Lucky Imaging Diffraction Limited Imaging From the Ground

Transcription

1 University of Ljubljana Faculty of Mathematics and Physics Department of Physics Gregor Kladnik Seminar Lucky Imaging Diffraction Limited Imaging From the Ground Advisor: prof. dr. Tomaž Zwitter Ljubljana, april 2007 Abstract Common goal in observational astronomy is to detect ever fainter objects with apropriate spatial resolution. It is long known that resolution is limited by diffraction of light and is inversely dependent on the diameter of the telescope. In practice ground based telescopes suffer from several different atmospheric effects which alter the light from distant sources (i.e. stars). Turbulent layers cause the images of stars to appear blurred, thus worsening theoretically achievable telescopic resolution. Lucky Imaging is a new imaging technique which can produce diffraction limited images from star light affected by atmospheric turbulence.

2 Contents Abstract 1 1 Introduction 3 2 Resolution of images Optical diffraction How atmospheric turbulence affects wavefronts The Kolmogorov model of turbulence Simulating atmospheric effects on astronomical imaging 8 3 Images at high frame rates (Lucky Exposures) New CCD technology, L3CCD or EMCCD Lucky criteria and test results Usage and Results 14 5 Conclusion 15 References 16 2

3 1 Introduction Optical devices are mainly built for two purposes: accumulating more light, thus enabling us to see fainter (and usualy more distant) objects and improving angular resolution of images. In this paper I will concentrate on a new high angular resolution imaging technique called Lucky Imaging. It deals with the problem of obtaining nearly diffraction limited images in visible from the Earth s surface. High resolution images find use in a number of scientific programmes, both galactic (eg. binary candidates, brown dwarfs, globular cluster cores) and extragalactic (eg. quasar host galaxies). [1] In theory an ideal optical system s (telescope and light aquisition device, usually a CCD camera) resolution is limited only by diffraction, which occurs because of the wave nature of light. In practical usage one has to [2, 3] cope with many other resolution-degrading effects, such as optical aberrations, statistical effects of photon counting, detector noise and certainly the turbulent nature of the atmosphere. [4] Optical aberrations can be minimized by clever design or precise craftmanship, detector noise can be minimized (to a negligible amount) with the recent development of new CCD technology such as EMCCD or L3CCD. The main problem is to cancel the effects of atmospheric blur. To understand how Lucky Imaging works we will have to take a brief look at how the atmosphere influences the light from distant objects. The phrase Lucky Exposures (nowadays Lucky Imaging ) was coined by D. L. Fried in 1978, when he thouroughly discussed and proposed this new imaging technique. Soon afterwards in the early 1980s several experimental results followed, which did not achieve the optimum performance of the technique due to the camera equipment available at the time. [5] It was not until the year 2000 when new CCD technology was developed by Marconi Applied Technologies, [6] which enabled astronomers to use these low-light-level gathering devices for actually successful lucky imaging. A very similar but less advanced technique is also known to amateur astronomers, who use common cheap webcams for imaging brighter planets or the Moon. Usually their equipment does not allow using this technique on fainter objects, mainly due to the less sensitive webcams used. 3

4 2 Resolution of images 2.1 Optical diffraction The Huygens Fresnel principle (named for Dutch physicist Christiaan Huygens, and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation (both in the far field limit and in near field diffraction). [2] It recognizes that each point of an advancing wave front is in fact the center of a fresh disturbance and the source of a new train of waves; and that the advancing wave as a whole may be regarded as the sum of all the secondary waves arising from points in the medium already [2, 3] traversed. This view of wave propagation helps to better understand a variety of wave phenomena, such as diffraction. The most common appli- Figure 1: Diffraction of incident plane waves on a simple aperture according to Huygens Fresnel principle. (Source: WikiMedia) cation of Huygens principle is for the case of a plane wave (usually light, radio waves, x-rays or electrons) incident on an aperture of arbitrary shape, figure (1). In this case, Huygens principle simply states that a large hole can be approximated by a collection of many small holes so each is practically a point source (whose contribution is easy to calculate). A point source generates waves that travel spherically in all directions. The wave that emerges from a point source has complex amplitude ψ at location r that is given by the solution of the wave equation for a point source. That is exactly the Green s function for the wave equation, which is in spherical coordinates [4] ψ(r) eikr r. (1) Therefore, if we approximate the amplitude from an aperture as coming from many point sources, we should sum together an infinite number of point 4

5 sources. But that just describes a surface integral. Thus, [4] Ψ(r) aperture e ikr r da (2) which is simply the spatial Fourier transform of the aperture. Huygens principle, when applied to an aperture, simply says that the far-field diffraction pattern is the Fourier transform of the aperture. The basic idea of Huygens-Fresnel diffraction was put forward in a more general mathematical form by Kirchhoff who calculated the Fresnel-Kirchhoff [2, 3] diffraction integral in the form ψ(p ) = Ai 2λ A e ik(r+s) rs [cos(n, r) cos(n, s)]ds, (3) where the integration is performed over the aperture A and r is the distance between the aperture and point P, s the distance between the aperture and the source of the incident wave. ψ(p ) is the complex wave amplitude at point P and n is the normal vector on the aperture A as shown in figure (2). Figure 2: Geometry of the (scalar) Kirchhoff diffraction theory. [2] The integral (3) can be solved for various approximations, the most used are Fresnel (near axis) and Fraunhofer (far-field) approximations. Next we shall take a quick look at the Fraunhofer approximation and the Fraunhofer diffraction pattern for the most common circular aperture. Fraunhofer approximation holds when the distance z between the aperture and the projecting screen is [3] z ρ2 λ, (4) where λ is the wavelength of diffracted waves and ρ the typical size of the aperture (i.e. for a circular aperture its radius). Obviously this is rarely 5

6 fulfilled in practice 1 but anyway enables us to conveniently calculate the [2, 3] resulting integral ψ(p (p, q)) 1 e ikz iλ z ψ A (ξ, η)e ik z (pξ+qη) dξdη (5) where ψ A (ξ, η) is equal 0 outside the aperture and 1 else. The intensity I is proportional to the square of the amplitude ψ(p ) and for a circular aperture with the help of equation (5) we get [2] ( ( kra )) 2 2J1 z I(P ) = I 0, P = P (r), (6) kra z where J 1 (x) is the first Bessel function, k the wave vector number k = 2π/λ, a the radius of aperture and z distance between the aperture and the projecting screen. This solution was first calculated by Airy and the diffraction pattern, figure (3), is named after him. Figure 3: Diffraction pattern of a point source for a circular aperture, equation (6). The Airy rings are enhanced for better view. 2.2 How atmospheric turbulence affects wavefronts It is first useful to give a brief overview of the basic theory of optical propagation through the atmosphere. In the standard classical theory, light is treated as an oscillation in a field ψ. For monochromatic plane waves arriving from a distant point source with wave-vector k ψ 0 (r, t) = Ae i(φ 0+2πνt+k r) (7) 1 Here we have opical systems in mind such as telescopes, where the equation (4) does not hold entirely. Instead the near-axis condition is satisfied which allows us to use this [2, 3] approximation. 6

7 where ψ 0 is the complex field at position r and time t, with real and imaginary parts corresponding to the electric and magnetic field components, φ 0 represents a phase offset, ν is the frequency of the light determined by ν = c k /(2π), and A is the amplitude of the light. The photon flux in this case is proportional to the square of the amplitude A, and the optical phase corresponds to the complex argument of ψ 0. As wavefronts pass through the Earth s atmosphere they may be perturbed by refractive index variations in the atmosphere. Figure (4) shows schematically a turbulent layer in the Earth s atmosphere perturbing planar wavefronts before they enter a telescope. The perturbed wavefront ψ p may be related at any given instant to the original planar wavefront ψ 0 (r) in the following way [5] ψ p (r) = ( ) χ a (r)e iφ a(r) ψ 0 (r) (8) where χ a (r) represents the fractional change in wavefront amplitude and φ a (r) is the change in wavefront phase introduced by the atmosphere. It is important to emphasize that χ a (r) and φ a (r) describe the effect of the Earth s atmosphere, and the timescales for any changes in these functions will be set by the speed of refractive index fluctuations in the atmosphere. Figure 4: Schematic diagram illustrating how optical wavefronts from a distant star may be perturbed by a turbulent layer in the atmosphere. The vertical scale of the wavefronts plotted is highly exaggerated. [5] The Kolmogorov model of turbulence A description of the nature of the wavefront perturbations introduced by the atmosphere is provided by the Kolmogorov model developed by Tatarski in 7

8 1961, based partly on the studies of turbulence by the Russian mathematician A. Kolmogorov. [3, 7, 8] This model is supported by a variety of experimental measurements and is widely used in simulations of astronomical imaging. The model assumes that the wavefront perturbations are brought about by variations in the refractive index of the atmosphere. These refractive index variations lead directly to phase fluctuations described by φ a (r), but any amplitude fluctuations are only brought about as a second-order effect while the perturbed wavefronts propagate from the perturbing atmospheric layer to the telescope. For all reasonable models of the Earth s atmosphere at optical and infra-red wavelengths the instantaneous imaging performance is dominated by the phase fluctuations φ a (r). The amplitude fluctuations described by χ a (r) have negligible effect on the structure of the images seen in the focus of a large telescope. The phase fluctuations in Tatarski s model are usually assumed to have a Gaussian random distribution with the following second order structure function [5, 7] D φa(ρ) = φ a (r) φ a (r + ρ) 2 (9) where D φa (ρ) is the atmospherically induced variance between the phase at two parts of the wavefront seperated by a distance ρ in the aperture plane, and... represents the ensembe average. The structure function of Tatarski can be described in terms of a single parameter r 0 (Fried parameter) ( ) 5/3 ρ D φa (ρ) = (10) r 0 indicates the strength of the phase fluctuations as it corresponds to the diameter of a circular telescope aperture at which atmospheric phase perturbations begin to seriously limit the image resolution. Typical r 0 values for I band (900nm wavelength) observations at good sites are 20 40cm. [5] Simulating atmospheric effects on astronomical imaging R. Tubbs (and others) did some simulations of short exposure (instantaneous) images of a distant point source through turbulent Kolmogorov atmosphere and a circular aperture (telescope). They neglected the effects of scintillation (χ a (r) = 1) and phase perturbations introduced into wavefronts by aberrations in the telescope (φ t (r) = 0). Equation (8) is slightly modified and now r 0 r 8

9 reads ψ p (r) = ( ) χ t (r)e iφ a(r) ψ 0 (r), (11) where χ t (r) describes the circular aperture of a telescope with primary mirror radius r p as { 1 if r r p χ t (r) = (12) 0 if r > r p The corresponding complex array ψ p was numerically evaluated using equation (11) and then Fourier transformed using a standard Fast Fourier Transform (FFT) routine to provide images of the point source as seen through the atmosphere and telescope. The image of a point source through an optical system is called the point-spread-function (PSF) of the optical system. For a simple optical arrangement with phase perturbations very close to the aperture plane, the response of the system to extended sources of incoherent light is simply the convolution of the PSF with a perfect image of the extended source. Figure 5: Typical short exposures through: a) a 20r 0 aperture; b) a 7r 0 aperture; and c) a 2r 0 aperture. All three are plotted with the same image scale but have different greyscales. [5] Figure (5) shows simulated PSFs for three atmospheric realisations having the same r 0 and image scales but with different telescope diameters. There are two distinct regimes for the cases of large (diameter d r 0 ) and small (d r 0 ) telescopes. Figure (5a) is a typical PSF from a telescope of diameter d = 20r 0. The image is broken into a large number of speckles, which are randomly distributed over a circular region of the image with angular diameter λ r 0, where λ represents the wavelength. With the slightly smaller aperture shown in figure (5b) the individual speckles are larger this is because the typical angular diameter for such speckles is 1.22 λ, equal to d the diameter of the PSF in the absence of atmospheric phase perturbations for a telescope of the same diameter d (i.e. a diffraction-limited PSF). For the small aperture size shown in figure (5c) the shape of the instantaneous 9

10 PSF deviates little from the diffraction-limited PSF given by a telescope of this diameter. The first Airy ring is partially visible around the central peak. It is useful at this stage to define a quantitive measure of image quality. One approach is to compare the PSF measured through the atmosphere with the diffraction-limited PSF expected in the absence of atmospheric aberrations. The ratio of the peak intensity in the PSF measured for an aberrated optical system to that expected for a diffraction-limited system is widely known as the Strehl ratio. [2 4] In this case we treat the atmospheric perturbations as the optical aberration, with the telescope itself assumed to be aberrationfree. The Strehl ratios of the exposures picked (median value chosen) were 0.024, 0.14 and 0.68 for figures (5a), (5b) and (5c) respectively. As the atmospheric fluctuations are random, one would occasionally expect these fluctuations to be arranged in such a way as to produce a diffractionlimited PSF, and hence good quality image. Fried (1978) coined the phrase Lucky Exposures to describe high quality short exposures which occur in such a fortuitous way. A perfectly diffraction-limited PSF will be extremely unlikely, but it is of interest to assess how good an image one would expect to occur relatively often during an observing run. If the speckle patterns change on timescales of a few milliseconds, and we are willing to wait a few seconds for our good image, then we can wait for a one-in-a-thousand Lucky Exposures. Figure 6: Short exposures through 20r 0, 7r 0 and 2r 0 aperture typical of those with the best (highest) Strehl ratios. The Strehl ratios for a), b) and c) are , and respectively. [5] For the definition of Strehl ratio see text. Figure (6) shows the short exposures with the highest 0.1% of Strehl ratios obtained out of several thousand random realisations of each PSF generated. 3 Images at high frame rates (Lucky Exposures) From simulations made by R. Tubbs we have seen how the turbulent atmosphere affects PSFs for different aperture sizes. The time scale of refractive 10

11 index fluctuations may be approximated with the Taylor assumption which argues that if the turbulent velocity within eddies in a turbulent layer is much lower than the bulk wind velocity then one can assume that the changes at a fixed point in space are dominated by the bulk motion of the layer past that point. [5] The timescale τ may then be approximated by τ r 0 v, (13) where r 0 is the Fried parameter and v the bulk wind velocity. The timescale of refractive index fluctations in the Earth s atmosphere are of the order 10 2 s for visual wavelengths (and much longer for longer wavelengths), which implies that in practice one should be able to make very short exposures to freeze this changes thus obtaining similar images as those from simulations shown in figures (5) and (6). [5] Such short exposures can be achieved with the newly developed high frame rate EMCCDs, which are high sensitive, low noise devices. 3.1 New CCD technology, L3CCD or EMCCD An electron-multiplying CCD (EMCCD, also known as an L3Vision CCD, Low-Light-Level CCD L3CCD or Impactron CCD) is a charge-coupled device in which a gain register is placed between the shift register and the output amplifier. The gain register is split up into a large number of stages. In each stage the electrons are multiplied by impact ionization in a similar way to an avalanche diode. Figure 7: Sample diagram illustrating the electron-multiplying gain register part of the new EMCCD cameras. The applied voltage across the multiplication register ranges from 5V to 50V and can be adjusted. [6, 12] (Source: WikiMedia) 11

12 The gain probability at every stage of the register is small (P < 2%) but as the number of elements is large (N > 500), the overall gain can be very high (g = (1 + P ) N ), with single input electrons giving many thousands of output electrons. [6] Reading a signal from a CCD gives a noise background, typically a few electrons. In an EMCCD this noise is superimposed on many thousands of electrons rather than a single electron; the devices thus have negligible readout noise. The low-light capabilities of L3CCDs are starting to find use in astronomy. In particular their low noise at high readout speeds makes them very useful for lucky imaging of faint stars, and high speed photon counting photometry. In future one may expect other technologies, such as CMOS, to overcome CCDs in readout speed, since the CMOS are essentially parallel readout devices. In terms of sensitivity and signal-to-noise ratio the CCDs are still the system of choice for most scientists. 3.2 Lucky criteria and test results As we are now capable to produce a large number of very short exposures 2, we can employ the new Lucky Imaging algorithm, to produce high resolution images: [5] Create several thousand short exposures using an EMCCD Choose the best 0.1% (1%) images with the highest Strehl ratio Shift and add these images to produce nearly diffraction-limited final image The criterion of the best 0.1% images was proposed by Fried (1978) for an aperture size of d = 7r 0. Obviously this criterion is flexible and because this is an offline technique 3, it allows researchers to adjust the criterion for attaining best trade-off between high resolution and signal-to-noise ratio images. In order to apply the Lucky Exposures image selection procedure to observational data taken on astronomical sources, one star in the field can be selected to act as a reference for measurement of the Strehl ratio and position of the brightest speckle. A small rectangular region in each short exposure which surrounds the reference star is then sincresampled to have four times as many pixels in each dimension. The Strehl ratio and position of the brightest speckle are then calculated from the resampled image region. The exposures 2 Frame rates differ from device to device, but most common values are larger than 30f ps (frames per second). 3 The Lucky Imaging algorithm can be used at a later time. 12

13 Figure 8: Example short exposure images of ζ Boötis: a) a typical exposure, having Strehl ratio of (close to median); b) exceptionally good exposure with Strehl ratio of [5] having the highest Strehl ratios are then selected for further processing: the full frame image for each of these short exposures is sinc-resampled, and then re-centred and co-added based on the location of the brightest pixel in the reference star image (calculated through maximum Strehl ratio). Figure 9: Lucky Imaging observations of V656 Herculis and ɛ Aquilae. Panels a) and b) show the best 1% of exposures shifted and added for V656 Herculis and ɛ Aquilae respectively, processed using the method described in the text. Beneath these panels are the respective averaged images in panels c) and d). These were generated by summing all of the short exposures without re-centring, and represent the conventional astronomical seeing disks at the times of the observations. The Strehl ratios and FWHM for the four images are: a) 0.21 and 80 94mas, b) 0.26 and 79 94mas c) and mas, d) and 380mas. [5] 13

14 4 Usage and Results A team of astronomers from Cambridge University built and successfully tested their Lucky Imaging setup (called LuckyCam) at the 2.56m Nordic Optical Telescope (NOT) site on La Palma, Canary Islands. After a testing period in the years 2003 and 2004, first relevant scientific data was gained by the group in the years to follow. They have discovered at least ten new [9, 10] close binary star systems in the visible light spectrum with the technique. Considering the rather small sample of (although carefully chosen) 48 nearby (< 40pc) cold red stars and the brief 8 hours telescope time the ten new discovered binary stars represent a huge success for the team and their technique. Despite seeing variatons from 0.5 to 1.2 (median 0.8 ), the final FWHM resolutions were in all cases better than 0.15 an improvement of factor between 3 to 8. These results present the aspired potential of the Lucky Imaging method. Figure (10) shows images of five newly discovered binary stars imaged through [9, 11] two different filters (SDSS i and z ). Figure 10: Five of the ten newly discovered binary stars are presented. Images are given for SDSS i and z filters (near infrared and infrared). Note the separation [9, 11] of the closest binary star discovered is only

15 5 Conclusion It is widely known that optical system s resolution is influenced by a number of different factors, like optical aberrations, atmospheric turbulence and detector noise. Theoretically achievable resolution is limited by the wave nature of light and the effect of diffraction. Recent development of high-gain CCD cameras reduced detector noise to a negligible amount, optical aberrations can be minimized (to a certain level) with adaptive optics or similar technical solutions. For minimizing atmospheric blur, caused by atmospheric turbulence, active and adaptive optics are heavily used, which are unfortunately marred by quite high cost. A new efficient and cheap technique was developed by astronomers called Lucky Imaging. Theoretical framework was done in 1960s and 1970s by Tatarski and Fried, but it was not until the 2000s when it was successfully used in observations. It builds on the fact that observing conditions change on a rate of few milliseconds and that statistically out of some thousand very short exposures a few best (depending on imposed criteria) can be chosen, shifted and added together to form (usually) nearly diffraction limited images. First results presented show the full potential of this technique, discovering ten new close binary systems in the visible. The team who developed the first version of the LuckyCam is currently working on an improved design, which could achieve even better angular resolution than the current setup. [12] 15

16 References [1] N. M. Law, C. D. Mackay, J. E. Baldwin, A&A 446, , (2006); arxiv:astro-ph/ , (2005). [2] M. Born, E. Wolf, Principles of Optics, Sixth Edition, A. Wheaton&Co. Ltd. (1986). [3] M. Bass, Handbook of Optics, Volume III, Optical Society of America (1995). [4] P. Lena, A. R. King, Observational Astrophysics, A&A Library (1988). [5] R. N. Tubbs, PhD dissertation, St Johns College, University of Cambridge, (2003). [6] C. D. Mackay, et al., SPIE Vol. 4306, (2001). [7] J. W. Goodman, Statistical Optics, Wiley Classics Library Edition (2000). [8] R. N. Tubbs, private communication (2006). [9] N. M. Law, S. T. Hodgkin, C. D. Mackay, MNRAS 368, , (2006); arxiv:astro-ph/ , (2005). [10] N. M. Law, et al., AN 326, , (2005); arxiv:astro-ph/ , (2005). [11] M. Fukugita, et al., AJ 111, 1748 (1996). [12] Lucky Imaging Web Site Home, optics/lucky Web Site/index.htm [13] R. N. Tubbs, Appl.Opt. 44, , (2005); arxiv:astro-ph/ , (2005). [14] S. K. Saha, arxiv:astro-ph/ , (2000). [15] J. L. Nieto, E. Thouvenot, A&A 241, (1991). [16] R. N. Tubbs, et al., A&A 387, L21-L24, (2002); arxiv:astro-ph/ , (2002). [17] J. E. Baldwin, et al., A&A 368, L1-L4, (2001); arxiv:astro-ph/ , (2001). [18] S. M. Flatte, Opt. Express 10, (2002). 16