Adaptive Optics

Based in part on CTIO AO tutorial,http://www.ctio.noao.edu/~atokovin/tutorial/intro.html

The Problem

From the ground, we view the Universe through the Earth’s atmosphere

The Problem

2.2 microns (K band) 0.7 sec integrations a=3.1 arcsec

Two Effects

Seeing – changes in position• Through small aperture – point source

appears to jump around• Through large aperture – point source blurs

Scintillation – changes in intensity• “twinkling” in eye/small aperture• Minimal effect in large aperture

Issues

• Spatial resolution• High contrast imaging• Limiting magnitudes / signal-to-noise

• Note: adaptive and active optics are not the same thing

References

• Babcock, H.W. 1953, PASP, 65, 229

• Beckers, J. 1993, Annual Reviews of Astronomy and Astrophysics, 31, 13

• Schroeder, D.J, 2000, Astronomical Optics, chapter 16

• Tokovinin, A. Adaptive Optics Tutorial at CTIO, http://www.ctio.noao.edu/~atokovin/tutorial/intro.html

Atmospheric Refraction• Light refracts as it passes through a medium with a varying index of

refraction n.• In air, n decreases with height from 1.000297 (STP) to 1.000 (vacuum)

– Let y (horizontal) and z (vertical) axes have origin on the Earth's surface.– An incoming photon is detected at a zenith angle α. – The curvature of the photon's path K = cos(α) dα/dz

Z n=1.00000

n=1.000297 Y

Atmospheric Seeing

• A consequence of the inhomogeneity of the atmosphere

Seeing. I

• Ideal performance requires parallel wavefronts. • Variations in n distort wavefronts• Variations in n are caused by temperature

differences or turbulence.

• Under adiabatic conditions, dn ∝ 2dT P/T2

– P: atmospheric pressure– T: atmospheric temperature

• The changes in n lead to speckles, or defocussing of the image.

Turbulence

• Phase of wavefront is perturbed by turbulence• Mean change in phase of wavefront = 0• Difference D between any two points is of

interest. D(r)=<[φ(x-r)-φ(x)]2>• Kolmogorov turbulence: D(r) = 6.88 (r/r0)5/3

– r0 : atmospheric coherence length = Fried parameter

• Achromatic perturbations: r0 ~ λ6/5 (at zenith)• FWHM β of PSF: β = 0.98 λ/r0

Turbulence with Height

Seeing. IICoherence length r0: the distance over which the wavefront remains coherent(the change in phase is < one radian). • r0 ∼ λ1.2(cos(z))0.6

– z: zenith distance. – A telescope with an aperture < r0 will be

diffraction limited. • At a good site (laminar airflow), such as a

mountaintop in the middle of the ocean, r0 ∼10 cm at 5500 Angstroms.

Seeing. IIIThe atmosphere acts as a series of incoherent lenses of aperture r0: λ/r0 is ~ the diameter of the seeing disk. – The ratio of the seeing disk to the Rayleigh criterion at

zenith is D/24 (5500A/λ)1.2 cm. – An optical telescope smaller than about 25 cm is

diffraction limited; larger telescopes are seeing-limited. – In the 2.2 μm K band, telescopes are diffraction-limited up

to about 3 m diameter. – In the near-IR, much of the seeing can be compensated for

by using active or adaptive optics, permitting diffraction-limited imaging in the 8m VLT and 10m Keck telescopes.

Telescopes above 25 km altitude are never seeing-limited.

Other Details

How fast does the atmosphere change?• Atmospheric time constant τ0 = 0.31 r0/V– V is wind velocity averaged over altitude– Typical V ~ 20 m/s -> τ0 is a few msec

• Isoplanatic angle θ0 = 0.31 r0/h (maximum separation for 2 stars to have same wavefront distortions)

– h is characteristic turbulence altitude, ~5 km– Θ0 ~ 1” in optical– related to the isokinetic angle

Examples of Seeing• Images from SMARTS

1.3m/Andicam.• V band• 0.369ʺpixels• Upper image:

σ=8.1ʺFWHM=9.5ʺ• Lower image:

σ=0.8ʺFWHM=1.0ʺ

Measures of Image Quality

• Ideal PSF is an Airy function• First minimum is at 1.22 λ/D• Image is convolution of Airy Function with

each point in field.• If resolution = Airy Disk, image is diffraction-

limited

1.22 λ/D radians = 0.25 λ(μm)/D(m) arcsec

Measures of Image Quality

• Ideal PSF is an Airy function• First minimum is at 1.22 λ/D• Image is convolution of Airy Function with

each object in field.

Measures of Image Quality

Point Spread Function P characterizes image• FWHM (β = 0.98 λ/r0)• Strehl ratio: P(0)/P0(0)• Encircled Energy (at some radius)

Strehl ratio

The ratio of the peak intensity to the theoretical diffraction-limited peak intensity.S > 0.8 is generally considered diffraction-limited.

Solutions. I.

Speckle Imaging• Image faster than the timescale for changes in

the atmosphere• Shift and add images• Requires bright speckles• Best with a telescope smaller than the

coherence length

Solutions. II.

• Adaptive Optics• Flatten the wavefront by real-time

observations of a bright point source– Natural stars– Laser guide stars

Schematic

Natural Guide Stars• Any star bright enough to measure in the desired time

interval (<τ0)– Measurement accuracy depends on sqrt(N)– V=0: 996 ph/s/cm2/A– N ~ 2.512-V

• Must be close to the target – within isoplanatic patch

Constraints limit number of NGS

Keck AO: needs R<13.5, within 30” of targetThis excludes 99% of the sky

Laser Guide Stars

Artificial star created by laser fluorescence• Two types:• Sodium (SGS)• Rayleigh (RGS)

Sodium Guide Stars• Sodium layer ~ 90km altitude in mesosphere.

– Na-rich layer from ablating meteorites• Fluoresced by shining 5892A laser

• Advantages– Fluoresces in narrow, 10 km wide layer

• Disadvantages– Narrowband – needs tuned laser– Can blind pilots – needs spotter / FAA approval– Na layer changes with time

• Example – Keck– 10 – 14W pulsed laser– Equivalent to 9.5 < V < 11 mag

Na D profile variations

Na LGS + backscatter cone

Rayleigh Guide Star

• Back-scattering from broad-band laser• Usually in blue; scattering ~ λ-4• Height set by pulse gating

• Advantages– Laser easier to construct– Can use UV laser

• Disadvantages– Distance << infinity

Rayleigh Guide Star

• Usually focused at 15-25 km• Ground Layer Adaptive Optics (GLAO)

• Example: Wm Herschel telescope– 25W pulsed laser– Focused to 10 cm spot at 20 km

Deformable Mirrors

Deformable Mirrors

LGS

• Not in focus• Not point source• Affected by focal isoplanatism• More sensitive to poor seeing than NGS – laser blurs both ways

• Need an NGS for tip-tilt and focus

Reference (Keck): http://www2.keck.hawaii.edu/optics/lgsao/(WHT): http://www2.keck.hawaii.edu/optics/lgsao/

Correcting the Full Wavefront

• Aberrations are defined in terms of Zernike Polynomials

• Correction to the diffraction limit requires phase shifts < 1 rad.

• The number J of Zernike modes that must be solved: J ~ 0.24 (D/r0)1.92

Zernicke PolynomialsOrders:• 1: piston• 2,3: tip, tilt• 4: defocus• 5,6: astigmatism• 7,8: coma• 9,10: trefoil• 11: spherical aberration• 12,13: 5th order astigmatism• 14,15: ashtray

Wavefront Correction Techniques

• Tip-Tilt (2d)• Tip-Tilt-Focus (3d)• Deformable Mirrors

Sensors must • Work with white-light incoherent sources• Work with Chromatic light• Be linear for phases > 2π

Shack-Hartmann Test

Shack-Hartmann Sensors

• Distortions proportional to wave-front slopes

• Slopes x,y from wavefrontcentroids

• Commonly-used; proven technology; stable

Wave-Front Reconstruction• WFS data can be represented by a vector. The unknown wave-front

can be specified as phase values on a grid or as Zernike coefficients. • The relation between the measurements and unknowns is assumed

linear, at least in the first approximation S = Aφ.

• A is the interaction matrix. In real AO systems A is determined experimentally. A reconstructor matrix B performs the inverse operation, retrieving wave-front vector from the measurements: φ = BS

• The number of measurements is typically more than the number of unknowns, so a least-squares solution is useful.

• Look for such a phase vector that best matches the data. The resulting reconstructor is B=(ATA)-1AT

Wave-Front Reconstruction• ATA is generally singular, so direct inversion is problematic.

The first Zernike mode (piston) cannot be determined from the slope measurements.

• In practice, use Singular Value Decomposition to remove the indeterminate (or poorly determined) parameters.

• N subapertures lets one recover N Zernike coefficients.• the minimum expected residual phase variance (hence

maximum Strehl ratio), yields a reconstructor matrix similar to a Wiener filter.

• In case of one-dimensional signals, the Wiener filter in frequency space is:– S,N = signal, noise

power spectra

Multi-Conjugate AO

• 3-d AO• Extends corrections to wide (~1’) field

On-Chip Tip-Tilt: ODI

Orthogonal Transfer CCDs• CCD divided into 8x8 OT arrays• Charge from bright star monitored in real-time• Shift applied to all OTs (and all CCDs) in array

Essentially real-time tip-tilt correction on detector

Examples