimum AO performance that can be achieved. Usually, the two most significant layers of turbulent air
are the turbulent boundary between low altitude wind and the stationary ground, i.e. the ground layer and the boundary layer between the jet streams flowing in different directions at the top of the troposphere.
The important effects of the turbulence are usually summarized by three parameters for the purposes of AO instrumentation. Other details, such as the turbulence profiles, wind speed profiles and annual variations are useful for detailed planning before construction, but are not required for most estimates.
• Fried Parameter r0:. The Fried parameter describes the spatial scale over which the RMS error in wavefront phase is one square radian. It is related to the Cn2(h) profile and zenith angle ζ as,
r0 =
16.699λ−2sec(ζ) Z ∞
0
Cn2(h)dh −3/5
. (1.7)
Physically, r0 is the aperture size that will give a diffraction limited with the same FWHM as the seeing FWHM. A seeing of 100at 500 nm corresponds to anr0 of 10 cm.
r0 is considered as the fiducial size of wavefront sensing for an AO system. If one can exactly measure and correct the wavefront for all scales equal to or larger thanr0, then we can achieve a moderate correction (Strehl ratio = 37%). Asr0 ∝λ6/5, correcting at visible wavelengths requires the sampling of the wavefront to be performed at correspondingly smaller spatial scales.
• Isoplanatic Patch θ0: The wavefront aberrations between two directions on the sky rapidly decorrelates as the the directions are separated. The angular scale of decorrelation (θ0) is where the RMS error in wavefront phase is one radian and is given by,
θ0=
115λ−2(secζ)8/3 Z ∞
0
Cn2(h)h5/3dh −3/5
(1.8) The isoplanatic patch is the radius at which the correction applied is still reasonably valid hence the wavefront measurement reference and the science target must be within this patch. It can be seen that the isoplanatic patch is significantly reduced by high altitude turbulence (due to theh5/3 weighting). If all the turbulence is concentrated in the ground layer, a very large isoplanatic patch can be achieved, greatly enhancing the science capabilities of AO. Observing locations in Antarctica are of great interest for this reason (among a few others).
• Coherence Timeτ0: The timescale of wavefront changes is given approximatelyr0/v wherev is the averaged wind-speed. τ0 is the timescale difference at which the RMS phase variations are one square radian. For useful AO performance, the correction bandwidth must be higher than the frequency 1/τ0.
Telescope Laser Science Target
10 – 20 km
Tip-‐8lt star
Ground Layer Upper Atmosphere
WFS Wavefront
Corrector
Distorted Wavefront from the Telescope
Science + Laser
Control Loop
Laser Light
IR Cam
Vis Cam Corrected Wavefront
Figure 1.3. A schematic depiction of the working of an adaptive optics system. Left Panel: The laser light propagates through the turbulence and forms a laser ‘beacon’ or guide star at a certain altitude. The down scattered light from the beacon propagates through the atmosphere and measures most of the turbulence that the starlight also propagates through. Right Panel: Inside the instrument, a dichroic sends the laser light to the wavefront sensor (WFS) which sends appropriate commands to the deformable mirror (DM) that corrects the wavefront deformation and forms a diffraction limited image on the science camera.