Ghost Imaging Using Optical Correlations

REVIEW ARTICLE www.lpr-journal.org Ghost Imaging Using Optical Correlations

Paul-Antoine Moreau, Ermes Toninelli, Thomas Gregory, and Miles J. Padgett*

of imaging system. Figure 1 displays Ghost imaging uses optical correlations to enable an alternative and intriguing the ghost image captured by Shih and image acquisition technique: even though information from either one of the co-workers.[2] detectors used for the acquisition does not yield an image, an image can be The process of parametric downcon- obtained by harnessing the optical correlations. This Review describes a version occurs when a light beam is incident upon a nonlinear optical crystal.[5] variety of both quantum and classical ghost imaging techniques, and seeks to The nonlinearity of the crystal refers to point out where these techniques may have practical applications. the functional relationship between the applied electric field (associated with the incident light) and the resulting polarisation of the material, the sinusoidal form 1. Introduction of the incident optical field resulting in a non-sinusoidal polarisation. This non-sinusoidal response cor- Conventional camera systems use light that is transmitted by, responds to a harmonic distortion of the electric field and hence or back-scattered from, the object to form an image on a film an incident light beam with frequency of ω results in some light or focal-plane detector array. By contrast, ghost imaging systems emitted with frequency 2ω, a phenomenon called second har- use spatial correlations between separated optical fields to obtain monic generation which is widely used in commercial systems to images without the need to record the image itself. In a ghost convert an infrared laser into one with a visible emission. More imaging system, the principal detector is a single-element (or subtly, a similar process can take an incident pump beam of fre- single-pixel) device that measures the interaction between a sin- quency ωp and downconvert it into signal and idler beams of fre- gle photon position (or optical pattern) and the unknown object. quencies ωs and ωi ,whereωp = ωs + ωi . In terms of photons it The other, spatially correlated photon, does not interact with the is useful to think of a single pump photon being downconverted object but it own position is measured using an imaging sys- into two photons, termed signal and idler.[6,7] This downconver- tem. This article seeks to explain both quantum ghost imaging sion process is subject to the standard energy and momentum techniques – i.e. ghost imaging techniques where correlations conservation laws. are quantum in their origin, and classical ghost imaging tech- In addition to constraining the frequencies of the downcon- niques – i.e. ghost imaging techniques that harness classical cor- verted light, the conservation of energy means that the downcon- relations. The article will also highlight where these techniques verted signal and idler photons are generated at the same position might have practical applications. within the crystal. Simultaneously, the conservation of transverse momentum imposes a momentum anticorrelation between sig- ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ nal and idler photons (i.e., kp = ks + ki ,wherekp , ks and ki are 2. Ghost Imaging Using Parametric the transvers wave-vectors of the pump, signal and idler beams Downconversion respectively). It is these position or momentum correlations that lie at the heart of the quantum ghost imaging technique. A quan- The term “ghost imaging” was introduced in the description of tum ghost imaging system uses these signal and idler photons to the work reported by Shih and co-workers[1,2] in the 1990’s fol- illuminate the imaging detector and the object respectively, see lowing theoretical works by Klyshko.[3,4] They used the quantum Figure 2. correlation between the signal and idler beams produced as a re- By configuring the crystal to give collinear phase-matching the sult of parametric downconversion to create an alternative kind signal and idler beams resemble spatially incoherent extended sources similar to what might be used in a conventional imaging system. However, the signal and idler still need to be separated from each other into different optical paths. In the case of type- II phase-matching, the signal and idler have orthogonal polarisa- Dr.P.-A.Moreau,E.Toninelli,T.Gregory,Prof.M.J.Padgett SUPA tions and this separation can be achieved using a polarising beam School of Physics and Astronomy splitter to create the two paths with high efficiency. However, in University of Glasgow the case of type-I phase-matching, the signal and idler polarisa- Glasgow, G12 8QQ, UK tions are identical and, if also near degeneracy, the only option to E-mail: [email protected] separate them is to rely on a simple beam splitter to separate the signal and idler photons with an inherent separation efficiency The ORCID identification number(s) for the author(s) of this article of 50%. can be found under https://doi.org/10.1002/lpor.201700143 DOI: 10.1002/lpor.201700143

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Paul-Antoine Moreau is a Marie Skłodowska-Curie Research Fellow at the University of Glasgow working in the Optics group. He received a PhD in Optics and Photonics from the Uni- versity of Franche-Comt´e in 2015 and has previously worked as a postdoc- toral researcher at the university of Bristol within the Centre for Quantum Photonics. His current research interests are in quantum metrology and technologies, quantum optics and general photonics in the space domain.

Ermes Toninelli is a PhD student in the Optics Research Group at the Univer- sity of Glasgow. In 2013 he researched Figure 1. First ghost image reconstruction by Shih and co-workers. a) The organic ﬁeld-eﬀect phototransistors, UMBC logo is reconstructed using b) the ghost imaging technique by exploiting correlated photons detected using single photon avalanche pho- under the supervision of Prof. Qingxin todiodes that are translated so as to reconstruct a spatially resolved im- Tang, at the Northeast Normal Uni- age. Reproduced with permission.[2] Copyright 1995, American Physical versity, Changchun, China. In 2017 he Society. worked on quantum state tomography and modal decomposition using vec- tor beams, under the supervision of Prof. Andrew Forbes, at the University of Witwatersrand, Jo- hannesburg, South Africa. His current research interests are in quantum optics, quantum metrology, and orbital angular momentum.

Thomas Gregory is a PhD student in the Optics Research Group at the Uni- versity of Glasgow. He has previously worked on creative camera systems with Dr Matthew Edgar at the Uni- versity of Glasgow in 2015 and on the development of computer vision sys- Figure 2. Quantum Ghost imaging setup, ‘Image plane’ configuration. tems for object tracking in industry in Correlated photons are generated in a nonlinear crystal pumped by an UV 2016. His current research interests laser. The photons are separated by a beam splitter. The center of the crys- are in quantum optics and quantum tal is imaged on both the object and the camera. In order to retrieve the metrology. ghost image, the coincidences between the Single-photon avalanche diode (SPAD) and the camera are recorded. As indicated by the broken arrows Miles Padgett holds the Kelvin Chair the position of the SPAD relatively to the object is unimportant, the only requierment being that it collects the full ligth beam to act as a ‘bucket of Natural Philosophy in the School detector’. of Physics and Astronomy at the Uni- versity of Glasgow where he heads an 3. Ghost Imaging Using Position or Momentum Optics Research Group. He is a Fellow both of the Royal Society of Edinburgh Correlations and the Royal Society. In 2017 he was In one possible configuration, the plane of the crystal is imaged awarded the Max Born Award of the to the plane containing the object after which a large-area, single- OSA. His research group studies in the field of optics and in pixel detector records any transmitted idler photons. This single- particular of optical angular momentum. His contributions pixel detector is often called the “bucket detector” to acknowledge include an optical spanner for spinning micron-sized cells, both its large area and non-imaging nature. In a separate optical use of orbital angular momentum to increase the data ca- path, the plane of the crystal is also imaged to a plane containing pacity of communication systems and an angular form of the an imaging detector that measures the position of the signal pho- quantum Einstein-Podolsky-Rosen (EPR) paradox. tons. In the early systems, this detector relied upon raster scanning over the field of view, limiting the collection efficiency of the system to be the reciprocal of the number of scan positions that made up the image.[1,2,8] Clearly, neither the single-pixel, nor imaging detectors by themselves acquire sufficient data to de- duce an image of the object. Taking the data from the imaging detector alone we see that all signal photons emitted from the

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Figure 3. Reconstruction of a ghost image by summing single frame with a few photons obtained when the camera is gated conditioned on the detection of a photon by the SPAD in both image plane and far field configurations. source have an equal chance of being detected and hence the image formed is simply an image of the source (in this case with a spatial extent corresponding to the diameter of the pump beam). Figure 4. Quantum Ghost imaging setup, ‘far-field’ configuration. The Taking the data from the bucket detector alone we see that only spatial spectrum of the light generated inside the crystal (Fourier plane) is imaged onto both the object and the camera. The position of the SPAD those photons that were transmitted by the object are recorded, relatively to the object is unimportant. but since their spatial position is unknown, again no image can be reconstructed. However, if the measurements from the two detectors are correlated together, an image can be reconstructed. ghost imaging must itself be a uniquely quantum phenomenon. One embodiment of this approach is when the bucket detector is However, this is not the case[9] and it has been a long-standing de- used to gate, or trigger, the recording of the position of the corre- bate whether or not ghost imaging could be demonstrated by har- lated photon. This position recording can be accomplished using nessing only classical correlations, i.e., using a scheme in which a triggered camera (as described later), and the image is formed one can explain ghost imaging of an object by using only a classi- by summing over many single-photon events, as seen in Figure cal light theory and a semi-classical theory for photodetection as 3. Alternatively, the outputs of the two detectors can be combined opposed to having to use a full quantum light theory and quan- such that only the coincident events are recorded as image infor- tum photodetection theory. A number of studies have been con- mation, with the image being the sum of these many correlated ducted to determine whether or not one could reproduce quan- events. tum ghost imaging features classically.[10–18] Note that the original demonstration of ghost imaging[2] was That ghost imaging does work is solely a result of the correla- based on a slightly different setup than that presented in Figure 2. tion between the transverse spatial position of the signal and idler Originally a lens in one of the arms was used to demonstrate the photons,[17,18] albeit this correlation can be a result of the entan- existence of a peculiar two-photon thin-lens equation, resulting glement between the photons. As an alternative to the entangle- in the acquisition of a ghost image. ment inherent in parametric downconversion one could instead In essence within figure 3 the confirmation of transmission conceive of photon-pair sources based on purely classical, albeit by the bucket detector is used to identify a sub-set of photons complicated, scanning systems that emit position-correlated pho- recorded by the imaging detector, and this sub-set of photons de- tons. Similarly a redesign of the classical system could also emit fines the image (akin to post selection). Essential in this imaging position anticorrelated photon pairs. In this respect the downcon- process is that the transverse position of the signal photons on the version source is different in that the photon pairs from the same imaging detector are highly correlated with the positions of the source are found to be correlated in either position or momentum idler photons in the plane of the object. This condition is ensured depending only on the basis of the measurements made.[19,20] since, as described above, the plane of the crystal is imaged both Therefore the capability of the downconversion source to give to the imaging detector and the object (note that since the bucket either upright or inverted images respectively in the image or detector is non-imaging it can be positioned in any plane subse- Fourier plane of the crystal is a consequence of an EPR-type en- quent to the object, providing that its aperture is sufficiently large tanglement between the photons.[21] In this last respect a ghost to collect the transmitted photons). However, there is an alterna- imaging system based on downconversion does rely upon quan- tive to this image-plane configuration. Rather than imaging the tum correlations.[22] It is also to note that using quantum correla- plane of the crystal to the object and imaging detector, the object tions for imaging presents an advantage in term of noise reduc- and the imaging detector can both be positioned in the far-field tion as discussed below.[23] of the crystal, see Figure 4. In the far-field, it is the momentum anticorrelation that means the signal and idler photons are now anticorrelated in their trans- 4. Improving the Detection Efficiency of a Ghost verse position. Interestingly, the switch from correlation to anti- Imaging System correlation means that the reconstructed image is now inverted with respect to the object see Figure 3. As discussed above, using a scanning detector as the imaging It is tempting to believe that since parametric downconversion detector is inherently inefficient, having a maximum detection is widely used as a source of quantum-entangled photons, then efficiency equal to the reciprocal of the number of independent

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Figure 5. Ghost images obtained using quantum correlations for diﬀerent numbers of photons per image, as indicated by the number above each column. The two lines of images correspond to a) summed binary images of photon detections b) images obtained through image regularisation of the above ghost images. The colorbars correspond to the number of photons per pixel. The images are composed of 256 × 256 pixels.

pixels in the image. Obviously, a preferred solution would be to basis of de-noising algorithms where spatial frequencies of low use a detector array covering the entire field of view such that no magnitude (and hence likely to be noise) are removed from an photons are missed. Until recently, camera systems did not have image. The de-noising approach is further aided by, in this case, the performance required to identify a single photon against the having exact knowledge of the noise model, allowing the sparsity background noise. However, modern time-gated intensified CCD condition to be precisely applied to reconstruct an estimate of the (ICCD) arrays are capable of detecting single photons against a image with highest sparsity (in the spatial frequency domain) that virtually noise-free background, providing that their gate-time is is still consistent with the measured data given the known noise kept short to the order of a few nanoseconds. In this respect, IC- type. Remarkably it is possible to reconstruct a reasonable quality CDs are the ideal cameras for inclusion into a ghost imaging sys- grey-scale image with fewer than one measured photon per im- tem where the single-pixel detector heralds precisely the arrival age pixel, which corresponds to an intensity orders of magnitude time of the correlated photon. Gating the intensified camera in less than a traditional imaging system.[25–27] Note that other com- this way allows an image to be accumulated photon-by-photon, putational imagers employing similar hypotheses and conven- potentially acquiring thousands of individual photons.[22,24] Since tional optical setups have been used for photon-efficient forma- most intensified cameras allow the intensifier to be triggered tion of high-quality reflectivity and depth images.[28,29] Figure 5 many thousands of times a second before the data is read out shows the reconstruction of photon-sparse ghost images using from the array, the spatially isolated photon detection events quantum correlations between photon pairs. The configuration dominate over the much smaller number of rogue noise events. used to record these ghost images is similar to the one presented in Figure 4 and the detection scheme as reported in [27], uses an Intensified CCD camera triggered by a single avalanche photodiode (SPAD) acting as the bucket detector. The photons in the 5. Reconstructing Images from Photon Sparse camera arm travel through an image preserving delay line such Data that the camera gating time, as triggered by the SPAD is syn- chronised with the arrival of the camera arm photon. The indi- Even images containing several thousand photons appear ex- vidual frames containing the detected photons are then summed tremely noisy. This noise arises since there are only a handful to obtain the presented images. The images are subsequently de- of photons per pixel and the reconstructed image of the object is noised using a search algorithm aiming to maximize the sparsity subject to large pixel-to-pixel fluctuations due to the inherent shot of the contributing spatial frequencies in the images while main- noise resulting from the finite number of discrete photon detec- taining the likelihood of the resulting image within the bounds tions. In the limit of high photon number, an expectation of n √ set by the Poissonian statistics of the original data. photons per pixel has an uncertainty of ± n in accordance with Gaussian statistics. For lower photon number the distribution of photon counts is described by a Poissonian (no negative values). However, real images are not random collections of pixel values, 6. Ghost Imaging and Wavelength Conversion typically there are strong correlations between neighbouring pixels in the image meaning that the spatial Fourier-transform of the Beyond the recording of images with very few photons, ghost image is sparse (i.e., has many zeros and/or very small values). imaging based upon parametric downconversion can do other This sparsity lies at the heart of image compression algorithms things too. Perhaps implicit in the discussion so far is that the used to store images within a minimum amount of memory such downconversion process is degenerate, namely that the signal as JPEG, and it is also used widely in image processing as the and idler photons have the same wavelength, i.e., ωs = ωi =

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ωp /2, but this need not be the case. While still satisfying the source that is directed through the object and back to the down- conservation of energy, the ratio between the signal and idler conversion crystal.[8] The crystal then acts as a mirror and re- wavelengths is set by the conservation of linear momentum flects the light to the camera, which in turn records an image. within the nonlinear crystal. Depending upon the crystal type, This image is exactly what would have been obtained from the changing the angle or the temperature of the crystal changes ghost imaging system (at least in the case of a degenerate sys- the refractive indices of the crystal, thereby changing the opti- tem). However, there are clearly differences too. For example, cal momentum of the signal and idler photons and hence chang- in the downconversion system the two detected events are si- ing their wavelength ratio. Even when far away from degener- multaneous in time, with the potential to be separated both in acy, such that the idler and signal have wavelengths in the in- space and time. Indeed it is the observation of simultaneous, frared and visible respectively, the photons are still strongly cor- space-separated correlations that lie at the heart of quantum me- related in their position and anti-correlated in their transverse chanics and is the essence of its “spookiness”. By contrast in the momentum.[30] This means that the ghost imaging approach still back-projection system, the emission process and subsequent de- works, but that the object can be probed using a different wave- tection are sequential. However, within the Klyshko picture, the length than measured by the camera. For example, the object can back-projection system illustrates beautifully the upright and in- be probed in the short-wave infrared and the transmitted photons verted nature of the images, corresponding to the object/camera detected using a single-element detector, whereas the camera de- being in the image plane or far-field of the crystal (mirror). In ex- tector is exposed to higher energy, visible photons at a wavelength perimental practise the change between the downconversion and for which the camera has single-photon sensitivity.[31] the back-propagation systems is easy to implement. It is usual for Although this form of wavelength conversion still requires a the single-pixel detector to be fibre coupled, so that a change from detector operating at the longer wavelength, it need not be a spa- detector to source simply involves the reconnection of one fibre, tially resolving one. However, there is an alternative configura- launching light directed to the crystal. The facet of the downcon- tion in which even the single-pixel detector is not required.[32] version crystal then acts as a mirror redirecting the light to the When using two downconversion crystals they can be arranged camera. As such, the back-projection system, in addition to be- such that both are pumped from the same source, and the idler ing a classical simulator of a quantum system, can also be used photons from one source pass through the other. Under these as a handy alignment tool.[35] circumstances, the actual source of the idler photons is indistin- guishable, as it could be either of the two crystals. This indis- tinguishability between the sources of the idler photons implies 8. Ghost Diffraction an interference between the two possible sources of the signal photons, and the subsequent routing of the signal photons at In addition to ghost imaging it is possible to configure these type a beam splitter. Placing an object in the idler path between the of systems for ghost diffraction.[1,36] In the early demonstrations two downconverted photon sources imposes a spatial variation the ghost diffracting object was placed in an arbitrary position, on the distinguishability of the idler source and hence a spatial but it is also possible to obtain the exact Fourier diffraction variation on the routing of the signal photons at the beam split- pattern[37] by placing the object and the camera in the image and ter. Hence the spatial variation of the object as illuminated by far-field plane of the downconversion crystal respectively (or vice the idler is mapped to the signal even though the idler itself is versa), see Figure 6. Again, the back-propagation concept dis- never detected. In this system, the only detector that is required cussed above gives insight to the required configuration. When is an imaging array sensitive to the signal wavelength. Despite trying to observe a classical diffraction pattern, for example from this seemingly quantum process, as for quantum ghost imaging a double slit, one needs to ensure that the illuminating light is itself, it has been suggested that this scheme also has a classical spatially coherent. This spatial coherence needs to be mimicked counterpart with which it shares a number of features.[33] Ghost imaging configurations based upon differing signal and idler wavelengths create other opportunities too. Beyond imaging systems, wavelength conversion can be applied to other sensor systems, the most obvious being spectroscopy. With similar advantages, as for the imaging system, the sample can be illuminated at one wavelength and the correlated spectral information recorded at a more convenient wavelength for which the multi- channel spectrometer is more sensitive.[34]

7. A Classical Simulator for Ghost Imaging The behaviour of a ghost imaging system based upon parametric downconversion can be analysed through standard quantum Figure 6. Quantum Ghost diﬀraction scheme. The object is placed in the techniques, however, as recognised shortly after its conception, Fourier plane of the crystal and the camera is placed in the image plane the results of such a system could also be predicted using classi- of the crystal, or conversely the object and camera are placed in the image cal optics. Referred to as a Klyshko picture or back-projection, plane and the Fourier plane respectively. The ghost diﬀraction images are the single-pixel detector is notionally replaced with an optical obtained in the same manner as in the quantum ghost imaging schemes.

Laser Photonics Rev. 2017, 1700143 1700143 (5 of 11) C 2017 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.lpr-journal.org in the downconversion system too, hence the bucket detector plies divergence) and has a standard deviation given by needs to be modified such that it detects only a single spatial- mode. In practise this modification is often quite simple. Under σc ≈ 2 2L/(πkp ) normal use, the bucket detector comprises a single-photon avalanche detector, coupled to the detection plane using an opti- Alternatively if the imaging system is configured with the ob- cal fibre. If this fibre is a multi-mode fibre then the detector acts ject and camera in the far-field then the position correlations be- as a large area, multi-mode bucket detector. Alternatively, using a tween signal and idler photons stem from the momentum corre- single-mode fibre converts the bucket detector to a single-mode lations in the source which in turn arise from the transverse mo- detector, hence enabling ghost diffraction.[37] Clearly, incorporat- mentum uncertainty in the pump beam. This momentum con- ing this additional degree of modal selection into the bucket de- sideration gives a spatial correlation between the signal and idler tector reduces the data rate of the system, meaning the diffraction photons in the far field given by pattern takes longer to acquire than the equivalent image would. In this respect, it is no different from a classical system where σc ≈ 4 f / kp wp spatially filtering an incoherent light source results in a reduction in intensity in proportion to the modal selectivity of the filter. Note Notwithstanding the strength of either of these correlations, finally that hybrid schemes using both diffraction and imaging the resolution of the ghost imaging system cannot exceed the allow heralded phase-contrast imaging to be performed through resolution by which the plane of the nonlinear crystal (or, for mo- introducing a phase filter in a Fourier plane of the crystal.[38] mentum correlations, its far-field) is imaged to either the object or the camera. In other words, if the resolution of the imaging system is to be limited only by the intrinsic correlation strength 9. Resolution Limits of Ghost Imaging of the source,[39] then the relay optics between source and camera/detector need to be of sufficient aperture to collect and image Quantum ghost imaging relies upon the spatial correlation or all of the downconverted light.[18] momentum anticorrelation between the signal and idler photons A further consideration as to the resolution of such a system is as produced by the parametric downconversion process. If the ob- to recognise that a ghost imaging system can be set up in a num- ject is positioned in an image plane of the crystal then the field of ber of seemingly equivalent configurations. Taking the case when view σv of any resulting imaging system depends upon the width object and imaging detector are positioned in image planes of the of the pump beam, σv = ωp . Alternatively, if the object is placed nonlinear crystal, one could use additional optics to introduce ad- in a far-field of the crystal, the field of view depends upon the di- ditional intermediate image planes within both the camera and vergence of the downconverted light and the focal length, f ,of the single-pixel detector beam paths. In principle an image could the lens creating the far-field plane. This divergence of light from be recorded with the object positioned in any one of these planes. the crystal is a function only of the wavelength of the light and the If the object is placed in any image plane in the beam path of the length of the nonlinear crystal. If one assumes when the signal single-pixel detector, then the system is a ghost imaging system, and idler photons are exactly collinear with the pump photons and the resolution limit is set as discussed above. If however, the that the phase matching is perfect (kp = ks + ki ), then an angu- object is positioned in the beam path of the camera detector, then lar deviation, α, for the signal and idler from co-linearity brings an image is still recorded but the purpose of the single-pixel de- a phase mismatch of tector is now simply to gate (i.e. herald) the camera, such that stray light and other noise sources may be supressed.[27] An inter- 2 kz = kp − 2cos(α) ks ≈ (1 − cos (α)) kp ≈ α /2 kp esting question one could ask is: “Do the two types of configuration give rise to an image of the same resolution?”. As discussed For efficient generation of signal and idler over the length, L, above, in the case of a ghost imaging system, the resolution is of the nonlinear crystal this phase mismatch should not exceed ultimately limited by the strength of the spatial correlations be- ≈ π meaning that the divergence of the downconverted light is tween the signal and idler photons. Clearly, the observed strength approximately given by cannot exceed the resolving power of the optics, it can however be lower, according to the limits imposed by the employed nonlin- 2 α 2π/(kp L) ear crystal. In contrast to this, for a heralded imaging system, the only factor governing the resolution is the resolving power of the Giving a far-field field of view optics in the camera beam path, i.e., the optics placed between the object and the camera itself. σv ≈ f 2π/(kp L) With regards to pseudothermal ghost imaging, systems utilis- ing classical correlations from a pseudothermal light source, the More subtle than the limitations on the field of view are the fac- resolution is limited by the speckle size in the object plane. The tors covering the resolution of the ghost imaging system. Leaving transverse coherence radius of the speckle pattern may be de- aside the normal resolution limitations arising from the diffrac- scribed by the Van Cittert-Zernike theorem[40,41] thereby allowing tion limit associated with finite aperture lenses, the resolution the spatial resolution of the pseudothermal ghost imaging sys- cannot exceed the strength of the spatial correlation between the tem to be determined.[42] This coherence radius is found to be to [39] signal and idler photons. In the image plane of the crystal this be of the order dx = λL/a0, in which the radius dx is determined correlation is closely related to the divergence of the downcon- by the wavelength λ of the pump used to illuminate the diffuser, verted light (since spatial localisation in the transverse plane im- the radius of the beam in the diffuser plane a0, and L the distance

Laser Photonics Rev. 2017, 1700143 1700143 (6 of 11) C 2017 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.lpr-journal.org of the object plane from that of the diﬀuser plane, or source of the speckle pattern.[42,43] For other theoretical consideration on the resolution limits in Ghost imaging the reader can refer to [44].

10. Other Imaging Systems Based on Parametric Downconversion Sources In a ghost imaging system, the accumulation of data occurs photon by photon with each signal photon being correlated to an idler photon detection. The ability to image the position of a single signal photon relies upon the heralding of its arrival through the detection of the correlated idler photon. This timing precision allows a system where the camera is active only at the instant of photon arrival and thereby largely eliminates dark-counts from Figure 7. Sub-shot noise imaging of a low absorption sample. Repro- the image. Alternatively, if a camera system is placed in both sig- duced with permission.[49] Copyright 2011, M. Genovese, G. Brida, A. nal and idler paths then no such heralding signal is possible and Meda, I. R. Berchera, “Exploiting PDC spatial correlations for innovative the cameras can no longer unambiguously detect a single photon quantum imaging protocols”, Proc. SPIE 8163, Quantum Communica- against the background noise. However, if the camera exposure tions and Quantum Imaging IX, Society of Photo-Optical Instrumentation Engineers. is set to such that many photon pairs are present in each read- out frame then correlations arising from the photon pair nature of the light can still be observed. This can be detected either as imaging scheme enabled by the presence of quantum correla- an overall correlation in the random intensity fluctuations of the tions. On the other hand classical ghost imaging can be defined signal and idler fields[45] or at the level of a small number of pho- as being the ensemble of ghost imaging techniques that harness tons pairs.[46,47] In this latter case by setting the flux such that the classical correlations. Ghost imaging systems using parametric number of photon pairs recorded is comparable to the number downconversion are a particular type of quantum ghost imaging of dark events per frame it is possible to observe the overall cor- and can be based either upon the spatial correlation in the image relation (in the image plane) or anti-correlation (in the far-field) plane of the downconversion crystal or the spatial anticorrelation within a single frame from the camera. in the far-field. If all that is required is a functioning imaging Such twin imaging systems can also be used for imaging system then either of these is sufficient, without the need of EPR where one camera directly records an image of the object, and the quantum correlations. A specific example of a classical ghost other camera (or different part of the same camera chip) records imaging system is one that uses classical correlations from a [11,13,51] only the illumination. Since the spatial distribution of the pho- thermal light source, one will see later that other classes of tons in the two arms is high correlated, the illumination pattern classical ghost imaging exist, e.g., computational ghost imaging is also almost identical (photon by photon) to that used to illu- techniques. minate the object. It follows that the subtraction/or division of In principle, any random light field possess a defined inten- the two intensity distributions, in principle, can return the trans- sity structure. A beam splitter inserted into the beam creates two mission of the object with a precision that is better than the shot- identical (classical) copies with an experimental ‘copy-accuracy’, noise limit of the detected image alone.[23,48] Figure 7 illustrates as observed after detecting the intensities of the copied beams, the principle of such experiments and shows the denoising of the that is limited in principle only by the Poissonian statistics as- image of a low absorption object using fluctuation subtraction.[49] sociated with the detected photon-number. In this picture, one This type of sub-shot-noise imaging once again relies upon the copy of the beam is allowed to propagate to a detector array that P x, y spatial correlation that arises as a result of parametric downcon- measures the exact pattern of the intensity distribution, ( )i , version. More specifically, this imaging technique exploits spatial whereas the other copy propagates through exactly the same dis- x, y entanglement, without being itself a proof of it. tance to the object, 0( ); meaning that the object is illuminated Finally, parametric down conversion applications are not lim- with exactly the same intensity distribution as recorded by the de- ited to intensity imaging, it has been shown that sub shot noise tector array, see Figure 8. It should be noted that the use of an imaging of phase objects can be obtained in the context of mi- imaging system after the beam splitter in Figure 8 is unneces- croscopy by using NOON states generated by parametric down sary, as the only requirement is that the camera and the object [13,15] conversion.[50] are positioned in the equivalent plane. The total power of the light transmitted, or backscattered, by the object is measured by a single-pixel detector; this amount constitutes the signal, Si . Combining the pattern information with the signal information 11. Ghost Imaging Based on Non-Quantum N Correlations, Using Thermal Light Source corresponding to many, , patterns allows one to estimate the image, I(x, y), of the object, as: The quantum nature of ghost imaging was for a number of years N a debated topic, but the issue is now largely resolved. We can I (x, y) = (P(x, y)i − P (x, y)) × (Si − S) practically define quantum ghost imaging as being any ghost i=1

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Figure 8. Pseudo-thermal ghost imaging. A coherent light beam passes through a rotating ground glass plate (GGP) thereby generating a series of speckle patterns. The beam is then separated in two using a beam splitter. Finally, the same plane is imaged onto both the object and the camera to ensure that the camera records the pattern projected onto the object. Figure 9. Computational ghost imaging scheme. An SLM is illuminated by Light transmitted by the object is detected by a photodiode that acts as a coherent light source. The spatially modulated light-beam is then sent a bucket detector, its exact position after the object is unimportant. Note through or reflected from an object, and a photodiode then detects the that the use of an imaging system after the beam splitter is unnecessary, overall beam intensity transmitted by or reflected from the object. The Pat- as the only requirement is that the camera and the object are positioned tern projected onto the object can be deduced from the phase pattern dis- in the same plane. played by the SLM. It should be noted that the position of the photodiode is unimportant. In practise, the patterns associated with a true thermal source vary too quickly for a detector array to follow. For this reason, from knowing the pattern displayed on the SLM. In the original one instead usually adopts a pseudo-thermal source, such as proposal, the SLM was used to create a field of known intensity that formed by passing a laser through a rotating ground glass and phase, such that the resulting intensity structure, P(x, y)i , screen. Unlike the photon-pair case, where P(x, y)i is effectively could be calculated in any subsequent plane. To stress that the the position of a single photon and Si is a binary signal, for correlations involved are between the light field and the SLM, this thermal case, P(x, y)i is a spatially extended function and this technique was called “computational ghost imaging”, in an Si is an analogue signal. The result of this continuous nature attempt to emphasise its non-quantum nature.[43,61] means that the thermal case produces low contrast images, albeit A more simple configuration that does not require the subse- ones that can be dramatically improved by numerical background quent computation of the illumination field is to use an intensity subtraction.[15,52] The higher flux of a pseudo-thermal source modulator, where the plane of the modulator is explicitly imaged means that once processed, the performance of this classical sys- to the plane of the object, again giving a known intensity pat- tem can exceed those based upon downconversion.[53] Moreover, tern, but this time without need for calculation, see Figure 9.Us- it has been shown that the image quality obtained through this ing spatially structured light to create imaging systems has been technique is resilient to slow acquisition in which many speckle known and utilised under various guises for many decades, per- patterns (ࣈ25) are transmitted over the course of a single expo- haps the most prevalent being the raster scanning of a laser beam sure, thereby allowing the technique to be used with rapidly fluc- over an object, with the use of a single element detector to mea- tuating patterns.[54] Some works have claimed that there exists sure the backscattered signal. However, if the source is spatially another practical advantage in that, although the presence of tur- extended then it cannot be focused to form a scanning spot and bulence can degrade the quality of the reconstructed images, ac- another approach is needed. Using a spatial light modulator, the cording to these studies, in some cases, thermal ghost imaging spatially extended source can still be structured, thereby allowing schemes can be designed to be robust against turbulence,[55–57] the illumination of the object with a known intensity pattern. which is claimed to make it a good candidate in remote sens- Measuring the backscattered signal in response to arbitrary il- ing scenarios.[58] Finally, various reconstruction techniques have lumination patterns holds further advantages over raster scan- been proposed to enhance the signal to noise ratio of the classical ning. For a raster-scanned system, the number of measurements ghost imaging methods compared to conventional ghost imaging required is exactly equal to the number of pixels in the desired reconstruction.[59,60] image, but if the object is illuminated with a series of extended intensity patterns then the same image can be reconstructed with 12. Ghost Imaging Based Correlations with fewer measurements required. The technique used here is simi- Structured Light Fields lar in concept to JPEG image compression which works because in the spatial frequency domain typical images are sparse, i.e. Rather than using a ground glass screen, or similar, it is possible they have many frequency components with very low weightings to make spatially structured fields using a programmable spatial that need not be stored.[62,63] A similar efficiency can also be ap- light modulator (SLM). In this case there is no need for a beam plied to measurement, allowing an excellent reproduction of an splitter and a detector array to measure the intensity of the light image to be obtained from fewer measurements than there are field in the plane of the object, since this pattern can be calculated pixels in the image.

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13. Structured Illumination Imaging Systems Structured illumination is used in various types of imaging systems, for example in the creation of Moiré patterns from which surface shapes can be deduced. However, here we restrict the discussion to the use of structured illumination as a means of single-pixel imaging. Perhaps the most obvious use for structured illumination lies in illumination at wavelengths where traditional imaging based upon focal plane detector arrays would be expensive. Recently we have adopted this approach for imaging methane gas at ࣈ1.6 μm.[64] Such systems are viable since spatial light modulators work over a wide wavelength range, and cost effective single-pixel detectors are similarly available over a range of different wavelengths. Another possibility is in the use of multiple detectors, either to extend the wavelength range or to collect light backscattered in different directions. Using multiple detectors located about the object and with known position is akin to photometric imaging. In such a setup, a single structured light source may be used to illuminate an object and the resulting backscattered signal is measured using the multiple detectors located about the object. This technique enables the gradients of the surface of the object at each pixel to be deduced and subsequently integrated to Figure 10. Single-pixel camera’s scheme. Coherent or incoherent light is used to illuminate an object. The object plane is imaged onto a DMD that produce a surface profile. This is achieved by processing a set spatially modulates the intensity of the incoming light-beam. A photodi- of simultaneously obtained images of the reflected light gener- ode is used to detect the overall beam intensity reflected by the DMD. It ated as a result of illuminating the object with a structured light should be noted that the position of the photodiode after the DMD can be source, where images that are obtained from a series of detectors arbitrary, as the only requirement is that the full beam is detected. at known position relative to the object. Unlike traditional photometric stereo which requires sequential illumination due to there being multiple light sources and a single detector, projected illumination requires only sequential patterns and multiple, but simultaneous detections.[65]

14. Single-Pixel Cameras

Although the physics community has developed what we term Figure 11. Single-pixel images. a) Conventional image of a black-and- computational ghost imaging, this whole field is closely related to white letter ‘R’. b) Single-pixel camera reconstructed image from M = 1, the broader field of single-pixel cameras as pioneered by Duarte 300 random measurements. c) Colour reconstruction of a printout of the et al.[63] In computational ghost imaging, the data arises from the Mandrill standard test image using a single photomultiplier tube sensor spatial correlation between the structured illumination and the and RGB colour filters (M = 6500 random measurements). Reproduced [63] object. However, the same concept can be employed in a differ- with permission. Copyright 2008, IEEE. ent manner to produce an equivalent result: rather than struc- turing the illumination, it is possible to structure the detection, simply by replacing the light source with a detector. In this con- from the plane of the modulator to the plane of the object means figuration, any light source can be used to illuminate the object that a simpler configuration is to explicitly image the plane of that is imaged to the plane of the spatial light modulator or a the object to the plane of the modulator and then measure Digital Micromirror Device (DMD) see Figure 10. The detector the corresponding intensity correlations with the single-pixel now measures the correlation of the backscattered image with detector. the programmed pattern.[65] Figure 11 shows reconstructed im- Single-pixel cameras offer a cost-effective imaging solution for ages, with data captured using a single pixel camera by Duarte operating at wavelengths were focal plane arrays are expensive or et al.[63] The optimum choice of patterns and algorithms to in- not available. The approach also opens options for hyperspectral vert the data formed from both the patterns and the measured imaging, in which multiple single-pixel detectors operating with signals to obtain an image are largely identical in the computa- different wavelength sensitivities can be incorporated into the tional ghost imaging and single pixel configurations. However, same camera system.[66–68] In addition, it has been demonstrated for the single-pixel camera configuration using a DMD there is that a single-pixel camera allows the imaging of a scene with now less control over the light source, specifically no knowledge a detector that is not placed in direct view of the object.[69] of the phase, so calculating the propagating field is problem- Finally, the high temporal-resolution of single-pixel detectors, atic. The inability to calculate the general propagation of the field compared to focal plane arrays, means that when combined with

Laser Photonics Rev. 2017, 1700143 1700143 (9 of 11) C 2017 by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.lpr-journal.org stroboscopic illumination, it is possible to obtain depth informa- imaging with short-wave infra-red light while recording the data tion from the scene, effectively creating an imaging LIDAR.[70–72] with a visible wavelength camera. However, there is no funda- mental reason why this technique could not be extended to even more inaccessible wavelengths. For example, could this be a route 15. Ghost Imaging Concepts Extended to Other to a low-cost, high-resolution THz camera[84]? Domains With respect to single-pixel cameras, there seems to be oppor- tunity to relax the “single” to include a small number of detec- Since its discovery, the concept of ghost imaging has been ex- tors, perhaps to create UV-vis-IR images from a single camera. tended to domains outside of the capture of spatial proprieties of Alternatively, to incorporate additional detection channels to give light and beyond the usual optical domain. simultaneous information relating to polarisation, timing/range A particularly active research area is that of ghost imaging in data or even just to provide complimentary data to a traditional the time domain. It was first suggested theoretically[73,74] that focal plane array. a temporal objects could be recorded using a slow integrating Future systems seem set to exploit the various elements and al- ‘bucket’ detector. It was later demonstrated experimentally[75] us- gorithms discussed above, from optical-correlations both classi- ing a temporally incoherent source split into two beams so as to cal and quantum, compressed sensing and data inversion along- record the laser fluctuations on one arm, while the other beam side the ever improving spectral and temporal detector specifi- was used to probe the temporal object before being detected by cations. This future seems both an intellectual adventure and a a slow integrating detector unable to resolve the signal. This route to delivering a family of multidimensional cameras operat- demonstration was limited to record temporal signals that are ing across a range of wavelengths, timescales and length-scales. reproducible in time, since as in standard ghost imaging where a large number of patterns must be used in order to reconstruct a ghost image, the temporal signal had in this context to be re- Acknowledgements produced many times to allow the statistical reconstruction of the ghost object. The exact space–time transposition of ghost P.-A.M. and E.T. contributed equally to this work. This work was funded by imaging was subsequently reported when it was demonstrated the UK EPSRC (QuantIC EP/M01326X/1) and the ERC (TWISTS, Grant no. 340507). P.-A.M. acknowledges the support from the European Union’s that a non-reproducible signal can be recorded by computational Horizon 2020 research and innovation programme under the Marie ghost imaging through the use of spatial multiplexing instead of Sklodowska-Curie fellowship grant agreement No 706410. E.T. acknowl- temporal multiplexing.[76] As in the case of standard ghost imag- edges the financial support from the EPSRC Centre for Doctoral Training in ing, it was shown that together with structured light this tech- Intelligent Sensing and Measurement (EP/L016753/1). T.G. acknowledges nique may also exploit both thermal light classical correlations[77] the financial support from the EPSRC (EP/N509668/1) and the Professor and quantum correlations[78] to retrieve an image of the ‘ghost Jim Gatheral quantum technology studentship. object’. Finally ghost imaging concepts also apply outside of the usual optical domain. In recently reported experiments, ghost imag- Conflict of Interest [79–81] ing was demonstrated using X-rays and for atomic systems The authors have declared no conflict of interest. through exploiting quantum correlations exhibited by pairs of ul- tracold helium atoms.[82] Keywords ghost imaging, non-linear optics, quantum correlations, quantum imag- 16. Conclusions and Outlooks ing, single-photon imaging Since its inception in the 1990’s ghost imaging has intrigued re- searchers, many of whom have been inspired to study this tech- Received: May 28, 2017 Revised: October 2, 2017 nique, both in terms of the underpinning physics it reveals and Published online: December 15, 2017 for the possible applications it enables. In terms of the underpinning physics, the analysis of images within quantum systems is a study of high-dimensional Hilbert spaces. In terms of applications there seem to be two main avenues of work. Firstly, when [1] D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, Y. H. Shih, Phys. Rev. using a light source based upon parametric downconversion the Lett. 1995, 74, 3600. temporal correlation allows for a stringent rejection of noise, [2] T. B. Pittman, Y. H. Shih, D. V. Strekalov, A. V. Sergienko, Phys.Rev.A. thereby only requiring very few photons to obtain valuable infor- 1995, 52, R3429. mation about the object/image. Secondly, computational ghost [3] D. N. Klyshko, Sov. Phys. Uspekhi 1988, 31, 74. imaging and single-pixel cameras which only require a single- [4] D. Klyshko, Phys. Lett. A 1988, 132, 299. [5] S.E.Harris,M.K.Oshman,R.L.Byer,Phys. Rev. Lett. 1967, 18, 732. pixel detector to reconstruct the image of an object, allows opera- [6] D. A. Kleinman, Phys. Rev. 1968, 174, 1027. tion at wavelengths, and on timescales, that would not be possible [7] D. C. Burnham, D. L. Weinberg, Phys. Rev. Lett. 1970, 25, 84. with focal-plane detector arrays. [8] T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. A comparatively untapped feature of quantum ghost imaging Sergienko, Y. H. Shih, Phys.Rev.A1996, 53, 2804. is that the spatial correlations can be between photons with arbi- [9] R. S. Bennink, S. J. Bentley, R. W. Boyd, Phys.Rev.Lett. 2002, 89, trary different wavelengths.[31,83] This feature has been used for 113601.

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[10] A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, M. C. Teich, Phys. Rev. [47] P.-A. Moreau, F. Devaux, E. Lantz, Phys. Rev. Lett. 2014, 113, 160401. Lett. 2001, 87, 123602. [48] E. Brambilla, L. Caspani, O. Jedrkiewicz, L. Lugiato, A. Gatti, Phys. [11] A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, Phys. Rev. Lett. 2004, Rev. A 2008, 77, 053807. 93, 093602. [49] M. Genovese, G. Brida, A. Meda, I. R. Berchera, in Proceedings [12] Y. J. Cai, S. Y. Zhu, Phys.Rev.E2005, 71, 056607. SPIE 8163, Quantum Communications and Quantum Imaging IX,San [13] A. Valencia, G. Scarcelli, M. D’Angelo, Y. Shih, Phys.Rev.Lett. 2005, Diego, USA 2011, p 816302. 94, 063601. [50] T. Ono, R. Okamoto, S. Takeuchi, Nat. Commun. 2013, 4, 2426. [14] F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, L. A. Lugiato, [51] F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, L. A. Lugiato, Phys.Rev.Lett. 2005, 94, 183602. Phys. Rev. Lett. 2005, 94, 183602. [15] G. Scarcelli, V. Berardi, Y. Shih, Phys.Rev.Lett.2006, 96, 063602. [52] R. S. Bennink, S. J. Bentley, R. W. Boyd, J. C. Howell, Phys. Rev. Lett. [16] B. I. Erkmen, J. H. Shapiro, Phys.Rev.A2008, 77, 043809. 2004, 92, 033601. [17] B. I. Erkmen, J. H. Shapiro, Adv. Opt. Photonics 2010, 2, 405. [53] A. Gatti, E. Brambilla, M. Bache, L. A. Lugiato, Phys. Rev. A 2004, 70, [18] J. H. Shapiro, R. W. Boyd, Quantum Inf. Process. 2012, 11, 949. 013802. [19] J. C. Howell, R. S. Bennink, S. J. Bentley, R. W. Boyd, Phys.Rev.Lett. [54] P. Zerom, Z. Shi, M. N. O’Sullivan, K. W. C. Chan, M. Krogstad, J. H. 2004, 92, 210403. Shapiro, R. W. Boyd, Phys.Rev.A2012, 86, 063817. [20] M. D’Angelo, Y.-H. Kim, S. P. Kulik, Y. Shih, Phys.Rev.Lett. 2004, 92, [55] J. Cheng, Opt. Express 2009, 17, 7916. 233601. [56] R. E. Meyers, K. S. Deacon, Y. Shih, Appl. Phys. Lett. 2011, 98, 111115. [21] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 1935, 47, 777. [57] R. E. Meyers, K. S. Deacon, Y. Shih, Appl. Phys. Lett. 2012, 100, [22] R. S. Aspden, D. S. Tasca, R. W. Boyd, M. J. Padgett, New J. Phys. 2013, 131114. 15, 073032. [58] B. I. Erkmen, J. Opt. Soc. Am. A-Opt. Image Sci. Vis. 2012, 29, 782. [23] G. Brida, M. Genovese, I. R. Berchera, Nat. Photonics 2010, 4, 227. [59] F. Ferri, D. Magatti, L. A. Lugiato, A. Gatti, Phys.Rev.Lett. 2010, 104, [24] R. Fickler, M. Krenn, R. Lapkiewicz, S. Ramelow, A. Zeilinger, Sci. Rep. 253603. 2013, 3, 1914. [60] B. Sun, S. S. Welsh, M. P. Edgar, J. H. Shapiro, M. J. Padgett, Opt. [25] M. Sonnleitner, J. Jeffers, S. M. Barnett, Optica 2015, 2, 950. Express 2012, 20, 16892. [26] L. Mertens, M. Sonnleitner, J. Leach, M. Agnew, M. J. Padgett, Sci. [61] Y. Bromberg, O. Katz, Y. Silberberg, Phys. Rev. A 2009, 79, 053840. Rep. 2017, 7, 42164. [62] O. Katz, Y. Bromberg, Y. Silberberg, Appl. Phys. Lett. 2009, 95, [27] P. A. Morris, R. S. Aspden, J. E. C. Bell, R. W. Boyd, M. J. Padgett, Nat. 131110. Commun. 2015, 6, 5913. [63] M. F. Duarte, M. A. Davenport, D. Takbar, J. N. Laska, T. Sun, K. F. [28] D. Shin, A. Kirmani, V. K. Goyal, J. H. Shapiro, IEEE Trans. Comput. Kelly, R. G. Baraniuk, IEEE Signal Process. Mag. 2008, 25, 83. Imaging 2015, 1, 112. [64] G. M. Gibson, B. Sun, M. P. Edgar, D. B. Phillips, N. Hempler, G. T. [29] A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Maker, G. P. A. Malcolm, M. J. Padgett, Opt Express 2017, 25, 2998. Shapiro, V. K. Goyal, Science 2014, 343, 58. [65] B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, [30] K. W. C. Chan, M. N. O’Sullivan, R. W. Boyd, Phys.Rev.A2009, 79, M. J. Padgett, Science 2013, 340, 844. 033808. [66] S. S. Welsh, M. P. Edgar, R. Bowman, P. Jonathan, B. Sun, M. J. Pad- [31] R. S. Aspden, N. R. Gemmell, P. A. Morris, D. S. Tasca, L. Mertens, gett, Opt. Express 2013, 21, 23068. M. G. Tanner, R. A. Kirkwood, A. Ruggeri, A. Tosi, R. W. Boyd, G. S. [67] D.-J. Zhang, H.-G. Li, Q.-L. Zhao, S. Wang, H.-B. Wang, J. Xiong, K. Buller, R. H. Hadfield, M. J. Padgett, Optica 2015, 2, 1049. Wang, Phys.Rev.A2015, 92, 013823. [32] G. B. Lemos, V. Borish, G. D. Cole, S. Ramelow, R. Lapkiewicz, A. [68] M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Zeilinger, Nature 2014, 512, 409. Mitchell, S. S. Welsh, M. J. Padgett, Sci. Rep. 2015, 5, 10669. [33] J. H. Shapiro, D. Venkatraman, F. N. C. Wong, Sci. Rep. 2015, 5, 10329. [69] Z. Zhang, X. Ma, J. Zhong, Nat. Commun. 2015, 6, 6225. [34] D. A. Kalashnikov, A. V. Paterova, S. P. Kulik, L. A. Krivitsky, Nat. Pho- [70] G. A. Howland, D. J. Lum, M. R. Ware, J. C. Howell, Opt. Express 2013, tonics 2016, 10, 98. 21, 23822. [35] R.S.Aspden,D.S.Tasca,A.Forbes,R.W.Boyd,M.J.Padgett,J. Mod. [71] N. D. Hardy, J. H. Shapiro, Phys.Rev.A2013, 87, 023820. Opt. 2014, 61, 547. [72] M.-J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, [36] E. da S. Fonseca, P. S. Ribeiro, S. Pádua, C. Monken, Phys.Rev.A1999, M. J. Padgett, Nat. Commun. 2016, 7, 12010. 60, 1530. [73] T. Shirai, T. Setälä, A. T. Friberg, J. Opt. Soc. Am. B 2010, 27, 2549. [37] D. Tasca, R. Aspden, P. Morris, G. Anderson, R. Boyd, M. Padgett, [74] T. Setälä, T. Shirai, A. T. Friberg, Phys. Rev. A 2010, 82, 043813. Opt. Express 2013, 21, 30460. [75] P. Ryczkowski, M. Barbier, A. T. Friberg, J. M. Dudley, G. Genty, Nat. [38] R. S. Aspden, P. A. Morris, R. He, Q. Chen, M. J. Padgett, J. Opt. 2016, Photon. 2016, 10, 167. 18, 055204. [76] F. Devaux, P.-A. Moreau, S. Denis, E. Lantz, Optica 2016, 3, 698. [39] K. A. Forbes, J. S. Ford, D. L. Andrews, Phys.Rev.Lett. 2017, 118, [77]F.Devaux,K.P.Huy,S.Denis,E.Lantz,P.-A.Moreau,J. Opt. 2016, 133602. 19, 024001. [40] P. H. Van Cittert, Phys. Utrecht 1934, 1, 202. [78] S. Denis, P.-A. Moreau, F. Devaux, E. Lantz, J. Opt. 2017, 19, 034002. [41] F. Zernike, Ibid 1938, 5, 785. [79] J. Cheng, S. Han, Phys.Rev.Lett. 2004, 92, 093903. [42] A. Gatti, D. Magatti, F. Ferri, Phys.Rev.A2008, 78, 063806. [80] D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, D. M. Paganin, Phys. Rev. [43] J. H. Shapiro, Phys. Rev. A 2008, 78, 061802. Lett. 2016, 117, 113902. [44] M. D’Angelo, A. Valencia, M. H. Rubin, Y. Shih, Phys. Rev. A 2005, 72, [81] H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, D. Zhu, Phys. Rev. Lett. 013810. 2016, 117, 113901. [45] O. Jedrkiewicz, Y.-K. Jiang, E. Brambilla, A. Gatti, M. Bache, L. Lugiato, [82] R. I. Khakimov, B. Henson, D. Shin, S. Hodgman, R. Dall, K. Baldwin, P. Di Trapani, Phys.Rev.Lett. 2004, 93, 243601. A. Truscott, Nature 2016, 540, 100. [46] M. P. Edgar, D. S. Tasca, F. Izdebski, R. E. Warburton, J. Leach, M. [83] S. Karmakar, Y. Shih, Phys.Rev.A2010, 81, 033845. Agnew, G. S. Buller, R. W. Boyd, M. J. Padgett, Nat. Commun. 2012, [84] V. V. Kornienko, S. A. Savinov, Y. A. Mityagin, G. K. Kitaeva, Opt. Lett. 3, 984. 2016, 41, 4075.

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