Chapter 1. Intensity Interferometry

Transcription

1 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 1 Chapter 1 Intensity Interferometry Dainis Dravins Lund Observatory, Box 43, SE Lund, Sweden dainis@astro.lu.se Intensity interferometry exploits the second-order coherence of light how intrinsic intensity fluctuations correlate between simultaneous measurements in separated telescopes. The optical telescopes connect only electronically (rather than optically), and the noise budget relates to electronic timescales of nanoseconds; light-travel distances of centimeters or meters. This makes measurements practically insensitive to either atmospheric turbulence or to telescopic optical imperfections, allowing very long baselines, as well as observing at short optical wavelengths. Kilometer-scale optical arrays of air Cherenkov telescopes will enable optical aperture synthesis with image resolutions in the tens of microarcseconds. 1. Highest resolution in optical astronomy Tantalizing results from current amplitude/phase interferometers begin to show stars as widely diverse objects, and a great leap forward will be enabled by improving angular resolution by just another order of magnitude. Bright stars with typical diameters of a few milliarcseconds require optical interferometry over hundreds of meters or some kilometer to enable surface imaging. However, phase interferometers need stability to within a fraction of an optical wavelength, while atmospheric turbulence and dispersion make their operation challenging for very long baselines. 2. Intensity interferometry Intensity interferometry exploits second-order optical coherence, that of intensity not of amplitude nor phase. It measures how random (quantum) intensity fluctuations correlate in time between simultaneous measurements in two or more separated telescopes. The method was pioneered by Robert Hanbury Brown and Richard Q. Twiss already long ago, for the original purpose of measuring stellar sizes. 1 The name intensity interferometer is actually somewhat misleading since nothing is interfering; the name originated from its analogy to amplitude interferometers, which at that time had similar scientific aims. Seen in a quantum context, it is a twophoton process, and is today seen as the first quantum-optical experiment. It laid 1

2 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 2 2 Dainis Dravins the foundation for experiments of photon correlations and for the development of the quantum theory of optical coherence. The great observational advantage (compared to amplitude interferometry) is that it is practically insensitive to either atmospheric turbulence or to telescopic optical imperfections, enabling very long baselines as well as observing at short optical wavelengths, even through large airmasses. Telescopes connect only electronically (rather than optically), and the noise budget relates to electronic timescales of nanoseconds (light-travel distances of tens of centimeters or meters) rather than those of the light wave itself. A realistic time resolution of perhaps 3 ns corresponds to 1 m light-travel distance, and the control of atmospheric path-lengths and telescopic imperfections then only needs to correspond to some fraction of that. Details of the original intensity interferometer and its observing program have been well documented. 1 5 The principles are also explained in various monographs and reference publications Following these early efforts, the method has not been used in astronomy since (but has found wide applications in particle physics since the same correlation properties apply to not only photons but to all bosons, i.e., particles with integer values of their quantum spin). Currently, there are considerable efforts to revive intensity interferometry in astronomy, applying high-speed electronics in arrays of large air Cherenkov telescopes. The following descriptions are based upon such ongoing studies, simulations and experiments. Fig. 1. Components of an intensity interferometer array. Spatially separated telescopes observe the same source, and the measured time-variable intensities, I n(t), are electronically cross correlated between different telescopes. Two-telescope correlations are measured as I 1 (t)i 2 (t), I 2 (t)i 3 (t), etc.; three-telescope quantities as I 1 (t)i 2 (t)i 3 (t), I 2 (t)i 3 (t)i 4 (t), etc. With no optical connection between the telescopes, the operation resembles that of radio interferometers. 3. Principles of operation The basic concept of an intensity interferometer is sketched in Figure 1. In its simplest form, it consists of two telescopes, each with a photon detector feeding one channel of a signal processor for temporally cross correlating the measured

3 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 3 Intensity Interferometry 3 intensities from two telescopes with the highest practical time resolution, probably around 1-10 ns. With telescopes sufficiently close to one another, the intensity fluctuations measured in both telescopes are more or less simultaneous, and thus correlated in time, but when moving them apart, the fluctuations gradually become decorrelated. How rapidly this occurs for increasing telescope separations gives a measure of the spatial coherence, and thus the spatial properties of the source Two-telescope observations The measurement provided by a two-telescope system is: I 1 (t)i 2 (t) = I 1 (t) I 2 (t) (1 + γ 12 2 ) (1) where γ 12 is the mutual coherence function of light between locations 1 and 2, the quantity commonly measured in amplitude/phase interferometers; denotes averaging over time. Compared to independently fluctuating intensities, the correlation between the intensities I 1 and I 2 is enhanced by the coherence parameter. This relation is valid for a classical concept of light as a wave, but fundamentally this is a quantum-optical two-photon effect, which presupposes that the light is in a state of thermodynamic equilibrium, obeying the Bose-Einstein statistics. Such ordinary chaotic (also called thermal, maximum-entropy or Gaussian ) light undergoes random phase jumps on timescales of its coherence time but that relation does not necessarily hold for light with different photon statistics (e.g., an ideal laser emits light that is both first- and second-order coherent, without any phase jumps, and thus would not generate any sensible signal in an intensity interferometer). Since the measured quantity is the square of the ordinary first-order visibility, it always remains positive, only diminishing in magnitude when smeared over time intervals longer than the optical coherence time of starlight. However, for realistic time resolutions (much longer than an optical coherence time in broad-band light of perhaps s), any measured signal is tiny, requiring very good photon statistics for its reliable determination. Large photon fluxes (thus large telescopes) are therefore required. For a given electronic time resolution, this dilution is smaller for lower-frequency electromagnetic radiation (with longer coherence time), and for long-wavelength infrared and radio, this additional variability due to fluctuations in the signal itself is more easily measured, and there known as wave noise Three or more telescopes Intensity interferometry lends itself to many-telescope operations. Since the signal is electronic only, it may be freely copied, transmitted, combined or saved, quite analogous to radio interferometry. In any larger array, the possible number of baselines between telescope pairs grows rapidly, without any additional logistic effort. With N telescopes, N(N 1)/2 baselines can be formed between different pairs of

4 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 4 4 Dainis Dravins telescopes. For triplets of telescopes, one may construct I 1 (t)i 2 (t)i 3 (t) = I 1 (t) I 2 (t) I 3 (t) (1 + γ γ γ Re[γ 12 γ 23 γ 31 ]) (2) The phase of the triple product in the last term is the closure phase widely used in amplitude interferometry to eliminate effects of differential atmospheric phase errors between telescopes since the baselines 1-2, 2-3, and 3-1 form a closed loop. Of course, intensity interferometry is not sensitive to phase errors but three-point intensity correlations permit to obtain the real (cosine) part of this closure-phase function which provides additional constraints that may enhance image reconstruction Optical aperture synthesis To enable true two-dimensional imaging, a multi-telescope grid is required to provide numerous baselines, and cover the interferometric Fourier-transform (u,v)-plane, analogous to radio arrays. However, compared to amplitude interferometers, a certain complication lies in that the correlation function for the electric field, γ 12, is not directly measured, but only the square of its modulus, γ Since this does not preserve phase information, the direct and immediate inversion of the measured coherence patterns into images is not possible. Fig. 2. Fourier-plane vs. image-plane information. The left column shows coherence maps ( diffraction patterns ) corresponding to the direct images at right. At top left, the Airy diffraction pattern originates from a circular aperture. At bottom left is an experimentally measured coherence pattern from an artificial asymmetric binary star, built up from intensity-correlation measurements over 180 baselines in an array of optical telescopes in the laboratory. At right is the image reconstructed from these intensity-interferometric measurements. 18,19 The circle shows the diffraction-limited resolution, thus realized by an array of optical telescopes connected through electronic software only, with no optical links between them.

5 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 5 Intensity Interferometry 5 Various techniques exist to recover the Fourier phases. Already intuitively, it is clear that the information contained in the coherence map (equivalent to the source s diffraction pattern) must place stringent constraints on the source image. Viewing the familiar Airy diffraction pattern in Figure 2, one can immediately recognize it as originating in a circular aperture, although only intensities are seen. Most imaging methods have been developed for other disciplines (e.g., coherent diffraction imaging in X-rays) but also for astronomical interferometry, demonstrating how also rather complex images can be reconstructed. One remaining limitation is the non-uniqueness between the image and its mirrored reflection. Figure 2 shows an example of imaging an artificial binary star with an array of small optical telescopes in the laboratory, operated as an intensity interferometer with 180 baselines. 18,19 5. Signal-to-noise in intensity interferometry No other current instrument in astronomy measures the second-order coherence of light and since its noise properties differ from those of other instruments they are essential to understand in defining realistic observing programs. For one pair of telescopes, the signal-to-noise ratio 1,27 for polarized light is given in a first approximation by: (S/N) RMS = A α n γ 12 (r) 2 f 1/2 (T/2) 1/2 (3) where A is the geometric mean of the areas (not diameters) of the two telescopes; α is the quantum efficiency of the optics plus detector system; n is the flux of the source in photons per unit optical bandwidth, per unit area, and per unit time; γ 12 (r) 2 is the second-order coherence of the source for the baseline vector r, with γ 12 (r) being the mutual degree of coherence. f is the electronic bandwidth of the detector plus signal-handling system, and T is the integration time Independence of optical passband Most of these parameters depend on the instrumentation, but n depends on the source itself, being a function of its brightness temperature. For a given number of photons detected per unit area and unit time, the signal-to-noise ratio is better for sources where those photons are squeezed into a narrower optical band. Indeed (for a flat-spectrum source), the S/N is independent of the width of the optical passband, whether measuring only the limited light inside a narrow spectral feature or a much greater broad-band flux. The explanation is that realistic electronic resolutions of nanoseconds are much slower than the temporal coherence time of optical light. While narrowing the spectral passband decreases the photon flux, it also increases the temporal coherence by the same factor, canceling the effects of increased photon noise. This property was exploited already in the Narrabri interferometer 28 to identify the extended emission-line volume from the stellar wind around the Wolf-Rayet star γ 2 Vel. This could also be exploited for

6 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 6 6 Dainis Dravins increasing the signal-to-noise by simultaneous observing in multiple spectral channels, a concept foreseen for the once proposed successor to the original Narrabri interferometer. 5,29,30 One major breakthrough in sensitivity can be expected once energy-resolving detectors become practical to use for also high photon count rates. This would enable straightforward parallel observing in multiple spectral channels, increasing the signal by a factor equal to the spectral resolution Dependence on source temperature Another S/N property is that high-temperature sources can be measured but, in practice, cool objects can not, no matter what their apparent brightness. To be a feasible target for long baseline interferometry, any source must provide both a significant photon flux, and have structures small enough to produce visibility over such baselines. This implies small sources of high surface brightness. Cool ones would have to be large in extent to give a sizable flux, but then will become spatially resolved already over short baselines. Seen alternatively, for stars with a given angular diameter but decreasing temperature (thus decreasing fluxes), telescope diameter must be increased in order to maintain the same S/N. When resolved by a single telescope, the S/N begins to drop (the spatial coherence decreases), and no gain results from larger telescopes. For currently foreseen instrumentation, practically observable sources would have to be hotter than about solar temperature. 6. Air Cherenkov telescopes A long-baseline intensity interferometer requires large telescopes spread over some square kilometer or more. Precisely such complexes are being erected for a different primary purpose: the study of gamma-ray sources through the observation of visual flashes of Cherenkov light emitted in air from the particle cascades triggered by gamma rays. Since these flashes are faint, the telescopes must be large but do not need to be more precise than the spatial equivalent of a few nanoseconds lighttravel time, that being the typical duration of these Cherenkov flashes. For good stereoscopic source localization, the telescopes need to be spread out over hundreds of meters, that being the typical extent of the light pool on the ground. These parameters are remarkably similar to the requirements for intensity interferometry and several authors have realized the potential for also this application The largest current such project is CTA, the Cherenkov Telescope Array. 34,35 CTA is planned to have up to 100 telescopes spread over several square kilometers, with a combined light-collecting area of some 10,000 m 2. Of course, it will mainly be devoted to its task of observing Cherenkov light in air but also other applications are envisioned; besides intensity interferometry, 36 searches for rapid astrophysical events, observing stellar occultations by distant Kuiper-belt objects or as a terrestrial ground station for optical communication with distant spacecraft. The impact on other Cherenkov array operations would probably be limited since

7 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 7 Intensity Interferometry 7 interferometry can be carried out during nights with bright moonlight which due to the faintness of the Cherenkov light flashes precludes their efficient observation. If baselines of 2 or 3 km could be utilized at short optical wavelengths, resolutions would approach 30 µas, an unprecedented spatial resolution in optical astronomy (An)isochronous telescopes For Cherenkov telescopes, a large field of view is desired while, in most telescopes, the image quality deteriorates away from the optical axis. Several telescopes have a Davies-Cotton design, 40 whose spherical prime mirror gives smaller aberrations off the optical axis compared to parabolic systems. These particular telescopes are not isochronous, i.e., light striking different parts of the entrance aperture may not arrive to the focus at exactly the same time. The signal-to-noise improves with electronic bandwidth f, i.e., with better time resolution for recording intensity fluctuations. The time spread in anisochronous telescopes acts like an instrumental profile in the time domain, filtering away the most rapid fluctuations. Fortunately, the gamma-ray induced Cherenkov light flashes last only a few nanoseconds, and thus the performance of Cherenkov telescopes cannot be made much worse, lest they would lose sensitivity to their primary task Telescopic image quality The technique does not demand good optical quality, permitting also flux collectors with point-spread functions of several arcminutes. Still, issues arise from unsharp images: in particular a contamination by the background light from the night sky. Such light does not contribute any net intensity-correlation signal but increases the photon-counting noise, especially when observing under moonlight conditions Focusing at infinity The foci of Cherenkov telescopes correspond to those atmospheric heights where most of the Cherenkov light originates, and the image of a cosmic object will be slightly out of focus. For a focal length of f=10 m, the focus shifts 1 cm between imaging at 10 km distance and at infinity. In order to minimize the night-sky background, it could thus be desirable to refocus the telescopes at infinity Telescope positions in an array The choice of telescopes within a larger array can be optimized for best coverage of the interferometric (u, v)- Fourier plane. As the source moves across the sky, projected baseline lengths and orientations between telescope pairs change, depending on the angle under which the object is observed. Telescopes in a repetitive geometric pattern cause the baselines to be similar for many pairs of telescopes. Since

8 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 8 8 Dainis Dravins celestial objects move from east towards west, baselines between pairs of telescopes that are not oriented exactly east-west will cover more of the (u, v)-plane During the night, stationary telescope pairs trace out ellipses in the Fourier plane as a function of the observatory latitude, the celestial coordinates of the source, and the relative placement of the telescopes. 41 The rotation of the Earth enables aperture synthesis in software and, of course, is the very principle used in much of radio interferometry Interferometry in space? Some ideas for space-based instruments have been proposed. Amplitude interferometers would avoid atmospheric turbulence, but still require extreme optical stability. To relax such requirements, concepts for intensity interferometry between free-flying telescopes have been proposed If the signals are stored for later analysis, there is no need to keep the telescopes precisely positioned, but only to know their positions at the time of observation. Such a system could enable the imaging of stellar surfaces in ultraviolet emission lines, delineating magnetically active regions Detectors Typical Cherenkov telescopes have point-spread functions in the focal plane on the order of 1 cm. Matching photon-counting detectors include solid-state avalanche diode arrays and vacuum-tube photomultipliers. In principle, only one detector pixel per telescope is required (although some provision for measuring the signal at zero baseline may be needed). In some telescopes, a pixel at the center of the large Cherenkov camera is electronically directly accessible, possibly suitable also for interferometry. An alternative concept is a dedicated detector mounted onto the outside of the mechanical cover of the ordinary camera. Bright sources, observed in broadband light with a large telescope, may generate count rates that are too great to practically handle. However, since the S/N is independent of the optical passband, the signal can be retained with a lower photon flux, if some color filter is used to reduce the flux to a suitable level, or perhaps a narrow-band filter tuned to some spectral feature of astrophysical significance Correlators The electronic correlator provides the averaged product of the intensity fluctuations, normalized by the average intensities. Current techniques permit to program electronic firmware units into high-speed digital correlators with time resolutions of a few ns or better. Equipment with such performance is also commercially available for various (non-astronomical) applications in photon correlation spectroscopy. Firmware correlators process data in real time, and eliminate the need for their further handling and storage. However, if something needs to be checked afterwards, the original data are no longer available, and alternative signal processing cannot

9 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 9 Intensity Interferometry 9 be applied. An alternative is to time-tag each photon count and store all data for later analyses off-line. However, this may require a massive computational effort and possible problems may not get detected while observations are still in progress. Correlator requirements are much more modest than for correlators at radio telescope arrays. Those amplitude interferometers measure not only the spatial but also the temporal coherence, achieving radio imaging with high spectral resolution. Optical spectra cannot realistically be obtained from intensity interferometry measurements: the spectral resolution comes from optical filters or the wavelength sensitivity of the detector. While correlators for radio arrays may be supercomputers to handle spatial and temporal correlations with many bits of resolution, a correlator for intensity interferometry can be a small table-top device controlled by a personal computer, merely correlating binary data streams of photon counts Delay units In the original Narrabri instrument, the telescopes were continuously moved during observation to maintain their projected baseline. Electronic time delays can now be used instead to assign successive measures of the spatial coherence to the appropriate baseline length and orientation. To follow a source across the sky, a variable time delay (in either hard- or software) has to compensate for the change of timing of the wavefront at the different telescopes of up to a few µs, corresponding to differential light travel distances of maybe half a km Experimental work As steps towards full-scale stellar intensity interferometry, laboratory and field experiments are pursued. A dedicated test facility ( StarBase ) has been developed in Utah, while Cherenkov telescopes of the VERITAS array in Arizona have been used to verify telescope connections for interferometry. Laboratory experiments and simulations are carried out by various groups to understand signal handling, correlator performance, and efficiency of image reconstruction algorithms. 18,19, Intensity interferometry on extremely large telescopes Extremely large optical telescopes (ELTs), with apertures in the m range, aim at diffraction-limited imaging using adaptive optics in the near-infrared. Although achievable resolution is coarser than with long baselines, ELTs will be attractive for intensity interferometry, once they are outfitted with an array of high-speed photoncounting detectors. 49,50 The telescope aperture would be optically sliced into many segments (analogous to a wavefront sensor across the entrance pupil), each feeding a separate detector. Cross correlations then realize intensity interferometry between pairs of telescope subapertures. This would be practical also when seeing conditions are inadequate for adaptive optics or when the segmented main mirror is only

10 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page Dainis Dravins partially filled. Achievable resolution in the blue will be superior to that feasible with infrared adaptive optics by a factor of 3 or 5. Steps towards such a facility have been taken in recent instrument constructions Although mirror segments on ELTs are smaller than Cherenkov telescopes, they offer certain advantages: image quality is arcseconds or better, which essentially eliminates background skylight, and permits the use of small detectors of very high quantum efficiency, such as single-photon-counting avalanche diodes. Precise optical collimation permits interference filters with very narrow bandpass to isolate spectral lines, and since the optical paths are isochronous, very fast electronics can be used to improve the signal-to-noise ratio. Although the finite size of the ELT aperture limits the extent of the interferometric (u,v)-plane covered, this can be sampled very densely, and an enormous number of baseline pairs can be synthesized, assuring a complete sampling of the source image, and its stable reconstruction. 8. Astronomical targets With optical imaging approaching resolutions of tens of microarcseconds (and also with a certain spectral resolution), we will be moving into novel and previously unexplored parameter domains, enabling new frontiers in astrophysics. With a foreseen brightness limit of perhaps m V = 7 or 8, for sources of a sufficiently high brightness temperature, initial observing programs have to focus on bright stars or stellar-like objects. 33,36,56 Promising targets for early intensity interferometry observations thus appear to be relatively bright and hot, single or binary O-, B-, and Wolf-Rayet type stars, perhaps with deformed shapes due to rapid rotating, or with dense stellar winds visible in emission lines. To reach fainter extragalactic sources appears to require multi-wavelength observations, probably achievable with some spectrally resolving detector array or, ultimately, with energy-resolving detectors. 9. Outlook The scientific potential of long-baseline optical interferometry and of stellar surface imaging was realized already a long time ago, and concepts for long-baseline optical amplitude/phase interferometers have been worked out for constructions at groundbased observatories, in Antarctica, as free-flying telescopes in space, or even placed on the Moon. All those proposals are based upon sound physical principles, yet do not appear likely to be realized in any immediate future due to their complexity and likely cost. However, progress in instrumentation and computing technology, building upon years of experience in radio interferometry, together with the growing availability of large flux collectors in the form of air Cherenkov telescopes, now appear to enable electronic long-baseline interferometry by software also in the optical, several decades after corresponding aperture synthesis techniques were first established in the radio domain.

11 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 11 Intensity Interferometry 11 References 1. R. Hanbury Brown, The Intensity Interferometer. Its applications to astronomy (Taylor & Francis, London, 1974). 2. R. Hanbury Brown, J. Davis and L. R. Allen, The stellar interferometer at Narrabri Observatory I. A description of the instrument and the observational procedure, Mon. Not. Roy. Astron. Soc. 137, (1967). 3. R. Hanbury Brown, J. Davis, L. R. Allen and J. M. Rome, The stellar interferometer at Narrabri Observatory II. The angular diameters of 15 stars, Mon. Not. Roy. Astron. Soc. 137, (1967). 4. R. Hanbury Brown, Photons, Galaxies and Stars (Indian Academy of Sciences, Bangalore, 1985). 5. R. Hanbury Brown, Boffin. A Personal Story of the Early Days of Radar, Radio Astronomy and Quantum Optics (Adam Hilger, Bristol, 1991). 6. A. Glindemann, Principles of Stellar Interferometry (Springer, Berlin, 2011). 7. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985). 8. R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, Oxford, 2000). 9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). 10. A. Labeyrie, S. G. Lipson and P. Nisenson, An Introduction to Optical Stellar Interferometry (Cambridge University Press, Cambridge, 2006). 11. S. K. Saha, Aperture Synthesis. Methods and Applications to Optical Astronomy (Springer, New York, 2011). 12. Y. Shih, An Introduction to Quantum Optics. Photon and Biphoton Physics (CRC Press, Boca Raton, 2011). 13. V. Malvimat, O. Wucknitz and P. Saha, Intensity interferometry with more than two detectors?, Mon. Not. Roy. Astron. Soc. 437, (2014). 14. P. D. Nuñez and A. Domiciano de Souza, Capabilities of future intensity interferometers for observing fast-rotating stars: Imaging with two- and three-telescope correlations, Mon. Not. Roy. Astron. Soc. 453, (2015). 15. A. Ofir and E. N. Ribak, Offline, multidetector intensity interferometers I. Theory, Mon. Not. Roy. Astron. Soc. 368, (2006). 16. A. Ofir and E. N. Ribak, Offline, multidetector intensity interferometers II. Implications and applications, Mon. Not. Roy. Astron. Soc. 368, (2006). 17. T. Wentz and P. Saha, Feasibility of observing Hanbury Brown and Twiss phase, Mon. Not. Roy. Astron. Soc. 446, (2015). 18. D. Dravins, T. Lagadec and P. D. Nuñez, Optical aperture synthesis with electronically connected telescopes, Nature Commun. 6:6852, 5 pp. (2015). 19. D. Dravins, T. Lagadec and P. D. Nuñez, Long-baseline optical intensity interferometry. Laboratory demonstration of diffraction-limited imaging, Astron. Astrophys. 580, A99, 13 pp. (2015). 20. R. B. Holmes and M. S. Belen kii, Investigation of the Cauchy-Riemann equations for one-dimensional image recovery in intensity interferometry, J. Opt. Soc. Am. A, 21, (2004). 21. R. B. Holmes, S. Lebohec and P. D. Nunez, Two-dimensional image recovery in intensity interferometry using the Cauchy-Riemann relations, Proc. SPIE, 7818, 78180O, 11 pp. (2010). 22. R. Holmes, B. Calef, D. Gerwe and P. Crabtree, Cramer-Rao bounds for intensity interferometry measurements, Appl. Opt. 52, (2013).

12 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page Dainis Dravins 23. P. D. Nuñez, R. Holmes, D. Kieda and S. LeBohec, High angular resolution imaging with stellar intensity interferometry using air Cherenkov telescope arrays, Mon. Not. Roy. Astron. Soc. 419, (2012). 24. P. D. Nuñez, R. Holmes, D. Kieda, J. Rou and S. LeBohec, Imaging submilliarcsecond stellar features with intensity interferometry using air Cherenkov telescope arrays, Mon. Not. Roy. Astron. Soc. 424, (2012). 25. J. J. Dolne, D. R. Gerwe and P. N. Crabtree, Cramér-Rao lower bound and object reconstruction performance evaluation for intensity interferometry, Proc. SPIE, 9146, , 12 pp. (2014). 26. D. Gerwe, P. Crabtree, R. Holmes and J. Dolne, Comparison of forward models and phase retrieval for image formation from intensity interferometer data, Proc. SPIE 8877, 88770F, 14 pp. (2013). 27. R. Q. Twiss, Applications of intensity interferometry in physics and astronomy, Opt. Acta, 16, (1969). 28. R. Hanbury Brown, J. Davis, D. Herbison-Evans and L. R. Allen, A study of Gamma-2 Velorum with a stellar intensity interferometer, Mon. Not. Roy. Astron. Soc. 148, (1970). 29. J. Davis, A proposed successor to the Narrabri stellar intensity interferometer, in Multicolor Photometry and the Theoretical HR Diagram, Dudley Obs. Rep. 9, (1975). 30. R. Hanbury Brown, A review of the achievements and potential of intensity interferometry, in: High Angular Resolution Stellar Interferometry, Proc. IAU Coll. 50, 11-1, 7 pp. (University of Sydney, Sydney, 1979). 31. S. LeBohec and J. Holder, Optical intensity interferometry with atmospheric Cerenkov telescope arrays, Astrophys. J. 649, (2006). 32. S. LeBohec, M. Daniel, W. J. de Wit, J. A. Hinton, E. Jose, J. A. Holder, J. Smith and R. J. White, Stellar intensity interferometry with air Cherenkov telescope arrays, in: High Time Resolution Astrophysics: The Universe at Sub-Second Timescales, AIP Conf. Proc. 984, (2008). 33. D. Dravins, S. LeBohec, H. Jensen and P. D. Nuñez, Stellar intensity interferometry: Prospects for sub-milliarcsecond optical imaging, New Astron. Rev. 56, (2012). 34. B. S. Acharya, M. Actis, T. Aghajani, G. Agnetta, J. Aguilar, F. Aharonian, M. Ajello, A. Akhperjanian et al., Introducing the CTA concept, Astropart. Phys. 43, 3-18 (2013). 35. Cherenkov Telescope Array (CTA), (2016). 36. D. Dravins, S. LeBohec, H. Jensen, P. D. Nuñez, for the CTA Consortium, Optical intensity interferometry with the Cherenkov Telescope Array, Astropart. Phys. 43, (2013). 37. I. Klein, M. Guelman and S. G. Lipson, Space-based intensity interferometer, Appl. Opt. 46, ( 2007). 38. E. N. Ribak, P. Gurfil and C. Moreno, Spaceborne intensity interferometry via spacecraft formation flight, Proc. SPIE 8445, , 6 pp. (2012). 39. E. N. Ribak and Y. Shulamy, Compression of intensity interferometry signals, Exp. Astron. 41, (2016). 40. J. M. Davies and E. S. Cotton, Design of the quartermaster solar furnace, Solar Energy, 1, (1957). 41. D. Ségransan, Observability and UV coverage, New Astron. Rev. 51, (2007).

13 May 23, :36 World Scientific Review Volume in x 6.5in Handbook Dravins page 13 Intensity Interferometry D. Dravins and S. LeBohec, Toward a diffraction-limited square-kilometer optical telescope: Digital revival of intensity interferometry, Proc. SPIE 6986, , 10 pp. (2008). 43. C. Foellmi, Intensity interferometry and the second-order correlation function g(2) in astrophysics, Astron. Astrophys. 507, (2009). 44. E. Horch and M. A. Camarata, Portable intensity interferometry, Proc. SPIE 8445, 84452L, 7 pp. (2012). 45. D. Kieda and N. Matthews, HBT SII imaging studies at the University of Utah, Proc. SPIE 9907, (2016). 46. S. LeBohec, B. Adams, I. Bond, S. Bradbury, D. Dravins, H. Jensen, D. B. Kieda, D. Kress et al., Stellar intensity interferometry: Experimental steps toward longbaseline observations, Proc. SPIE 7734, 77341D, 12 pp. (2010). 47. C. Pellizzari, R. Holmes and K. Knox, Intensity interferometry experiments and simulations, Proc. SPIE 8520, 85200J, 17 pp. (2012). 48. J. Rou, P. D. Nuñez, D. Kieda and S. LeBohec, Monte Carlo simulation of stellar intensity interferometry, Mon. Not. Roy. Astron. Soc. 430, (2013). 49. C. Barbieri, D. Dravins, T. Occhipinti, F. Tamburini, G. Naletto, V. Da Deppo, S. Fornasier, M. D Onofrio et al., Astronomical applications of quantum optics for extremely large telescopes, J. Mod. Opt. 54, (2007). 50. D. Dravins, C. Barbieri, R. A. E. Fosbury, G. Naletto, R. Nilsson, T. Occhipinti, F. Tamburini, H. Uthas and L. Zampieri, QuantEYE: The quantum optics instrument for OWL, in: Instrumentation for Extremely Large Telescopes, MPIA spec. publ. 106, (2006). 51. I. Capraro, C. Barbieri, G. Naletto, T. Occhipinti, E. Verroi, P. Zoccarato and S. Gradari, Quantum astronomy with Iqueye, Proc. SPIE 7702, 77020M, 9 pp. (2010). 52. G. Naletto, C. Barbieri, T. Occhipinti, F. Tamburini, S. Billotta, S. Cocuzza and D. Dravins, Very fast photon counting photometers for astronomical applications: From QuantEYE to AquEYE, Proc. SPIE, 6583, 65830B, 14 pp. (2007). 53. G. Naletto, C. Barbieri, T. Occhipinti, I. Capraro, A. di Paola, C. Facchinetti, E. Verroi, P. Zoccarato et al., Iqueye, a single photon-counting photometer applied to the ESO New Technology Telescope, Astron. Astrophys. 508, (2009). 54. G. Naletto, C. Barbieri, E. Verroi, I. Capraro, C. Facchinetti, S. Gradari, T. Occhipinti, P. Zoccarato and V. da Deppo, Upgrade of Iqueye, a novel photon-counting photometer for the ESO New Technology Telescope, Proc. SPIE, 7735, , 12 pp. (2010) 55. L. Zampieri, G. Naletto, C. Barbieri, M. Barbieri, E. Verroi, G. Umbriaco, P. Favazza, L. Lessio and G. Farisato, Intensity interferometry with Aqueye+ and Iqueye in Asiago, Proc. SPIE 9907, (2016). 56. S. Trippe, J.-Y. Kim, B. Lee, C. Choi, J. Oh, T. Lee, S.-C. Yoon, M. Im and Y.-S. Park, Optical multi-channel intensity interferometry Or: How to resolve O-stars in the Magellanic Clouds, J. Korean Astron. Soc. 47, (2014).