Gary Chanan Department of Physics and Astronomy University of California, Irvine 4 February 2000

1. Atmospheric optics A brief introduction to turbulence A guide to the relevant mathematics Derived optical properties of the atmosphere 2. Wavefront sensing Shack-Hartmann wavefront sensing Curvature sensing Outline

Wind Tunnel Experiments Time (sampling units) v (arbitrary units) Frequency (s -1 ) Power (arbitrary units) Slope = -5/3 L = 72 m Re = ~ 10 V = 20 m sec VL 7 Gagne (1987)

Kolmogorov’s Law (1941) Energy cascades down to smaller spatial scales r, higher spatial frequencies  = 2  /r. In inertial range: l 0 r L 0 Power spectrum:  (  )   - 5/3 v  (  )   - 11/3 v   (cm -1 ) Slope = -5/3 Power (arbitrary units) or 10 cm air jet (Champagne 1978) For atmosphere: ~ 1 mm 10 m - 10 km < ~ < ~ inner scale outer scale

Oboukov’s Law (1949) Fluctuations in scalar quantities associated with this flow (passive conservative additives) inherit this same power spectrum.  s (  )   -5/3

Kolmogorov’s Law describes behavior in Fourier space; we are often interested in real space. Consider structure function: There is a Fourier-like relation between  s and  s :  s ( r ) = 2   s (  ) ( 1 - e i  r ) d  3 Structure Function  s ( r ) = 2 mean squared fluctuation

We write  s ( r ) = C r s 2 2/3 structure constant Power laws predominate. Integral of a power law is a power law. Power law indices are easy to calculate; numerical coefficients are hard. Thus:  s (  )    s ( r )  r -5/32/3 Mathematical Notes

n - 1 = 79 x P T in Kelvins in atmospheres  n = -79 x P  T T 2 [Neglect pressure fluctuations.] For typical night-time atmosphere (0.1 to 10 km): C ~ 10 m 2 T N -2/ C ~ 10 m 2-2/3 Thus on meter scales  T ~ 10 mK,  n ~ 10 ! -8 Index Fluctuations Application of Oboukov’s Law to the (pre-existing) large scale temperature gradients in the atmosphere => - law for T fluctuations => - law for n fluctuations

Height (km) C n (m - 2/3 ) 2 C n Profile 2

The index fluctuations are well-characterized. What are the corresponding fluctuations in the accumulated phase? h D = 2R  n ( r ) = C n r 22/3 Central Problem

We will do a first-order treatment, which gives a surprisingly good accounting of the typical astronomical situation (esp. for large telescopes): All points on the wavefront travel straight down, but are advanced or retarded according to:  (x,y) =   n(x,y,z) dz 22 First Order Treatment

This neglects diffraction effects. Valid when: For large telescopes, Characteristic vertical scale of atmosphere is h ~ 10 4 meters; so the near field approximation is usually well-satisfied in practice. R 2 Typical diffraction angle ( ) h h << R R 2 lateral displacement of ray Near - Field Approximation  10 6 meters.

Note that it is precisely these diffraction effects which give rise to scintillation or twinkling. Different parts of the diffracted wavefront eventually interfere with one another. Thus there is no scintillation in the near-field approximation. But for the dark-adapted eye R ~ 4 mm and: ~ 30 meters The inequality turns around and the stars appear to twinkle. R 2 Scintillation

  ( r ) ~ ( ) C n h r Our central propagation problem can be elegantly stated in Fourier space: Given the 3-dim spectral density of n, what is the corresponding 2-dim spectral density of the phase , which is proportional to the integral of n? …and elegantly solved by the following theorem:  (  x,  y ) = 2  h  n (  x,  y, 0)  ndz The phase structure function follows directly: 2 5/3 22 2

  ( r ) = 6.88 ( ) We write: 5/3 r r o where r o is the diameter of a circle over which rms phase variation is ~ 1 radian. Fried’s Parameter r 0 ( ) C n h 2 22 2 r ~ { } -5/3 Fried’s parameter o For C n ~ m - 2/3 h ~ 10 4 m ~ 0.5  m 2 we have r o ~ 10 cm.

r 0 - Related Parameters r 0 6/5 Fried parameter 20 cm120 cm (coherence diameter)  0 ~ 6/5 c oherence time 20 ms120 ms  0 ~ 6/5 isoplanatic angle 4" 24"  fwhm ~ - 1/5 image diameter 0.50" 0.38" N act ~ 12/5 r equired no. of actuators S ~ - 12/5 uncorrected Strehl ratio 4 x r0r0 v r0r0 D2D2 r0r0 2 2 r0r0 D2D2 r0r0 h QuantityScaling Name Value (at 0.5  m) Value (at 2.2  m)

Expansion of the Phase in Zernike Polynomials An alternative characterization of the phase comes from expanding  in terms of a complete set of functions and then characterizing the coefficients of the expansion:   (r,  ) =  a m,n Z m,n (r,  ) piston tip/tilt focus astigmatism Z 0,0 = 1 Z 1,-1 = 2 r sin  Z 1,1 = 2 r cos  Z 2,-2 =  6 r 2 sin2  Z 2,0 =  3 (2r 2 - 1) Z 2,2 =  6 r 2 cos 2 

Z 1,-1 Z 1,1 Z 0,0

Z 2,-2 Z 2,0 Z 2,2

Z 3,-3 Z 3,-1 Z 3,1 Z 3,3

Z 4,-4 Z 4,-2 Z 4,0 Z 4,4 Z 4,2

Atmospheric Zernike Coefficients Zernike Index RMS Zernike Coefficient (D/r o ) 5/6

Shack Hartmann I Shack-Hartmann Test

Shack Hartmann II Shack-Hartmann Test, continued

UFS Ref Beam Ultra Finescreen Reference Beam Exposure

UFS Image (before) Ultra Fine Screen Image (Segment 8) (0.44 arcsec RMS)

UFS C. offsets (before) Centroid Offset Summary Info: Translation from Ref: Rotation from Ref (rad): 0.271E-03 Scale change from Ref: KEY: arcseconds per pixel Scale Error: % Enclosed Energy: % Enclosed Energy: 7.66 RMS Error: 3.14 Max Error (pixels):11.11 Subimage With Max Error: 209 Centroid Offset Display 15 pixels

Curvature Sensing Concept (F. Roddier, Applied Optics, 27, , 1998) Laplacian normal derivative at boundary    2  rr    RR I +  I  I + + I   I  ( r )  I + ( r )   (r) 

Difference Image Z 1,-1

Z 2,-2 Difference Image

Z 2,0 Difference Image

Z 3,-3 Difference Image

Z 4,0 Difference Image

Random Zernikes