The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of
n cloned photons, resulting in a
1/n reduction of the diffraction limit. In the limit of
nth-order collective state are shown to propagate on particle like trajectories, reproducing the 0th-order ray-optics picture. These results are discussed in terms of the linearity of quantum mechanics and Heisenberg’s space–momentum uncertainty principle. Finally, a general concept of coherence based on photon density is developed that in first order is equivalent to the conventional wave-based picture.
A promising approach for direct detection of quantum entanglement in solid state materials is probing with entangled particles. X-rays are ideal for characterization of many materials but have not been generated as entangled pairs in useful numbers, even at the most powerful synchrotron sources. This proposal describes how x-rays in fully entangled N00N states (with N = 2) can be produced by applying Hong-Ou-Mandel interferometry at an x-ray synchrotron source. The technical requirements for indistinguishable photons, x-ray beamsplitters, and interferometer design are met with existing resources. The entangled x-ray biphotons can be filtered from the unavoidable single photon background by simple diffraction from an analyzer crystal, resulting in intense beams suitable for characterizing quantum materials.
We present the results from a non-diffracting optical beam experiment that utilizes extremely low power levels (single-photon). The non-diffracting beam has a Bessel spatial distribution and demonstrates interesting single-photon self-interference effects such as spatial confinement. The single-photon Bessel beam is generated using two means: (1) an attenuated HeNe laser beam that statistically provides a single photon flux through the optical system, and (2) one photon from a pair of quantum entangled twin photons produced by spontaneous parametric down-conversion (SPDC) in a Beta Barium Borate (BBO) crystal pumped by a UV laser. The entangled nature of the single-photon Bessel beam using the SPDC source provides a high level of discrimination from ambient background noise photons that would otherwise severely limit the utility of such a technique to dark enclosures. The HeNe laser on the other hand, provides higher photon count rates and is more convenient to work with in contrast to the entangled photon source. We verify that a single-photon Bessel beam reforms itself beyond a circular obscuration by measuring the transmitted spatial distribution.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
A method for generating two-photon wave packets (biphotons) with a small width of the spatial intensity correlation function (correlation radius) has been considered. We propose to prepare a two-photon wave packet inhomogeneously broadened in the angle by means of spontaneous parametric down-conversion in an inhomogeneous crystal. Such biphotons are not diffraction limited; i.e., their correlation radius is much larger than the inverse width of the spatial spectrum. However, the correlation radius of such a biphoton decreases as the biphoton propagates in free space, and transverse compression occurs at a certain distance from the crystal; i.e., the biphoton becomes diffraction limited.
A comprehensive review is given of investigations, both theoretical and experimental, that demonstrate interference to take place even when the fields involved are produced by independent sources. The physical situation is rather different for coherent and incoherent fields. In the first case, where the sources are lasers, a conventional interference pattern can be observed which is adequately described by the classical theory. In the present paper, special attention is paid to the question of whether this interference persists when the two laser beams become strongly attenuated. In the second case, the sources are individual atoms excited by a pumping mechanism, that emit spontaneously and, hence, independently from each other. In those circumstances, no interference pattern can show up. However, it becomes evident from both the classical and the quantum-mechanical theory that interference effects can still be established by observing intensity correlations rather than the intensity itself. This point is discussed in greater detail. The pioneering experiments of Forrester, Gudmundson and Johnson, and Brown and Twiss are reviewed in this context. Especially interesting from the theoretical point of view is the case of two emitting atoms, since then the classical and the quantum-mechanical description differ significantly, the quantum theory predicting the intensity correlations to be distinctly stronger than those following from classical considerations. This specific quantum-mechanical feature is shown to be intimately connected with the corpuscular aspect of light.