Traditional optical imaging faces an unavoidable trade-off between resolution and depth of field (DOF). To increase resolution, high numerical apertures (NA) are needed, but the associated large angular uncertainty results in a limited range of depths that can be put in sharp focus. Plenoptic imaging was introduced a few years ago to remedy this trade off. To this aim, plenoptic imaging reconstructs the path of light rays from the lens to the sensor. However, the improvement offered by standard plenoptic imaging is practical and not fundamental: the increased DOF leads to a proportional reduction of the resolution well above the diffraction limit imposed by the lens NA. In this paper, we demonstrate that correlation measurements enable pushing plenoptic imaging to its fundamental limits of both resolution and DOF. Namely, we demonstrate to maintain the imaging resolution at the diffraction limit while increasing the depth of field by a factor of 7. Our results represent the theoretical and experimental basis for the effective development of the promising applications of plenoptic imaging.Plenoptic imaging (PI) is a novel optical method for recording visual information [1]. Its peculiarity is the ability to record both position and propagation direction of light in a single exposure. PI is currently employed in the most diverse applications, from stereoscopy [1][2][3], to microscopy [4][5][6][7], particle image velocimetry [8], particle tracking and sizing [9], wavefront sensing [10][11][12][13] and photography, where it currently enables digital cameras with refocusing capabilities [14,15]. The capability of PI to simultaneously acquire multiple-perspective 2D images brings it among the fastest and most promising methods for 3D imaging with the available technologies [16]. Indeed, high-speed and large-scale 3D functional imaging of neuronal activity has been demonstrated [7]. Furthermore, first studies for surgical robotics [17], endoscopic application [18] and blood-flow visualization [19] have been performed.The key component of standard plenoptic cameras is a microlens array inserted in the native image plane, that reproduces repeated images of the main camera lens on the sensor behind it [1,15]. This enables reconstruction of light paths, employed, in post-processing, for refocusing different planes, changing point of view and extending depth of field (DOF) within the acquired image. However, a fundamental trade-off between spatial and angular resolution is naturally built in standard plenoptic imaging. If N tot is the total number of pixels per line on the sensor, N x the number of microlenses per line, and N u the number of pixels per line associated with each microlens, then N x N u = N tot . Essentially, standard PI gives the same resolution and DOF one would obtain with a N u times smaller NA. The final advantage is thus practical rather than fundamental, and is limited * francesco.pepe@ba.infn.it † milena.dangelo@uniba.it to higher luminosity (hence SNR) of the final image and parallel acquisition of m...

Plenoptic imaging is a promising optical modality that simultaneously captures the location and the propagation direction of light in order to enable three-dimensional imaging in a single shot. However, in standard plenoptic imaging systems, the maximum spatial and angular resolutions are fundamentally linked; thereby, the maximum achievable depth of field is inversely proportional to the spatial resolution. We propose to take advantage of the second-order correlation properties of light to overcome this fundamental limitation. In this Letter, we demonstrate that the correlation in both momentum and position of chaotic light leads to the enhanced refocusing power of correlation plenoptic imaging with respect to standard plenoptic imaging.

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We construct an analogue computer based on light interference to encode the hyperbolic function f (ζ) ≡ 1/ζ into a sequence of skewed curlicue functions. The resulting interferogram when scaled appropriately allows us to find the prime number decompositions of integers. We implement this idea exploiting polychromatic optical interference in a multi-path interferometer and factor seven digit numbers. We give an estimate for the largest number that can be factored by this scheme.To find the factors of a large integer number N is a problem of exponential complexity. Indeed, the security of codes relies on this fact but is endangered by Shor's algorithm [1], which employs entanglement between quantum systems [2]. In the present paper we report the optical realization of a new algorithm for factoring numbers, which takes advantage of interference only.A naive way of approaching the problem of factorization consists of dividing N by integers ℓ, starting from ℓ = 3 until N/ℓ is an integer. In the worst case this procedure requires √ N divisions before one would find a factor. On a digital computer division of large numbers is a rather costly process. However, in many physical phenomena division occurs in a rather natural way. For example, a wave of wavelength λ, propagating over a distance L, acquires a phase φ = 2πL/λ and therefore probes the ratio L/λ. In the optical domain λ is measured in nanometers (nm), that is λ = ℓ nm. When we also express the path length L in units of nm, that is L = N nm, the phase φ = 2πN/ℓ is sensitive to the ratio N/ℓ. For factors of N , φ is an integer multiple of 2π. Otherwise φ is a rational multiple of 2π.In order to enhance the signal associated with a factor relative to the ones corresponding to non-factors, we use interference of waves, which differ in their optical path length by an integer multiple. In this way we take advantage of constructive interference when ℓ is a factor of N , but destructive interference when ℓ is not a factor of N . The cancellation of terms is most effective when the individual optical paths increase in a nonlinear way. In this case, the intensity of the interfering waves is determined by the absolute value squared of a truncated exponential sum [3]. A polychromatic source of light, which contains several wavelengths λ = ℓ nm, allows us to test several trial factors simultaneously, taking advantage of the properties of truncated exponential sums [4] with continuous arguments.Our method is motivated by recent work on factorization using truncated exponential sums [5], which has been realized in several experiments [6]. However, it differs from the past realizations in three important points:(i) the division of N by the trial factors ℓ is not precalculated [7], but it is performed by the experiment it- self, (ii) all the trial factors are tested simultaneously in a single experiment, and (iii) a scaling property inherent in the recorded interferogram of a single number allows us to obtain the factors of several numbers.The optical setup used to implement th...

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We demonstrate a novel second-order spatial interference effect between two indistinguishable pairs of disjoint optical paths from a single chaotic source. Beside providing a deeper understanding of the physics of multiphoton interference and coherence, the effect enables retrieving information on both the spatial structure and the relative position of two distant doublepinhole masks, in the absence of first order coherence. We also demonstrate the exploitation of the phenomenon for simulating quantum logic gates, including a controlled-NOT gate operation.

Correlation Plenoptic Imaging (CPI) is a novel imaging technique, that exploits the correlations between the intensity fluctuations of light to perform the typical tasks of plenoptic imaging (namely, refocusing out-of-focus parts of the scene, extending the depth of field, and performing 3D reconstruction), without entailing a loss of spatial resolution. Here, we consider two different CPI schemes based on chaotic light, both employing ghost imaging: the first one to image the object, the second one to image the focusing element. We characterize their noise properties in terms of the signalto-noise ratio (SNR) and compare their performances. We find that the SNR can be significantly higher and easier to control in the second CPI scheme, involving standard imaging of the object; under adequate conditions, this scheme enables reducing by one order of magnitude the number of frames for achieving the same SNR.