Atmospheric turbulence

1. Atmospheric turbulence Eric Gendron

2. Wavefront, and image • The energy (= light rays) propagates orthogonally to the wavefront no real convergence point convergence point = centre of the sphere non spheric wavefront spherical wavefront

3. Aberrations • Difference between the actual wavefront, and the ideal one • Optical path difference varying across the pupil : d(x,y) x d(x,y) no real convergence point aberrated wavefront

4. Aberrations : examples convergence point (center of curvature) in a vertical plane • Astigmatism • d(x,y) = x2-y2 or d(x,y) = xy • Easily created by tiltinga lens in an optical system convergence point (center of curvature) in a horizontal plane

5. Aberrations : examples rays from pupil edge converge here • Spherical • wavefront curvature changes linearly with pupil radius • d(x,y) = r3 • Any simple lens creates spherical aberration rays from pupil centre converge here

6. Aberrations : examples convergence point (center of curvature) • Defocus • « wrong radius » • d(x,y) = x2+y2 • Easily created by moving a lens along the optical axis where convergence point was expected

7. Aberrations : examples convergence point (center of curvature) • Tilt • « image is not centered » • d(x,y) = x or d(x,y) = y • Easily created by moving a lens transversal to the optical axis where convergence point was expected

8. When aberrations depend on l blue wave converge here • Chromatic • Chromatic aberration of a single lens • mainly defocus (focal length is shorter at short l) • In general : wavefront shape depends on wavelength ! • can be anything : spherical in the red, and astigm in the blue red wave converge here

9. When aberrations depend on field position • Field curvature • defocus varies quadratically with field angle • Distorsion • tilt is introduced with field angle

10. Diffraction • Image = |electric field in focal plane|2 • Electric field in focal plane =F( electric field in pupil plane ) • Phase : • Electric field in the pupil : phase amplitude

11. Diffraction limit • For a « perfect » wavefront : the image is determined only by the pupil function of the instrument (assuming uniform amplitude) |F [A(x,y)] |2

12. Diffraction limit • For a circular aperture : Airy pattern angle a distance R in the focal plane a0 = 1.22 l/D R0 = 1.22 lf/D normalized intensity r (a orR)

13. Aberrations • With f(x,y)≠0 • image becomes wider than l/D, light is spread around • peak intensity is reduced • Relation between image quality and phase ? • How to measure image quality ?

14. Image formation depends on l • Image(u,v) = |F [ ] |2 • Same wavefront d(x,y), but different images : l=1 µm l=0.7 µm l=0.5 µm

15. Phase variance • The phase variance tells how degraded the wavefront is : • sf2=0 when the wavefront has no aberration • units : radians2 • proportional to l-2 • will allow us to transform quantities in terms of wavelength f(x,y) x

16. Strehl ratio • Ratio between • the intensity of the degraded image on the optical axis • the intensity of the diffraction-limited image on the optical axis 0 ≤ SR ≤ 1 SR>1 impossible !!! Idiff Ideg

17. Phase variance and SR • Approximation : • Usually ≈ok for sf2< 1 rd2 • True when phase is a white noise • Exercice : • SR(0.5µm) = 0.40. Determine SR at 1.65 µm.

18. Atmospheric turbulence • Turbulence is not sufficient to produce wavefront distorsion • wavefront is distorted because of random refractive index fluctuations • Temperature fluctuations are required (and/or water vapor concentration fluctuations) cold air warm air

19. Atmospheric turbulence • Air refractive index depends on wavelength • Air refractive index depends on temperature • optical path fluctuations are, at first order, independent of wavelength : wavefront shape d(x,y) is close to achromatic air refractive index 0°C 20°C wavelength

20. Atmospheric turbulence • Turbulent temperature mixing occurs mainly • close to the ground (0-40m) • at inversion layer (1-2 km) • at jet-stream level (8-12 km) • Most of it occurs at interface between air slabs • notion of « turbulent layers »

21. Atmospheric turbulence • Fractal properties • Change of spatial scale turns into amplitude factor • comes from Kolmogorov statistics (1941) : • statistical scale invariance of the cascade : sc aling arguments and dimensional analysis • a, b ?V = speede = energyL = distance

22. Atmospheric turbulence • 3-D phase structure function of refractive index : • True for l0 < r < L0 : the inertial regime • inner scale l0 • outer scale L0 • CN2 is called refractive index structure constant • depends on altitude h : CN2(h) • is expressed in m-2/3 • Phase variance will vary proportionally to CN2(h)

23. Atmospheric turbulence • 3-D phase structure function of refractive index : • True for l0 < r < L0 : the inertial regime • inner scale l0 • outer scale L0 • CN2 is called refractive index structure constant • depends on altitude h : CN2(h) • is expressed in m-2/3 • 3D power spectrum : kolmogorov L0=100m Von Karman L0=10m Von Karman version

24. Wavefront statistics • 2D phase power spectrum : Wiener spectrum • 2D phase structure function • r0 characterizes the amplitude of wavefront disturbance

25. same resolution l/r0 The Fried parameter • Fried, JOSA, 1966 : • r0 is the diameter of a diffraction-limited telescope having the same resolution as an infinitely large telescope limited by the atmosphere turbulence large telescope, limited by the atmosphere diameter r0

26. The Fried parameter • When D<r0: the telescope is limited by diffraction • wavefront is « nearly flat » over the aperture • When D>r0: the telescope is seeing-limited • r0 : area over which the wavefront can be considered as « flat » • with respect to l ! image width seeing-limited telescope diffraction-limited telescope l/r0 l/D telescope diameter D Order of magnitude of r0 ??... In the visible Exceptionally : 25 cm Astronomical site : 10 cm Meudon : 3 cm Horizontal propag : ~ mm r0

27. The Fried parameter • Expression of r0 : • Notice that • For a fully developped Kolmogorov turbulence : • sounds like a definition : r0=area over which phase variance ≈ 1 rd2 • Seeing :

28. Image properties • typical atmospheric-degraded image : • structure with speckles (short exposures) • Typ. size of a speckle • l/D • Typ. size of long exposure image • l/r0 • l/D • seeing = l/r0

29. Long-exposureoptical transfer function • One demonstrate that the long-exposure transfer function is the product between • the OTF of the telescope • an OTF specific to atmosphere H(u) spatial frequency u r0/l D/l

30. Exercice • On a 1m telescope, seeing is 3 arcsec at lvis=0.5µm.SR at l=10 µm ? • 3 arcsec = 1.45e-5 rd =lvis/r0 : r0(0.5µm)=3.4cm • sf2=1.03(D/r0)5/3 = 283 rd2 at lvis=0.5µm • sf2(0.5µm) 0.52 = sf2(10µm) 102 => sf2(10µm) = 0.71 rd2 • SR = exp(-0.71) = 0.49 • or ... scale r0 • r0(10µm) = r0(0.5µm) (10/0.5)6/5 = 1.25m • sf2(10µm) = 1.03(D/r0)5/3 = 0.71 rd2

31. Temporal evolution • One assumes that the layers move as a whole, with speed of inner eddies slower than the global motion(Taylor hypothesis) • One define a correlation time : • V is the average speed • t0 is proportional to l6/5

32. Angular anisoplanatism directn 1 • isoplanatic : when wavefronts are the same for the different directions in the field • If separated enough, 2 points of the field will see different wavefronts • One definesH is the average height • q0 is proportional to l6/5 directn 2 layer B hB layer A hA telescope pupil

33. Example

34. Modal decomposition of phase • f(x,y,t) not easy to handle • Decomposition on a modal basis • Zernike modes • defined on a circular aperture • analytic expression • orthogonal basis • look like first order optical aberrations • derivatives can be expressed as a simple combination of themselves • Fourier transform has analytic expression • coefficient ai

35. Zernike modes m=0 • Indexi refers to n and m, radial and azimutal orders of the polynomial • i is increasing with n and m, i.e. with spatial frequency m=1 n=1 m=2 n=2

36. Modal decomposition of phase • Phase variance • Setting one of the ai=0 • best way to flatten the wavefront

37. Modal spectrum tip and tilt low spatial freq • Noll, R.J., JOSA 66, (1976) high spatial freq

38. Residual error • Phase error after perfect compensation of J Zernike modes • Equivalence Zernike deformable mirror with Na actuators • Greenwood, JOSA, 69, 1979

39. Residual error • Re-writing Greenwood formula • sf2 will be kept constant if one keeps product (r0 na) constant naactu across the diameter Na actuators inside the pupil

40. Temporal spectrum • Temporal spectrum of ai(t) log(PSD) f-11/3 tip-tilt f-2/3 Power Spectral Density f0 higher orders f-17/3 frequency (hz) log(f) fc

41. Angular correlations normalized correlation tip-tilt high-order mode low order mode separation angle

42. Thanks for your attention