Wavefront errors due to atmospheric turbulence Claire Max Page 1
Kolmogorov turbulence, cartoon solar Outer scale L 0 Inner scale l 0 h Wind shear convection h ground Page
Atmospheric Turbulence generally has Kolmogorov Power Spectrum 3-D power spectrum of velocity fluctuations: k = / l 3D (k) k 11/3 See V. I. Tatarski, 1961, Wave Propagation in a Turbulent Medium, McGraw-Hill, NY L 0 = outer scale of turbulence Typically 10-100 m l 0 = inner scale of turbulence Typically a few mm Power (arbitrary units) Lab data /L /l 0 0 (cm -1 ) Page 3
Definition of C N, a measure of turbulence strength Need to parameterize variation of turbulence with height h Separate height dependence from wave-number dependence (k,h) = C N (h)k 11/3 Page 4
Expression for Fried Parameter r 0 in terms of atmospheric turbulence strength Measure the resolving power of the imaging system by R = df S ( f ) = df B ( f ) T ( f ) Define a circular aperture r 0 such that the R of the telescope (without any turbulence) is equal to the R of the atmosphere alone: df B ( f ) = df T ( f ) ( / 4 ) ( r 0 / ) r 0 = [ 0.43k sec dhc N (h)] 3/5 Page 5
Scaling of r 0 r 0 is size of subaperture, sets scale of all AO correction H r 0 = 0.43k sec C N (z)dz 0 3/5 6/5 ( sec ) 3/5 C N (z)dz -3/5 r 0 gets smaller when turbulence is strong (C N large) r 0 gets bigger at longer wavelengths: AO is easier in the IR than with visible light r 0 gets smaller quickly as telescope looks toward the horizon (larger zenith angles ) Page 6
r 0 sets the number of degrees of freedom of an AO system Divide primary mirror into subapertures of diameter r 0 Number of subapertures ~ (D / r 0 ) where r 0 is evaluated at the desired observing wavelength Example: Keck telescope, D=10m, r 0 ~ 60 cm at = μm. (D / r 0 ) ~ 80. Actual # for Keck : ~50. Page 7
AO Timescales: depend on r 0 and wind speed Time for wind to carry frozen turbulence over a subaperture of size r 0 (Taylor s frozen flow hypothesis): Typical values: 0 ~ r 0 / V =.0 μm, r 0 = 53 cm, V = 0 m/sec 0 = 65 msec Determines how fast an AO system has to run 0 = f 1 G = 0.10 k sec dz C N (z) V (z) 5/3 0 3/5 6/5 Page 8
What determines how close the reference star has to be? Turbulence has to be similar on path to reference star and to science object Reference Star Science Object Common path has to be large Anisoplanatism sets a limit to distance of reference star from the science object Turbulence z Common Atmospheric Path Telescope Page 9
Expression for isoplanatic angle 0 Strehl = 0.38 at = 0 0 is isoplanatic angle 0 =.914 k (sec ) 8/3 dz C N (z) z 5/3 0 3/5 0 is weighted by high-altitude turbulence (z 5/3 ) If turbulence is only at low altitude, overlap is very high. If there is strong turbulence at high altitude, not much is in common path Common Path Telescope Page 10
Isoplanatic angle, continued Isoplanatic angle 0 is weighted by z 5/3 C N (z) Simpler way to remember 0 0 = 0.314 cos r 0 h where h dz z 5/3 C N (z) dz C N (z) 3/5 Page 11
How to characterize a wavefront that has been distorted by turbulence Path length difference z where kz is the phase change due to turbulence Variance = <(k z) > If several different effects cause changes in the phase, tot = k <(z 1 + z +...) > = k <(z 1 ) + ( z )...) > tot = 1 + + 3 +... radians or (z)( = (z( 1 ) + (z ) + (z 3 ) +... nm Page 1
Notional error budget tot = fitting + anisop + time + measurement +... Page 13
Wavefront errors due to 0, 0 Wavefront phase variance due to 0 = f G -1 If an AO system corrects turbulence perfectly but with a phase lag characterized by a time, then timedelay = 8.4 Wavefront phase variance due to 0 If an AO system corrects turbulence perfectly but using a guide star an angle away from the science target, then angle = 0 5/3 0 5/3 Page 14
Deformable mirror fitting error Accuracy with which a deformable mirror with subaperture diameter d can remove aberrations fitting = μ ( d / r 0 ) 5/3 Constant μ depends on specific design of deformable mirror For segmented mirror that corrects tip, tilt, and piston (3 degrees of freedom per segment) μ = 0.14 For deformable mirror with continuous face-sheet, μ = 0.8 Page 15
We want to relate phase variance to the Strehl ratio Two definitions of Strehl ratio (equivalent): Ratio of the maximum intensity of a point spread function to what the maximum would be without aberrations The normalized volume under the optical transfer function of an aberrated optical system OTF S = aberrated ( f x, f y )df x df y OTF un aberrated ( f x, f y )df x df y where OTF( f x, f y ) = Fourier Transform(PSF ) Page 16
Examples of PSF s and their Optical Transfer Functions Intensity Seeing limited PSF Seeing limited OTF 1 / D / r 0 r 0 / D / -1 Intensity Diffraction limited PSF 1 Diffraction limited OTF / D / r 0 r 0 / D / -1 Page 17
Relation between phase variance and Strehl ratio Maréchal Approximation Strehl ~ exp(- ) where is the total wavefront variance Valid when Strehl > 10% or so Under-estimate of Strehl for larger values of So error budget can be used to predict Strehl directly: Strehl ~ exp(- tot ) Page 18