ASTR519 EPISODES 3 5. Given by Michael Lloyd-Hart. 1.Review of the physics of the imaging process

Transcription

1 ASTR519 EPISODES 3 5 Given by Michael Lloyd-Hart 1.Review of the physics of the imaging process 2. Adaptive optics system concept for fixing up image quality 3. Components of an adaptive optics system There will be a modest problem set handed out on 2/18/03, due back at the end of the semester. ASTR519 2/4/03 1 of 32

2 PHYSICS OF IMAGING Let the complex electric field potential at the aperture of our telescope be E(u). Then the field in the focal plane is given by the Fraunhofer far-field diffraction integral (see e.g. Born & Wolf, Principles of Optics) i ω t U x U 0 E u e k x u d u where U 0 is a constant related to the source brightness and the integral is taken over the whole aperture. Here, u is the vector coordinate in the pupil and x is the vector coordinate in the image plane. The intensity distribution (i.e. the image we actually record on our detector) is the squared modulus of this field I(x) = U(x) 2 If we spell out the complex nature of the field through Then we can say U(x) = M(x) exp [i θ(x)] I(x) = M(x) 2 Note that in recording the image, we have lost all information about the phase θ of the field. This will end up causing much grief later when we try to correct phase aberrations with adaptive optics. Look again though at the equation for U(x). It's just the Fourier transform of the field potential E(u) (times the usual phase factor exp[-iωt]). So if we can define the electric field everywhere in the aperture of the telescope, we know exactly what the image will look like it's just the power spectrum of that field. Mathematically this is very convenient and very easy to calculate. ASTR519 2/4/03 2 of 32

3 IMAGING AN OBJECT The job of a telescope is to form an image of a scene that's as faithful a rendition as possible of the scene itself (even spectroscopists have to make an image before the light can be sent through a slit/fibre). The process can be described through a convolution: I x O x P x O ξ P x ξ d ξ where I is the recorded image, O is the distribution of surface brightness on the object, and P is the point-spread function (PSF) of the imaging system. Imaging without turbulence If there's no aberrating atmosphere, then the optical system in question is just the telescope. If we're looking at a point source with no aberrating medium between it and the telescope, then E is very well defined. Let's write the complex field as E(u) = A(u) exp [i φ(u)] For an undisturbed plane wave, A(u) = 1 over the aperture and φ(u) = 0. We find then that the corresponding PSF is just the power spectrum of the telescope aperture function, and if that's a (nearly) filled circle as it normally will be, we get the well-known Airy pattern. ASTR519 2/4/03 3 of 32

4 (logarithmic stretch) This is the best image quality that we can ever expect our telescope to produce, limited only by the fundamental physics of wave diffraction. As such, it's the gold standard by which we judge the performance of adaptive optics systems. Even so, this PSF is not a delta function, and the convolution above implies that the recorded image will therefore be somewhat blurred. ASTR519 2/4/03 4 of 32

5 Imaging with turbulence If we put a turbulent atmosphere in our light path, the whole situation changes. Now E(u) is no longer constant over the aperture, but becomes a strong function of both u and time (and a weak function of wavelength, which we'll ignore for now). The job of an adaptive optics system is to remove image motion and distortion; in other words, to make P be as close as possible to a delta function. What does it mean to say that the image is moving or distorted? The electric field disturbance arising from the object at the aperture of the telescope is being altered by processes not in the object itself in this case the atmosphere and the telescope optics. The telescope we can't do much about; we don't get an image without it. But the atmosphere we can. Turbulence affects both amplitude A and phase delay φ. Amplitude errors It turns out that we can generally ignore these for two reasons: 1. Amplitude errors have less effect on the point-spread function than do phase errors. [I won't show this here, but the PSF depends on the structure function of the complex field in the pupil. The amplitude structure function saturates around unity (it's difficult to change the amplitude by more than 100%), but the phase structure function can grow without limit.] 2. Amplitude errors in the atmosphere are generally small anyway. Nevertheless, let's explore how they arise. Amplitude errors arise through diffraction at high-altitude turbulence. Mixing of turbules of air at different temperatures (=> different indices of refraction) leads to bending of light rays as through a prism or lens. ASTR519 2/4/03 5 of 32

6 Consider a thin scattering screen some distance away from the aperture: Example of real intensity variations in the pupil of an imaging system (from a recent experiment on the optics bench to demonstrate the use of a phase-diversity algorithm as a wavefront sensor). ASTR519 2/4/03 6 of 32

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8 Now recall r 0 α λ 6/5. Setting the constant of proportionality to C, we can say scintillation is not a problem provided that h < C 2 λ 7/5 Experimentally, we find C ~ 5.5 x 10 6 m -1/5 and the maximum height of turbulence is around 15 km. So scintillation can usually be ignored for wavelengths longer than about 250 nm (where the atmosphere is largely opaque anyway). N.B. In the limit of perfect phase correction, amplitude errors do limit image quality, and this needs to be addressed in so called extreme AO when we're looking for the ultimate contrast ratio e.g. searching for images of faint exo- Jupiters right next to screamingly bright parent stars. Stay tuned for a later lecture... ASTR519 2/4/03 8 of 32

9 Phase errors Phase errors arise from refraction of light which introduces variable optical path length between the telescope aperture and the source. We ignore scintillation from now on. Let us suppose that the atmosphere introduces phase delay Φ(u,z) at each point in the three-dimensional volume. Then we can calculate the aberration of the wavefront in the pupil by simply integrating along ray paths to the star: φ u 0 Φ u, z d z To the extent that we can treat the atmospheric turbulence as layered in a vertical series of thin screens, we can replace the integral by a sum over the layers. This is usually the case to a very good approximation. ASTR519 2/4/03 9 of 32

10 To a good approximation it is sufficient simply to add up the phase error contributions from turbulent layers at different altitudes to calculate the total phase error expected in the telescope aperture. ASTR519 2/4/03 10 of 32

11 Simulations with an 8.4 m diameter telescope under good seeing conditions. 1 micron images Phase error 5 micron images Uncorrected Corrected Uncorrected Corrected 1 arcsec Top row instantaneous snapshots Bottom row 5 second integrations N.B. Speckles are much smaller and more numerous at the shorter wavelength (same size as the diffraction limit) The uncompensated integrated image is actually slightly tighter at the longer wavelength. The corrected images are close to diffraction limited in each case, but the 1 micron images show more uncorrected scattered light. ASTR519 2/4/03 11 of 32

12 PHASE STRUCTURE FUNCTION Kolmogorov theory predicts that in the free atmosphere, the structure function of the turbulence will have the following power law behaviour: D φ ( x) = <[φ(x) - φ(x+ x)] 2 > = 6.88 ( x/r 0 ) 5/3 This is observed to be a reasonably good approximation to the truth much of the time. N.B. Ground-layer and dome turbulence though can sometimes depart markedly from the Kolmogorov spectrum because they're not driven by physical processes that allow a cascade of energy from large to small scales in an environment free of physical obstacles. Observed structure functions will look something like this: Measured at 2.2 µm wavelength r 0 = 1.2 m Structure function recorded at the old MMT in The bump at ~1 m suggests there are actually two distinct layers contributing here. The knee at 30 m reflects the baseline of the interferometer making the measurement (6 m), not a real break in the behaviour of the phase. ASTR519 2/4/03 12 of 32

13 A key (and obvious) thing to note about the structure function it gets bigger at larger spatial scales. This has two important consequences: phase errors on the largest scales have the biggest effect on the image. In fact, image motion (caused by global tip-tilt of the wavefront) is often the dominant effect. Secondly, if we are able to correct aberration on all scales larger than some critical value, we can ignore residual aberration on smaller scales. This critical scale is r 0. ASTR519 2/4/03 13 of 32

14 STREHL RATIO The degree of compensation can be quantified in a well-corrected imaging system by the Strehl ratio the ratio of peak intensity in the Airy core in the corrected image to the peak in the theoretically perfect image. Perfect image Partially corrected image ASTR519 2/4/03 14 of 32

15 PRINCIPLE OF WAVEFRONT PHASE COMPENSATION We've seen that the images made by a telescope are messed up largely because of refraction by the air which introduces variable phase delays in the rays from a star to the aperture. Adaptive optics sets out to restore high-quality imaging by removing the phase aberration. We do this by introducing aberrations of equal amplitude and opposite sign. Let a compensating phase φ c (u) be applied somehow at the telescope aperture. Then the corrected field E c (u) is E c (u) = A 0 exp {i[φ(u) - φ c (u)]} where we now take the amplitude A 0 to be constant. If φ(u) = φ c (u) then E c (u) reduces to the diffraction-limited case and we recover perfect imaging. In general though we won't be able to do quite this well, and we're left with some residual phase error φ r φ r (u) = φ(u) - φ c (u) This can provide a very useful statistical measure of image quality: σ 2 = < φ r (u) 2 > where <...> means averaging over both space and time. One finds that for σ 2 < ~1 radian 2, the Strehl ratio is well approximated by SR ~ exp {- σ 2 } N.B. This is clearly a VERY strong function of rms wavefront error σ you lose peak intensity rapidly as the quality of correction goes down. ASTR519 2/4/03 15 of 32

16 One consequence of this result is that adaptive optics becomes very much more difficult as the wavelength goes down. If our wavefront is corrected to some rms residual error s expressed in nm, then we see that σ = 2πs/λ and SR ~ exp {- (2πs/λ) 2 } Seeing limit α λ -1/5 Diffraction limit α λ For this small telescope, AO would be useful roughly in the range 1 to 4 microns. For λ > 4 microns, seeing doesn't limit our resolution For λ < 1 micron, AO gets very difficult as we shall see For larger telescopes, AO's usefulness extends to correspondingly longer wavelengths. ASTR519 2/4/03 16 of 32

17 AO at a glance HOW DO WE DO THIS? THE ADAPTIVE OPTICS SERVO LOOP The basic flow of information in an adaptive optics system. Aberrated light from a reference source in the sky impinges on a deformable optic that's shaped to straighten out the wavefronts again; a beam-splitting optic separates out some of the light to go to a high resolution imaging camera, while the rest goes to some sort of wavefront sensor. Signals from the sensor are massaged by a fast computer and turned into drive signals for the deformable optic. ASTR519 2/4/03 17 of 32

18 Flow of information in a typical AO system Timing sequence: Light arrives at the wavefront sensor, with mean age equal to half the integration time The sensor is read out There's some computation Commands are sent to the phase corrector. All these events are going on in parallel at various points in the servo loop. N.B. Integration is generally set by readout time of the wavefront sensor camera you clock out the detector as fast as possible. This sets the fundamental cycle time of the system. All this means there's typically a delay of ~2 cycles between the time information arrives at the detector and the time this information is fed back as a correction. ASTR519 2/4/03 18 of 32

19 COMPONENTS OF AN ADAPTIVE OPTICS SYSTEM There are 5 key components to an AO system (4 of which appear in the cartoon above). In order, they are: 1. A reference beacon in the sky, light from which samples the turbulent air in approximately the same way as light from the scientific target. 2. A phase-correcting optic of some sort to introduce precise and programmable phase delays. 3. Beam-splitting optics to separate out beacon light from target light. 4. A wavefront sensor that records some parameters of the aberrated wavefront. 5. A fast computer that turns output signals from the wavefront sensor into driving signals for the phase compensator. System approach With one exception, all AO telescopes around the world treat the AO as an addon. That is, a conceptually and physically separate box of optics is built to do the correction. Aberrated light from one of the telescope foci (usually Nasmyth or Coudé) goes in, and compensated light comes out. This approach has one major advantage, and two big drawbacks: Pro: As a separate gadget, the AO can be designed, built, tested, installed and operated with essentially no impact on the telescope. Cons: Extra optics in the light path reduce photon throughput. Thermal IR observations are compromised by emission from these optics Being an uncompromising bunch, the Steward AO group has built the MMT AO system differently, as we shall see. ASTR519 2/4/03 19 of 32

20 REFERENCE BEACONS We need a source of light above the turbulence we're trying to sense. Not just anything will do. Recall that any image we record will be a convolution of the object distribution O(x) with the PSF, P(x): I x O x P x What we're interested in is the effect of the phase aberration φ(u) on the PSF, but we can't tell if features in the image I are caused by the object or the PSF unless we already know O in some detail. For that reason, we need to work with unresolved, or barely resolved sources like stars, Galilean satellites, or backscattered light from a projected laser beam. Then we can say that O is approximately a delta function. We'll talk more about laser guide beacons later. For now we'll restrict ourselves to stars as reference sources. ASTR519 2/4/03 20 of 32

21 PHASE COMPENSATORS There are several ways we might think of to introduce programmable phase delay into the optics. The universally adopted solution in currently operating AO systems is to use a deformable mirror. These things are ubiquitous, and therefore given the acronym DM in the literature. The usual approach incorporates a DM of perhaps 15 cm diameter. A thin sheet of glass is bonded on the back to several hundred piezo-electric actuators. An image of the primary mirror is projected onto the DM so that corrections made there will appear from the point of view of downstream instruments to have been made at the primary. (N.B. Requires warm optics to image the primary, and more optics to relay the corrected focus to the science instrument.) These mirrors are readily available (e.g. Xinetics, Itek, Cilas). They come in three main flavours: Continuous facesheet, stacked actuator Segmented Bimorph Continuous facesheet Continuous facesheet DMs consist of a thin sheet of glass bonded on the back side to a large number of actuators, usually stacked piezo-electric. Cost is ~$1000 per actuator (e.g. Keck II AO system has 349 actuators, cost was about $350k.) Limited stroke (typically a micron) prevents tilt correction, so a separate fast steering mirror is needed to compensate image motion. PZT actuators have 10-20% hysteresis, making control difficult Actuator print-through on the optical surface acts like a diffraction grating, producing secondary peaks in the PSF. Most operational AO systems use this kind of DM. ASTR519 2/4/03 21 of 32

22 Facesheet DM geometry A 249 actuator deformable mirror made by CILAS (France) ASTR519 2/4/03 22 of 32

23 Image of a binary star made with the Air Force adaptive optics system on a 1.5 m telescope in Albuquerque. The DM is a continuous facesheet with the actuators arranged on a square grid. This geometry creates effectively a diffraction grating that gives rise to the secondary peaks in a square around the main Airy core. (This is ε Boo, separation is 2.8 arcsec) ASTR519 2/4/03 23 of 32

24 Segmented mirrors A 512-segment DM made by Thermotrex in San Diego. Each segment is glued to a three-axis PZT actuator that can move in tip, tilt, and piston, giving over 1500 degrees of freedom. Overall diameter is 22 cm. This is how segmented mirrors are driven to try to match the shape of the incoming wavefront. ASTR519 2/4/03 24 of 32

25 Segmented DMs have pros and cons too: Cost is comparable to continuous facesheet mirrors Lack of inter-actuator coupling means controlling them is very easy Gaps between segments make for unpleasant diffraction effects in the image Gaps also can easily double the mirror's thermal emissivity ASTR519 2/4/03 25 of 32

26 Bimorph mirrors A 36-actuator bimorph DM. Left optical surface; right contacts for the 36 addressing electrodes, which are arranged radially. The bimorph is a sandwich of two layers of piezo-electric material, oppositely poled, so that when an electric field is applied across the sandwich, one layer shrinks while the other expands. The result is an induced curvature. This kind of DM has been very successfully used in low-order AO systems at CFHT, Gemini North, and Subaru. ASTR519 2/4/03 26 of 32

27 Deformable secondary mirrors The one exception to the system approach in which the AO is added to an already complete telescope is the MMT. This system optimises photon efficiency and thermal performance by making the DM be one of the mirrors that has to be in the telescope anyway. How many PhDs does it take to install one (not-so-simple) mirror??? (July 2002) ASTR519 2/4/03 27 of 32

28 The world's first AO secondary installed at the MMT. A beautiful sight! (January 2003) ASTR519 2/4/03 28 of 32

29 Guts of the AO secondary, in an exploded view, poised below the MMT secondary hub and active alignment hexapod. Three electronics crates contain 168 DSPs A cold plate circulates liquid coolant to remove ~1 kw of waste heat 336 voice-coil actuators drive the deformable surface like little loudspeakers A rigid glass back plate provides ground-truth shape information The 2 mm thick deformable glass surface ASTR519 2/4/03 29 of 32

30 A few details about this DM, since some of you are likely to use it in the course of your work: The deformable part is a glass membrane 2mm thick, 64 cm in diameter. 336 permanent magnets are glued to the back surface of the membrane electric current in coils above each one provide driving force. Back of the glass membrane showing the actuator magnets attached. Actuators are arranged in a circularly symmetric pattern (minimises diffraction effects), have tens of microns of stroke (no need for a separate fast steering mirror), and have no physical connection to the glass they're bending (magnetic coupling only, so no print-through). Each actuator has an associated capacitive sensor which measures the local displacement of the glass at 40 khz. The AO system therefore always knows exactly what the real shape of the mirror is. Actuators have zero hysteresis. ASTR519 2/4/03 30 of 32

31 10 micron image with and without AO note the very clean PSF on the right when compared to the image from the Air Force AO system earlier. N.B. Measured emissivity of the MMT with AO attached is only 6-7% (Phil Hinz' measurement). This is as good as any telescope, and much better than any other AO telescope. ASTR519 2/4/03 31 of 32

32 Other kinds of phase compensators Other things have been proposed (and even tried, although never in real life at a telescope): Nematic liquid crystal arrays Can be made with > 10 5 pixels (actuators) High fill factor Generally only work for one polarisation Poor photon efficiency Limited range of correction ( < 1 wave) MEMS (micromachined mirror arrays) A variation on the deformable mirror concept Beginning to be available with many thousands of elements Necessarily very small Likely to be cheap Promising for the future Sonic standing waves Erez Ribak proposes a liquid filled cell in which transverse standing sound waves are excited. These create pressure gradients with associated refractive index variations. By tuning the position, amplitude and frequency of these waves, we could in principle reproduce not only φ(u), but also the full 3-d phase Φ(u,z). This has exciting possibilities for multi-conjugate AO that we'll discuss in a later lecture. ASTR519 2/4/03 32 of 32