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 cloned photons, resulting in a reduction of the diffraction limit. In the limit of th-order coherence, all photons within the th-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. A general concept of coherence based on photon density is developed that in first order is equivalent to the conventional wave-based picture.
© 2019 Optical Society of America
Few problems in physics have received more attention than the complex nature of light [1,2]. Our modern understanding of light was triggered by Planck’s empirical blackbody radiation formula [3,4]. Einstein can be credited with many significant contributions, especially his explanation of the photoelectric effect  and the derivation of Planck’s blackbody law by use of the concept of light quanta [6,7]. It is this concept that also defines what we refer to as “light” in this paper, owing to its detection through the photoelectric effect. The founders of quantum theory furthermore left us with key insights, such as Heisenberg’s uncertainty principle , which underlies the important diffraction limit and Schrödinger’s puzzling concept of entanglement , famously criticized by Einstein as “spooky action at a distance” . The existence of entanglement, arguably the most nonclassical manifestation of the quantum formalism, has by now been convincingly proven and underlies the evolving field of quantum information science [11–14].
1.1. From Single- to Multi-Photon Interference
The name photon was coined by Lewis in 1926  and was formally defined a year later through Dirac’s quantization of the electromagnetic field [16,17], where the photon corresponds to an elementary excitation of a single mode of the quantized field. Dirac’s field quantization into monochromatic modes, also referred to as number or Fock states, defined by their wavevector and polarization , was the basis for his famous and bold statement that photons interfere only with themselves .
In his statement, which has led to much discussion and confusion over the years , he loosely used the word “photon” instead of “photon probability amplitude.” The statement was meant to explain the formation of Young-type diffraction patterns in the detector plane through one-photon counts at a time , with the final pattern forming with time as the mosaic of the number of single-photon counts deposited at different positions (see Section 2.6). The crucial point Dirac tried to convey is that the conventional diffraction pattern, today referred to as the first-order pattern, consists of a binary assembly of 0 and 1 photon counts, corresponding to the destructive and constructive interference of a single photon (wavefunction = field) with itself.
This picture of diffraction pattern formation was famously discussed for electrons by Feynman et al.  and has been beautifully verified experimentally [22–24] as discussed in Section 2.6. Feynman clearly recognized and most eloquently articulated the essence of quantum interference in his space–time formulation of quantum mechanics [25–27]. What interferes in quantum mechanics are not particles or photons but their complex space–time probability amplitudes between birth and destruction (detection). We shall introduce and extensively utilize the important concept of probability amplitudes in this paper for the discussion of both one-photon and two-photon diffraction.
For independent single photons, the complex probability amplitudes assemble the same way as complex fields through superposition (addition) to a single complex amplitude. In both cases, its absolute value squared gives the real detection probability at a given point. Remarkably, Feynman showed  that the conventional wave description of light emerges from quantum electrodynamics (QED), the fundamental theory of light, in first order. This resolves the schizophrenic “wave–particle duality” through the equivalence of the classical wave concept and the QED first-order probability amplitude concept associated with independent photons.
The interference of single-photon probability amplitudes considered by Dirac and Feynman was extended in the mid-1960s by Glauber [28–30], who developed the modern quantum theory of coherence based on the photon detection process. His concept of higher order degrees of coherence may also be understood in terms of a perturbation series description of QED, where first order coherence is described by single photon probability amplitudes, second order by two photon amplitudes, and so on . Glauber also showed how the time-independent Heisenberg picture used in Dirac’s field quantization, resulting in monochromatic number or Fock states that are delocalized in time, may be phrased in the time-dependent Schrödinger picture, where through superposition of Fock states one may create polychromatic wavepackets , sometimes referred to as wavefunctions . This naturally allows the definition of a coherence time in terms of the energy width of the wavepacket. Interference between wavepackets exists when the wavepackets overlap, i.e., if there is no possibility of “welcher Weg” (which path) distinction from the creation (birth) to the destruction (detection) events [33–35].
The two most fundamental quantum optics experiments that proved the indivisibility of single photons and the destructive interference of the probability amplitudes of two photons were carried out in the mid-1980s by Grangier, Roger, and Aspect  and Hong, Ou, and Mandel , respectively. They are discussed in Section 4. The two-photon case naturally extends Dirac’s first order to Glauber’s second-order quantum description. In first order, 0 and 1 counts mean destructive or constructive interference of fields, or in Dirac’s language, 1 count represents the interference of a single-photon probability amplitude with itself. From a detection point of view, a single photon is an excitation of the electromagnetic field whose photon-number statistics has a mean value of one photon and a variance of zero. In second order, two photons can only lead to 0, 1, and 2 counts, and they correspond respectively to destructive, no, and constructive interference, respectively, of two-photon probability amplitudes at a single point in the detector plane.
Today, the term “photon–photon interference” or “multi-photon interference” has assumed common usage in the quantum optics community [33–35], as reflected by the title of this paper. Like Dirac’s statement, it needs to be understood as an abbreviation for “interference of two or more photon probability amplitudes.” The shorter expression simply reflects the fact that in th-order coherence theory, the diffraction pattern is defined through the coincident detection of photons. Similar to the single-photon case, the th-order diffraction patterns are formed with time as the mosaic of coincident counts.
In the optical and x-ray range, photons are detected through the created photoelectric charge. In the process the photon gets destroyed. This is mathematically treated through the action of a quantum mechanical operator. If photons are created simultaneously, their destruction is treated to occur simultaneously or in coincidence through the actions of destruction operators. This corresponds to the instantaneous collapse of the multi-photon wavefunction. Remarkably, in quantum mechanics, superposition and interference of photon wavepackets can exist even when their paths to two or more detector positions cannot be distinguished. In this case, interference is defined as the coincident count rate between detectors at different positions, and it is a purely quantum mechanical phenomenon. For this reason, the treatment of multi-photon detection through the photoelectric effect is a central part of Glauber’s theory.
Glauber’s and Feynman’s approaches explain the seminal Hanbury Brown–Twiss (HBT) experiment [38–42] in terms of the interference of two-photon probability amplitudes [2,27] rather than classical statistical intensity fluctuations [43,44]. Since intensity fluctuations naturally arise in QED through the photon-based graininess of light, the HBT result must naturally follow from the more fundamental quantum picture, and we shall derive it here using the probability amplitude concept. It just happens that for a chaotic source, the second-order quantum and corresponding statistical optics formalisms give the same answer. However, the quantum formulation includes experimental observations when two or more photons are involved that cannot be explained classically [33–35,45,46]. Here we will specifically review the evolution of diffraction from single independent photons to two correlated photons and highlight how the conventional diffraction limit can be overcome by two-photon interference and further reduced by multi-photon interference.
1.2. Objectives of the Paper
The present paper elucidates the concepts of coherence and diffraction by comparing the well-established wave or statistical optics [43,44] predictions to those of quantum optics [30,34,35,46]. In first-order coherence theory, the two approaches are shown to yield identical results, first pointed out by Mandel and Wolf in 1965 . In second order, the classical field-based and quantum photon-based approaches are known to differ . This is here directly shown for the cases of cloned and entangled biphoton diffraction. The concepts are elucidated by calculations of the conventional and biphoton diffraction patterns for the fundamental case of a circular source with a uniform intensity distribution, which historically has been the basis for the definition of the conventional diffraction limit .
The quantum approach, based on the interference of photon probability amplitudes, also reveals a remarkable duality between partial coherence in first and partial entanglement in second order, which was first recognized by Saleh et al.  and experimentally verified by the same group a year later . While in first order the concept of chaoticity is the lack of first-order coherence, in second order the concept of entanglement is the lack of second-order coherence. Thus, the first-order concept of partial coherence with its chaotic and coherent limits has the second-order analogue of partial entanglement with its limits of entangled and cloned biphotons.
We furthermore explore the evolution of the far-field diffraction pattern with increasing order of coherence, following Glauber’s theory. The area of the far-field diffraction pattern, which determines the diffraction limit, is shown to decrease as with a concomitant increase of the central intensity by due to power conservation. For very large , diffraction effects disappear and the pattern in the far field exhibits the same area as the source. This reveals the remarkable equivalence of the 0th-order ray-optics and the th-order quantum-optics picture, where photons within the collective state may be envisioned to travel on particle-like trajectories. It supports a general view of coherence based on photon density, which in first order is equivalent to the conventional wave-based picture.
Another goal of the present paper is to introduce quantum optics and higher order coherence concepts to the broader x-ray community, which only recently has ventured into the exploration of effects outside the conventional first-order description of light. A broader understanding of the concepts of coherence is necessitated by the advent of x-ray free electron lasers (XFELs) [52–56]. It is envisioned that the exploration of higher order coherence in the x-ray regime may reveal new aspects of the complex nature of light, and it may well become a new paradigm for obtaining unprecedented structural resolution.
1.3. Formal Structure of the Paper
We start in Section 2 by reviewing the conventional concepts of “light,” defined in this paper as extending from the visible to the x-ray range, and the characteristic time scales associated with light and its detection. We then review the conventional concepts of coherence, photon degeneracy parameter, source brightness, and their link to the existence of a diffraction pattern. We point out that the conventional concepts provide only a first-order description and correspond to the detection of independent, noninteracting photons. We close the section by defining the diffraction limit as the central paradigm of optics and show that it follows from both the classical wave-based and quantum mechanical photon-based formalisms.
Section 3 presents an overview of the evolution of the study of two-photon phenomena. We then discuss the controlled generation of two photons, so-called biphotons, whose correlation in space–time is created by their simultaneous birth through a nonlinear process. We review biphoton creation by spontaneous parametric downconversion (entangled biphotons) and stimulated resonance scattering (cloned biphotons). We point out that stimulated emission and the cloned biphoton concept may also be viewed classically as a parametric amplification process, while the concept of entangled biphotons is truly quantum mechanical.
In Section 4 we review fundamental one- and two-photon experiments and the associated one- and two-photon detection schemes. In particular, we use Feynman’s two-photon probability amplitude concept to explain the HBT experiment. We then outline seminal experiments with entangled and cloned biphotons that reveal some of the unique interference and diffraction features relative to conventional experiments with independent photons.
In Section 5 we formally introduce the concept of first-order coherence in the frameworks of statistical and quantum optics. We review Zernike’s powerful statistical optics theorem for the propagation of coherence, and derive it quantum mechanically by use of one-photon probability amplitudes. This formally proves the equivalence of the field and photon descriptions of coherence in first order. We then apply Zernike’s theorem to the pedagogical case of a Schell model source, and derive the Fraunhofer diffraction formula for a coherent source and the van Cittert–Zernike formula for the coherent part of the intensity distribution for a chaotic source. The chaotic and coherent diffraction patterns are directly compared through the concept of a first-order coherence matrix. The equivalence of the wave and particle pictures is then established through the introduction of a single-photon probability amplitude or wavefunction. Partial coherence, i.e., the evolution between the chaotic and coherent limits, is elucidated through the example of a circular Schell model source.
Second-order coherence is introduced in Section 6. We start by formally extending the description of first-order coherence to second order and discussing their link. We then discuss the essence of two-photon probability amplitudes and how they are formed from one-photon amplitudes using Feynman’s rules. The general second-order propagation law for two correlated photons (biphotons) is derived and compared to Zernike’s first-order law for independent photons. The difference between the statistical and quantum optics formulations is emphasized. We illustrate the quantum approach based on the interference of two-photon probability amplitudes through the example of two photons that are correlated by their simultaneous birth and emitted either as an indistinguishable cloned pair or an entangled pair. We show how the concept of partial coherence in first order has its analogue in partial entanglement in second order. Partial entanglement is shown to reduce to the limits of entangled and cloned (i.e., second-order coherent) biphotons. The evolution of the two-photon diffraction pattern for the general case of partial entanglement is discussed for the case of a circular Schell model source, leading to the entangled and cloned limits.
In Section 7 we discuss how multi-photon interference overcomes the conventional diffraction limit. We start with a simple picture of entangled and cloned biphotons that reflects their diffraction behavior. We then generalize the case of cloned biphotons to cloned photons, and show that the one-photon diffraction limit is lowered by . In the collective th-order coherent case, the individual photons are shown to propagate on particle-like parallel trajectories. The reduced diffraction limit is discussed in terms of Heisenberg’s uncertainty principle. Finally, we present a remarkably simple picture of lateral coherence based on the creation and propagation of spatial photon density. The paper is summarized in Section 8.
2. Conventional Concepts of Light and Diffraction
What we refer to as “light” in this paper is distinguished from longer wavelength electromagnetic (EM) radiation through its detection process. It covers the range illustrated in Fig. 1(a).
The frequency of light is sufficiently high that direct detection of any induced charge oscillation, utilized for example in the microwave regime, is too fast to be observable. Light detection is accomplished through the photoelectric effect, and this process must be taken into account in the description of the measured intensity of the light signal. We start with a brief overview of typical parameters that characterize light sources.
2.1. Time Scales of Light and Photon Density of Sources
In classical electromagnetism, EM radiation consists of self-propagating fields that have separated from an accelerated charge. Such “acceleration fields” are distinguished from the Coulomb or “velocity fields” that remain attached to charges moving with a constant velocity . EM radiation needs to be defined at least over one wavelength , and one may associate the volume with the minimum birth volume of photons. Figures 1(a) and 1(b) summarize some key properties of the light spectrum emitted by different sources, such as the wavelength range, the quality of a source defined by the photon density it produces, called the photon degeneracy parameter, given here as the number of photons generated in a birth volume of [59,61], and the fundamental oscillation time of the EM field, .
The second time scale is set by the (longitudinal) coherence time , illustrated in Figs. 1(c) and 1(d) for two different processes of light generation. In Fig. 1(c) we illustrate the case of characteristic atomic emission due to electronic decays from an upper to a lower electronic state. The coherence time is defined by the 1/e lifetime width of the electronic transition according to , where is the FWHM energy-width of the emitted Lorentzian line shape. After a time , the probability of finding the atom in the upper state has reduced to 1/e or to of the start value. In Fig. 1(d) the coherence time of thermal light sources is illustrated. At a given temperature, an oscillating electron emits an EM wave until its emission is interrupted during a collision, as schematically indicated by vertical lines in Fig. 1(d). If the collision is elastic, the emission will resume with the same frequency but a different phase until another collision occurs. In general, the coherence time is then given by the average time interval between collisions. The existence of collision intervals adds frequency Fourier components that cause a finite energy bandwidth of Gaussian shape.
The range of coherence times is shown in blue in Fig. 1(e). For optical sources, the shown range extends from conventional sources to optical lasers with coherence times as long as . The dark blue region indicates the range for biphotons created by parametric downconversion discussed in this paper. For x-ray sources, the light blue range covers the inherent source coherence times up to the time scales reached by insertion of a monochromator shown in dark blue.
The temporal properties of modern synchrotron radiation (SR) sources are illustrated in Fig. 2. The electron bunches in a storage ring that generate SR have a typical length of about 3 cm or and consist of electrons. The temporal x-ray pulse length is about the same as the electron bunch length. The individual electrons radiate coherent wavetrains over a time as for a bending magnet source and as for an undulator source owing to their longer path through about 100 undulator periods. In both cases, however, the wavetrains emitted by individual electrons are not correlated, so that the total SR pulses are globally chaotic [62,63].
The temporal nature of x-ray pulses in a self-amplified spontaneous emission (SASE) XFEL is schematically illustrated in Fig. 3. In this case a linear accelerator is used to compress the electron bunches to micrometer lengths, and the x-ray pulse length is shortened to femtoseconds. Most importantly, the interaction of the emitted radiation with the electrons leads to a self-ordering effect in the electron bunch. The ordered micro-regions within a bunch emit coherently so that the x-ray pulse contains coherent intensity spikes of width. As a rule of thumb, the number of micro-regions scales with the total bunch length, and for a 10 fs pulse there are about 10 spikes, as shown in Fig. 3(b). There is no correlation between the spikes and the individual single pulses are globally partially coherent. The structure and intensity of the spikes varies from pulse to pulse, with an average over many pulses assuming a Gaussian distribution, as shown by the dashed black curve. The temporal average renders XFEL sources, despite their name, to be statistically chaotic . It has recently been shown that external seeding of the FEL process, which is presently only possible in the XUV range, indeed generates pulses that statistically behave like a laser . By use of ultrashort electron bunches of low charge or passing a bunch through a narrow aperture [52,65], it is possible to produce individual near transform-limited pulses of about 1 fs duration as shown in Fig. 3(c). Note, however, that the statistical average of such pulses is still chaotic.
2.2. Detector Time Scales
The third important time scale in modern quantum optics experiments is defined by the detector. In general, the most relevant of the various time scales associated with the detection process itself and the electronics depends on the experiment. Detector time scales also need to be considered in conjunction with other detector characteristics, such as the detection efficiency, dark count background, linearity of response, and the ability to provide photon energy and photon number resolution.
Photon detectors involve the conversion of a photon into an electrical signal. The timing characteristics of a detector are determined not only by the photon/electron conversion process but also by the associated electronics. Today, the two most-used detectors in quantum optics are semiconducting avalanche photodiodes (APDs) and superconducting single-photon detectors (SSPDs) [66–69].
APDs are typically made of Si, Ge, or InGaAs and offer low dark count rates, high detection efficiencies, and high count rates. Most APDs can operate at room temperature and, due to their low expense and common availability, have increasingly replaced photomultiplier tubes and microchannel plates . In SSPDs, the detector element (typically a NbN nanowire) has to be kept well below the superconducting critical temperature, adding cost and experimental complexity at the benefit of the fastest response times (see below). Since the energy gap across which electrons are excited is about 2 to 3 orders of magnitude lower in a superconductor than in a semiconductor, for the same incident photon energy the avalanche electron charge is considerable larger in SSPDs than APDs.
For the one- and two-photon diffraction experiments discussed in this paper, the most important detector time scales may be summarized as follows.
- • The dead or recovery time, , is the interval after a detection event during which the detector is unable to reliably register a new incoming pulse. The main contribution to may be either the detector element itself or the counting electronics. The dead time limits the maximum count rate of the detector, which is of the order of 10 MHz for APDs and 1 GHz for SSPDs .
- • The jitter time, , corresponds to the average interval between the absorption of a photon and the generation of an electrical output pulse. The timing jitter of the detector sets the minimum time interval for gating or time-stamping and typically sets the minimum clock time of a photon counting experiment. Typical values for are about 400 ps for APDs and only about 30 ps for SSPDs , as shown in green in Fig. 1(e).
- • Position sensitive detectors are typically charge-coupled devices (CCDs) that accumulate the locally deposited charge in individual pixels. The frame read-out time, , is the time required to completely read out all pixels. For x-ray imaging applications, for example, is a few milliseconds [70–72].
Typically, APDs and SSPDs give only a binary response of 0 and 1 so that a multiphoton pulse triggers the same output signal as a single photon. This limitation is overcome by photon-number resolving detectors, which can distinguish between the arrival of single and two photons within a time window of about a nanosecond [67,73,74]. They have been less used for quantum optics experiments relative to APDs, but we will encounter an interesting application of such a detector in Section 4.4 below.
According to Fig. 1(e), the shortest time jitter, , falls into the range of optical coherence times, but it is orders of magnitude longer than x-ray coherence times. For SR and XFEL sources one may, however, utilize their pulsed nature to correlate signals observed at different positions within the duration of each pulse, with a temporal average obtained by summing over many pulses, as pioneered by Vartanyants and collaborators [53,62,63].
2.3. Fields, Intensities, and Photons
In the conventional framework based on the wave–particle duality, light is born locally as a photon, propagates as a wavefield, and is detected in a local detector pixel as a photon that generates charge through the photoelectric effect. From a wave point of view, in the detection process, energy equivalent to that of a single photon is extracted locally from the total electromagnetic field. The process is governed by the electric field of the EM wave, and the measured intensity of dimension [energy / (area × time)] is linked to the field strength, number of photons, and their energy through the relation
2.4. Nature of Incoherent Sources
A source or secondary source is called “incoherent” or “chaotic” when the intensity distribution in a distant detector plane shows no structure, i.e., no distinct diffraction pattern. Classically, the mathematical description of a chaotic source is based on the fundamental notion that radiation can propagate away from the generating electric charge only if it is defined at least over the dimension of the mean wavelength [75,76].
In practice, the description of an incoherent source is treated as the limiting case of a partially coherent source whose coherence area shrinks to the size of the wavelength. The description of this limit is facilitated by Wolf’s concept of quasi-homogeneity [61,77]. In quasi-homogeneous sources, the intensity distribution varies slowly while the coherence may vary fast from region to region. In practice one assumes that such a source has a well-defined macroscopic intensity distribution while it contains uncorrelated microscopic coherence regions. Hence, from a global perspective, the source is spatially chaotic.
The concept of quasi-homogeneity is typically used together with the assumption that the correlation (coherence) function of all microscopic regions has the same functional form. For example, a blackbody source with a broad bandwidth spectrum actually contains small regions with a sinc-function field–field correlation between two points separated by [43,78,79]. The best description of a quasi-monochromatic source that emits a characteristic line is through an Airy correlation function of the form [75,78,79].
A SR or XFEL source, on the other hand, contains microscopic Gaussian electron distributions, , that emit x-ray pulses with no discernible internal coherence structure for a storage ring source but coherent “spikes” for an XFEL source, as shown in Figs. 2 and 3. The radiation is also emitted preferentially in the forward direction due to relativistic effects, as addressed later in Section 7.6.
Mathematically, the description of lateral incoherence is facilitated by treating the small-area limit of any peaked field correlation function as a cylindrically symmetric 2D Dirac -function. We adopt the notation that the dimensionality of the -function is defined by the dimensionality of its argument, which in our case is a 2D vector . Denoting the microscopic coherence area of any peaked function as , we may write the following explicit expressions for 2D Gaussian, flat-top, and Airy distribution functions:
2.5. Diffraction of a Sample Illuminated by an Incoherent Source
The existence of diffraction when a sample is inserted into a beam emitted by an incoherent source is a consequence of coherence propagation, as illustrated in Fig. 4. While the longitudinal coherence time remains unchanged upon propagation in vacuum, the lateral micro-coherence areas in the source expand upon propagation, and their overlap defines the size of the coherence areas at the sample position shown as a red circle in Fig. 4(a). For a crystalline sample, the circle typically covers several lattice unit cells. The periodicity of the lattice then leads to the same diffraction patterns for micro-regions that are individually illuminated coherently, without the requirement of longer range x-ray coherence between micro-regions .
When the sample is non-periodic, as shown in Fig. 4(b), the interference patterns from small domains in the sample are no longer identical and therefore the diffraction pattern is blurred. The conventional small angle scattering pattern corresponds to the incoherent superposition of the coherent patterns from different domains. The intensity distribution consists of a ring-like pattern located at a momentum transfer , where is the statistical correlation length of the domains.
By moving the sample away from the source, as shown in Fig. 4(c), one increases the effective coherence area at the cost of coherent intensity. If the entire sample is coherently illuminated, the scattered intensity recorded by an imaging detector is “speckled.” The speckle pattern contains the complete information on the real space structure within the illuminated area. For a circular aperture, the central bright spot has a diameter determined by the diffraction limit, as discussed in Section 2.9. In practice, the inversion of the reciprocal space diffraction pattern into real space requires the use of holographic reference beams [81,82], the exploitation of the strong dependence of the optical constants near a resonance [83,84], or computer-based phase retrieval algorithms .
2.6. Diffraction and the Wave–Particle Duality
To account for interference and diffraction, we still use the magical Huygens–Fresnel principle that each point may be viewed as the origin of a spherical wave of uniform amplitude and an outward traveling phase front, referred to as a wavelet. The concept of spherical waves around all possible source points, e.g., atoms in a sample, has been a great aid since it produces the experimentally observed diffraction pattern and it underlies the first Born approximation .
Remarkably, one can use the Huygens–Fresnel principle of spherical waves emanating from points even when calculating the diffraction pattern of an illuminated hole where there are no real atoms that can scatter! In this case, one may invoke Babinet’s principle as an explanation, which states that the far-field diffraction pattern is the same for an opaque object surrounded by “nothing” and for a hole (“nothing”) replica of the object surrounded by opaque matter. From a fundamental point of view, both the Huygens–Fresnel and Babinet principles are unsatisfactory, since the complete intensity distribution in the detector plane is simply calculated by forming the absolute value squared of the field interference pattern without the need to consider photons, which after all are the building blocks of QED, the complete theory of light.
The first demonstration that the diffraction pattern accumulates through the addition of single photon counts was carried out in 1909 by Taylor , who recorded the diffraction pattern of the tip of a needle by using an extremely weak gas flame source that was so weak that it required three months to record the pattern. In 1961, Jönsson  repeated Young’s double-slit experiments with electrons. It appears that this experiment was not known to Feynman, who prominently discussed it in his 1962 lectures as a “Gedanken” experiment, published later in Volume III of his lecture series . Jönsson’s experiment showed that the use of electrons gave the same pattern as photons. Improvements of the experiment clearly revealed how the diffraction pattern assembles with an increasing number of electrons [23,24]. It has also been repeated with increasing number of photons , neutrons , and large organic molecules  with similar results. As shown in Fig. 5, the diffraction pattern builds up probabilistically through single “particle” processes.
These experiments have provided clear evidence that the diffraction pattern emerges as a point-by-point mosaic of single particle detector clicks. Young’s double-slit experiment therefore proceeds by single-photon processes whose probabilistic repetition leads to the buildup of the diffraction pattern as shown in Fig. 5. The yellow curve is the intensity distribution in the limit of recording a large number of counts. The experimental results support Dirac’s famous statement: “…each photon interferes only with itself. Interference between different photons never occurs” . It is likely that Dirac knew about Taylor’s experiment.
In 1909 Einstein derived a fluctuation formula for Planck’s blackbody radiation that contained a wave and a particle term and suggested an inherent wave–particle duality in the nature of light that combines two seemingly different concepts . By using such a semiclassical combination of the wave and particle concepts, one may naively envision three steps leading to a diffraction pattern, as indicated at the bottom of Fig. 5. A photon is created by an electronic transition in an atom. It then expands into space with the speed of light and acts as a wave that may be diffracted by a sample put in its path. The shape of the diffraction pattern is determined by wave interference. However, a single photon is detected at only a single point in the detector plane, where an energy packet is locally taken from the total wave field. The wave collapses into a photon again. Local photon to electron conversion in a detector pixel then leads to a “bright spot.” While one can envision the build-up of the diffraction pattern by such a naive semiclassical model, it has the flavor of science fiction where “phase shifting” between particle and wave is allowed. Below we will discuss an entirely quantum mechanical derivation of diffraction.
2.7. Diffraction and Quantum Electrodynamics
Today, quantum electrodynamics (QED) is regarded as the fundamental theory of light and its interactions with electronic matter. It can account for the most sophisticated experimental results, such as the Lamb shift of the energy states of hydrogen and the anomalous magnetic moment of the electron. In principle, it must therefore be able to account for all conventional laws of optics. In practice, the optical laws governing absorption, reflection, refraction, and diffraction are typically derived from a classical wave picture. While this is partly due to the historical development of optics, the general theory of QED also appears too complicated and intimidating. Many of the QED complications may be avoided, however, by Feynman’s space–time formulation of quantum mechanics . In lowest order perturbation theory, the space–time formulation leads to simple rules regarding the addition and multiplication of single-photon probability amplitudes , which are used in Feynman’s 1985 book  to derive the conventional laws of optics. In the same book, Feynman also reveals the amazing fact that within first order, individual photons and electrons are both described in QED by the same single particle probability amplitudes, which explains the experimental observations discussed in the previous section.
Feynman’s treatment is remarkable in that it exposes the very origin of the schizophrenic wave–particle duality, which suggests that light acts sometimes like a wave and sometimes like a particle. In QED, this ambiguity does not exist since light is fundamentally granular or quantized and composed of photon building blocks. The remarkably simple solution of the wave–particle dilemma is that the complex QED theory may be approximated in first order by a wave theory with its own governing principles. Examples of these principles are the magical Huygens–Fresnel principle and the concept of spherical waves. Although light is fundamentally composed of photons, the wave picture with its own associated physical principles or postulates may then simply be used as a convenient approximation.
We will use Feynman’s quantum approach to diffraction for different cases below. In Section 2.9 we will derive the fundamental diffraction limit by associating independent probability amplitudes with each photon. In this case, the individual complex probability amplitudes for each alternative space–time path between the birth and destruction of photons are added at a given detector point. The detection probability at a given point is then calculated as the absolute value squared of all superposed (added) probability amplitudes. This quantum approach is obviously analogous to the superposition of complex fields in wave optics, with the intensity calculated as the absolute value squared of all superposed fields.
The equivalence of the photon and wave calculations in first-order QED breaks down in second and higher order, which is a key topic discussed in Section 6. Historically, this became apparent through Glauber’s treatment of higher order coherence, which was largely motivated by the HBT experiment discussed in Section 4.1. In second order, one accounts for a quantum correlation between two photons that are born simultaneously in space–time. In this case, single-photon probability amplitudes are first multiplied to obtain a two-photon probability amplitude, reflecting a concomitant propagation and simultaneous (coincident) detection. As for the first-order single-photon case, the correlated (multiplied) two-photon amplitudes for alternativetwo-photon space–time paths are then added. The absolute value squared of all summed two-photon amplitudes constitutes the two-photon detection probability. We will show in Section 4.2 that this quantitatively accounts for the HBT effect.
Today, diffraction calculations based on particle concepts are sometimes based on alternative formulations of quantum mechanics by Madelung , de Broglie , and Bohm [94–96], referred to as the pilot-wave or quantum trajectory method [97–100]. A key ingredient of the theory is Bohmian mechanics [95,96], which through a deterministic or “causal” interpretation of quantum mechanics can provide information about particle trajectories. The particles perform quantum mechanical fluctuations around the trajectories that give rise to Heisenberg’s position–momentum uncertainty. The fluctuations may be viewed as arising from the interaction of the particles with the stochastic zero-point field and also lead to a finite coherence length. The trajectories of single photons in a double-slit experiment have recently been reconstructed from experiment .
It has been pointed out, however [102,103], that the quantum trajectory calculations for the double-slit experiment  disagree with experimental data recorded with different types of particles and photons, which all give the same diffraction pattern. In the quantum trajectory calculations, the intensity of the central peak is too large and of the side peaks too small. This suggests that the quantum trajectory method overestimates the particle-like behavior. This may be explained, as discussed in Section 7.5, by higher order QED contributions to the quantum trajectory method.
2.8. Link of Diffraction Pattern, Degeneracy Parameter, Brightness, and Coherence
Figure 1(b) reveals the radical change in the photon degeneracy parameter in the optical regime with the advent of the laser and in the x-ray regime from x-ray tubes to XFELs. Before lasers and XFELs, the degeneracy parameter was less than unity, so all early diffraction experiments with conventional optical sources and even the best SR sources were carried out one photon at a time. This naturally supports the assembly of diffraction patterns illustrated in Fig. 5. It raises the question, however, whether the diffraction pattern changes with the increase of the photon degeneracy parameter.
Experimentally, one finds that Young’s experiment carried out with the slits illuminated by a distant monochromatic conventional optical source or a laser give the same pattern. Similarly, x-ray tubes, synchrotron, and XFEL radiation with degeneracy parameters differing by a factor of all give the same Bragg diffraction patterns! It therefore does not matter whether the photons arrive in the detector plane one after the other or all together. In both cases, their energy is deposited probabilistically to create the final pattern, as shown in Fig. 5. This agrees with Dirac’s statement that photons do not interfere with each other, and we can therefore state that the conventional diffraction pattern does not depend on the photon degeneracy parameter.
The physics behind this statement, however, is non-trivial and we need to explore it in more detail since the title of our paper indicates that “multi-photon interference” exists and may indeed change the diffraction pattern. Dirac’s quantization of the EM field [16,17] resulted in so-called number or Fock states . Photons in these states occupy the same wavevector and polarization modes, and the photon degeneracy parameter is nothing but the number of photons in the same mode and . They can be envisioned to be born in a source volume of order . As the name implies, number states have a well-defined number and corresponding field amplitude, but their quantum phase is random [45,104]. While for a small number a single observation may still produce interference [105,106], it increasingly washes out when one averages over many observations, corresponding to forming a statistical ensemble average or a quantum mechanical expectation value. We shall see below that the interference washout does not exist in other forms of light called coherent states, as discussed in conjunction with cloned biphotons in Section 7.2.
The photon degeneracy parameter is directly linked with the concept of spectral brightness or brilliance of a source, which was introduced for the quantitative description of the quality of SR sources by Kim , following an earlier paper that focused on x-ray coherence . The brightness is defined as [109–112]7.4). The photon degeneracy parameter listed in Fig. 1(b) for various kinds of sources corresponds to . The photons are assumed to be birthed in the box volume , defined by the lateral area and the longitudinal length , where is the oscillation time of the EM wave.
The so-defined brightness also defines the coherence of the source. At first sight this connection is puzzling since coherence is usually defined by assuming that light behaves as waves and then demanding that the waves are in-phase. In contrast, the brightness expression does not contain any phase information, and on the right side of Eq. (4) is expressed simply as the density of photons. This indicates that coherence may be phrased either in a wave or a photon picture. In the latter, the coherence of a source is simply determined by the number of photons it produces in the coherence volume . We will see later in Section 7.5 that the coherence concept based on photon density or energy density of the EM field is the most general description of coherence.
It turns out that the concepts of coherence, brightness, degeneracy parameter, and diffraction limit constitute only a first-order description of the properties of light. “First order” is most elegantly defined by the quantum theory of light, where it means non-interacting and non-interfering photons. In quantum optics, the classical field and its complex conjugate are replaced by field operators and (see Section 5), which act as raising or creation and lowering or destruction operators. In quantum mechanics, photons are created by these operators out of the zero-point quantum vacuum and destroyed back into it. In first order, treated by Dirac, the detection process consists of the destruction of a single photon.
Glauber, in a series of papers [28–31,113], went a step further and considered the possibility that two or more photons were simultaneously destroyed, leading to a substructure of light where th order of coherence means indistinguishable photons or clones. In second order, one considers the simultaneous creation of two photons and their simultaneous destruction at a later time. At a given point in the detector plane, there are only three possible outcomes within a time interval defined by the two-photon creation process in the source, corresponding to the arrival of 0, 1, or 2 photons. If we detect zero photons (0 counts), the two photons must arrive elsewhere; if we detect one photon (1 count), the other photon must arrive elsewhere; and the last possibility is that we indeed detect both photons (2 counts). The three cases correspond to destructive interference, no interference, and constructive interference of the two-photon probability amplitudes at a given point in the detector plane, respectively. More generally, in th order, photons are simultaneously born and simultaneous destroyed at the same or different points in the detector plane.
Before going into the details of multi-photon phenomena, we need to briefly review a key concept of conventional optics, the definition of the conventional diffraction limit. It arises in a quantum picture by treating all photons as being independent.
2.9. Central Paradigm of Conventional Optics: the Diffraction Limit
The central dilemma of wave optics or first-order diffraction theory is the finite width of the central Airy disk in a distant detector plane produced by a uniformly illuminated circular aperture, as illustrated and defined in Fig. 6. This problem has haunted astronomers for centuries, since the finite size of the aperture of a telescope necessarily limits the resolution due to diffraction. Through the years, opticians have tried to reduce the width of the detector plane Airy disk through modification of the intensity distribution across the aperture, e.g., through non-uniform coatings. It was found that because of power conservation, the narrowing of the width of the central Airy disk always comes at the expense of an increase of the intensity of the outer rings , which reduces the sharpness of the total image in a telescope.
The diffraction pattern of a uniformly illuminated circular aperture of area in a detector plane at a large distance is the shown Airy diffraction pattern. It is typically derived by use of the Huygens–Fresnel principle, where each point in the circular aperture emits a spherical wave . The diffracted intensity is obtained by considering the interference of the different waves at a given detector point. The diffracted intensity is found to change with the distance from the optical axis as (see Section 5.6.a)6(b) in blue that the intensity of the pattern in the detector plane is reduced if a uniform film is inserted into the aperture. The reduction is given by the absorption in the film, described by a factor , where is the linear absorption coefficient of the material and its thickness. In the figure we have chosen . The figure also indicates that the flat-top function (red) with a radius of contains the same power as the total Airy pattern (including all outer rings). The solid emission angle defines the coherence cone in geometrical ray optics.
2.10. Photon Probability Amplitude Formulation of the Diffraction Limit
To show the complete equivalence of the classical wave-based and the quantum mechanical photon-based derivation of the conventional diffraction pattern, we use Feynman’s space–time formulation of quantum mechanics [25–27,115] and calculate the diffraction limit for the example of a circular source of uniform average intensity. This is done as illustrated in Fig. 7 by use of the concept of single-photon space–time probability amplitudes , which are associated with the birth of a photon at position and the destruction of a photon at position in a distant detector plane.
The single-photon amplitude for the path length may be written as116] to account for its attenuation with distance. It then has the form of a spherical wave in classical optics or the Green’s function for a point source. However, in this paper we consider only the typical case where the source–detector separation is large, so that one may approximate the general propagator as . Note that this so-called Fraunhofer approximation cannot be made for the rapidly changing exponential phase term, which leads to interference [see Eq. (9) below]. Keeping the path-independent attenuation term in mind, we will here simply use one-photon probability amplitudes of the form of Eq. (6). The unification of our quantum formulation and the classical spherical wave approach underlying the Huygens–Fresnel principle is given in Section 5.4.
The birth is defined through a phase constant that depends on the place of birth , and the propagation phase depends on the propagation path length . The fact that the source and propagation amplitudes in Eq. (6) are multiplied follows from Feynman’s rule that the amplitudes of consecutive events are multiplied, corresponding to the addition of associated phases . Owing to the constant speed of light, the propagation may also be expressed by the travel time , hence the term “space–time.” Each one-photon probability amplitude is normalized to reflect a single photon according to .
Different source points and a given detection point are linked by alternative space–time paths. The final one-photon amplitude , which determines the detection or destruction probability at point according to , is therefore determined by alternative places of birth and propagation paths according to27]. The approach is completely analogous to the classical wave superposition (addition) concept used in the Huygens–Fresnel principle. In practice, the addition is replaced by an integration over all source points within the effective source area according to
At large average source–detector separation , the so-called Fraunhofer diffraction limit, we can express the path length in Fig. 7 by assuming , the so-called paraxial approximation, and obtain8) contains in an exponential, we need to consider all terms in Eq. (9). However, the integral significantly simplifies if we can assume , corresponding to the assumption of a small source. We can then neglect the term in Eq. (9) and obtain
As discussed in Section 2.4, for a chaotic source there are only microscopic coherence regions of average size , so that the integration over the source in Eq. (10) can be conveniently expressed by a Dirac -function. The cases of a chaotic and coherent source then can be distinguished by averaging over many emission events, corresponding to the expectation values:5). The equivalence of the classical wave-field and quantum probability-amplitude concepts shown here for the limiting cases of chaoticity and first-order coherence also holds for the general case of a partially coherent source as derived in Section 5.4.
The diffraction limit has been overcome in the past in the imaging of biological specimens by utilizing the nonlinear response of fluorophores emitters. In 2014, the Nobel Prize in Chemistry was awarded to Betzig, Moerner, and Hell for “the development of super-resolved fluorescence microscopy” . These developments are examples of the general principle developed in this paper that the diffraction limit can be overcome by use of nonlinear electronic response in the object whose diffraction pattern or real space image is of interest. As the name implies, nonlinear interactions with the sample induce photon correlation effects that appear only in second and higher order.
In this paper we will specifically address the fundamental case illustrated in Fig. 6, which defines the diffraction limit as a result of first-order coherence, and demonstrate how the conventional diffraction limit, defined by the area of the central Airy peak, can be overcome with a concomitant disappearance of the weak outer rings. As an introduction of the formal treatment, we shall review the case where instead of two photons being born independently at two points, their birth process is correlated in space–time.
3. Overview of Two-Photon Interference and Diffraction
While conventional diffraction is based on individual non-interacting photons, there are interesting cases, where two photons are born simultaneously in space–time and maintain some kind of correlation upon propagation to the detector, where they arrive simultaneously at the same point or two different points. In contrast to the one-photon case in which the photon probabilistically deposits its energy at a single point in the detector plane, in two-photon diffraction, the two photons may deposit their energy simultaneously at one or two points. The correlation of the two-photon destruction events contains cases that have no classical analogue. In this section we first give a short overview of the historical development of two-photon experiments and the controlled generation of biphoton probability amplitudes or wavepackets.
3.1. Historical Overview of Two-Photon Experiments
The two-photon (boson) correlation case is of similar fundamental importance as the two-electron (fermion) correlation case, with the two cases being distinguished by the symmetry of the total wavefunction according to the symmetrization postulate . The case of two electrons, first studied for the two-electron He atom and molecule, famously laid the foundation for Pauli’s exclusion principle . Similarly, the case of two photons has allowed fundamental tests of the nature of light and quantum theory. The two-photon case also exhibits the most dramatic departures from the classical field-based description of light.
Schrödinger already recognized in 1935 that two electrons (fermions) or photons (bosons) may form states that he referred to as verschränkt or entangled . Entanglement can exist in different physical properties, such as position, momentum, spin, energy, and polarization. For example, spatial entanglement between two particles described by one-particle wavefunctions and exists if the total two-particle wavefunction can be written only in the form , but not in the position-factored form .
The existence of entanglement was famously related to the question of the completeness of quantum mechanics by Einstein, Podolsky, and Rosen (EPR) . The first investigations of the concept of entanglement and the EPR paradox formulated in terms of Bell’s inequalities  were carried out by using two photons created in atomic cascade emission by Clauser and co-workers [119–121]. Many studies of entanglement have since followed, as reviewed in Refs. [13,14,33–35,46,122]. They convincingly demonstrated the existence of entanglement, with the latest study showing spatial entanglement over ground–satellite distances of up to 1200 km .
Two-photon physics was experimentally pioneered in the late 1950s by Hanbury Brown and Twiss, who extended a radio-astronomy method of correlating the signals in two antennas to the optical range where the correlation was measured between two photodetectors [38,39,41,42]. A nice review of the experiment from a classical point of view has been given by Wolf , and we will discuss the experiment in Section 4.1 and give its quantum derivation in Section 4.2.
3.1a. Spontaneous Parametric Downconversion—Entangled Biphotons
Two-photon experiments were revolutionized by the discovery of spontaneous parametric downconversion (SPDC) in 1967 [124–126]. In SPDC, discussed in more detail below, a nonlinear process in a crystal without inversion symmetry is used to simultaneously generate two lower energy photons through fission of an incident photon [127,128]. For example, an incident photon of energy may be split into two photons of energy . Quantum correlations between SPDC photon pairs were first observed in 1970 by Burnham and Weinberg . In the 1970s and ’80s, SPDC was utilized by Mandel et al. [130–134] to investigate how the interference of two photons changes as a function of their temporal separation. These studies, reviewed by Ou , culminated in the study by Hong, Ou, and Mandel  that demonstrated the destructive interference of two temporally overlapping photon probability amplitudes, as discussed in Section 4.4.
From a spatial interference or diffraction point of view, two-photon experiments with SPDC photons were pioneered by Klyshko, who in 1982 gave the first description of their transverse correlation , and in 1988 suggested the concept of coincidence imaging and coined the term “biphotons” . Coincidence imaging was experimentally demonstrated by Shih and collaborators in 1995 , and the related technique of “ghost imaging” was also introduced in the same year . It was later shown to be possible with chaotic sources, as well [138,139].
In ghost imaging, photons are separated by a beam splitter and propagate to different detectors, where they are detected in coincidence. Remarkably, the image of a sample located in one beam is recorded by scanning the detector in the other beam that does not contain a sample. Ghost “images” can be recorded in either real or reciprocal space [140,141], and the technique has recently been extended to x rays [142,143]. The potential of ghost imaging in the x-ray regime lies in the fact that sample damage may be avoidable by subjecting the sample to the weaker part of an entangled beam while recording its image with a stronger one that does not traverse the sample .
In practice, experiments with entangled photon pairs, here referred to as entangled biphotons, are typically performed by coincidence correlation of the signals of two single photon detectors, although detectors exist today that can discriminate between the arrival of single and two photons [73,74], as discussed in Section 4.4. The coincidence method assures that the wavetrains of the two photons overlap in time. As illustrated in Section 4.5, the so-measured diffraction pattern of entangled biphotons is characteristic of the wavelength of the incident photon that is fractured, and not of the individual wavelength of the two downconverted photons. This was first demonstrated by Fonseca et al. in 1999  and led to the proposal of “quantum lithography” below the conventional diffraction limit [146,147]. The diffraction behavior has been attributed to the entangled biphotons forming a collective new “quanton” [148,149] state of wavelength , that interferes with itself. Owing to the close link of multi-photon states to Bose condensates or “de Broglie matter waves” , the entangled biphoton has also been called a “photonic de Broglie wave” .
3.1b. Stimulated Emission—Cloned Biphotons
Stimulated emission, conjectured by Einstein in 1916 [6,7], constitutes another nonlinear process where two photons are born correlated in space–time. It may also be explained classically as the complement to absorption, with the two processes respectively describing the decrease and increase of energy of a driven system . In modern optics, stimulated emission is a consequence of a nonlinear interaction, which may be pictured semiclassically as a parametric amplification process [153,154] and treated in terms of a third-order nonlinear susceptibility . The stimulation process has been studied in the time domain by superimposing a coherent laser beam onto one of the beams generated by SPDC . With increasing laser beam intensity, the effect of stimulation could then be directly observed by an increase of interference fringe visibility of the SPDC beam.
From a quantum perspective, stimulation leads to the simultaneous creation of two indistinguishable photons or cloned biphotons in an atomic decay [see Fig. 1(c)]. The maximum achievable cloning was quantitatively determined by a beautiful experiment carried out by Lamas-Linares et al.  in 2002. It proved the so-called “no cloning” theorem [133,158–160], which states that no quantum operation exists that can duplicate perfectly an arbitrary quantum state. The optimal cloning fidelity is 5/6 = 0.833, and the experiment yielded a value of .
The cloning limitation arises from consideration of emission by an atom in an excited state, whose decay probability scales with the well-known factor , where 1 represents the relative spontaneous decay probability induced by the zero-point field and expresses the relative stimulated decay probability driven by the presence of photons in the same mode. If only a single incident photon, , is available to stimulate the excited state decay, there is equal probability of spontaneous and stimulated emission. While the incident polarization is preserved in stimulated emission, it is arbitrary in spontaneous emission. Hence the presence of spontaneous emission negates perfect cloning of the incident photon. More generally, the essence of the no-cloning theorem is founded in the linearity of quantum theory.
The nature of cloned biphotons has been difficult to probe since they are indistinguishable in space–time. The question arises whether they behave differently than two photons in the same wavevector and polarization mode, i.e., the photons emitted on average by a source with degeneracy parameter 2? We have seen earlier that the diffraction pattern does not depend on the degeneracy parameter of the source, and therefore a source of degeneracy parameter 2 will produce the same pattern as a source of lower degeneracy parameter. Hence, the answer to our question depends on whether the diffraction pattern of a cloned biphoton source is different.
We will show below that the diffraction pattern indeed changes because photons in the same mode are only first-order coherent, while cloned biphotons are second-order coherent. The cloned biphoton diffraction pattern has the same spatial periodicity as the single photon one, reflecting the common wavelength , but the central intensity of the cloned biphoton image is increased by a factor of 2 and the image area is reduced by a factor of 2 below the conventional diffraction limit .
3.2. Creation and Properties of Biphotons
Biphotons are created simultaneously in space–time through a nonlinear fusion or fission process, which induces a “correlation” between them that remains after birth and manifests itself in the diffraction pattern. The creation of secondary biphoton sources for the entangled and cloned cases is illustrated in Fig. 8.
3.2a. Entangled Biphotons
In the SPDC process, illustrated in Fig. 8(a), entangled biphotons are produced by interaction of a pump laser with a non-centrosymmetric crystal. In the optical regime, atomic oscillators are driven into the nonlinear regime, and the downconverted photons created in the nonlinear process usually have a broad bandwidth corresponding to coherence times of the order of 10 fs that can be lengthened by use of energy filters. For an in-depth discussion of entangled biphoton physics, the reader is referred to the books by Ou [33,35] and Shih .
In the optical regime, the frequency and wavevector directions of the biphotons are determined by the nonlinear crystal response and momentum conservation. One distinguishes two types of SPDC processes. In Type I, the photons have the same polarization, which is opposite to that of the pump beam, and are typically emitted into different directions, although collinear emission can also be achieved by appropriately cut crystals. In Type II, the photons have opposite polarizations and, in practice, are most often created with parallel wavevectors so that they are emitted collinearly [33,128], as shown in Fig. 8(a). In all cases the two photons are mutually incoherent. The cross section of SPDC is linear in the incident intensity and may be described by the second-order nonlinear susceptibility [127,155]. Although the biphoton creation probability is very low (of order ), one may nevertheless eliminate the high-intensity pump beam by a polarization or energy filter, so that one can study the correlations between the two entangled photons emitted by the secondary source.
SPDC has also been demonstrated in the x-ray regime. Suggested by Freund and Levine in 1970 , the process was first demonstrated by Eisenberger and McCall in 1971 . This experiment, performed with a 17 keV Mo x-ray tube source and about two weeks of accumulation of coincidence events, constituted the observation of the first nonlinear x-ray effect, about 40 years before the advent of XFELs! Later experiments used brighter 8 keV x rays from a rotating Cu anode  and SR [164–167]. Following Yoda et al. , today diamond single crystals are typically used as the nonlinear medium.
Two-color experiments have also been performed using entangled x rays of very different energy  with a primary beam of 11.107 keV that produced 11.007 keV signal and 100 eV idler photons, and with 760 and 840 nm optical photons .
3.2b. Cloned Biphotons Produced in Stimulated X-Ray Resonant Scattering
Cloned biphotons are naturally created in stimulated emission, as discussed above. Spontaneous and stimulated emission may occur independent of the excitation process. For example, in the x-ray regime a core hole may be created by electron or ion beam impact on a sample. The hole may then be filled by decays of electrons from outer shells. They may be driven either by the Coulomb field of the hole, resulting in Auger decay, or by the EM zero-point fluctuations, resulting in spontaneous emission of a photon. In the presence of a photon of the right energy, the decay may be stimulated by the photon, which clones itself in the process, constituting stimulated emission.
In the soft x-ray region, one often uses resonant excitations where the incident photon energy is tuned to an atomic resonance and a core electron is excited to a rather localized valence state (orbital). The resonant process maximizes the excitation (absorption) cross section and provides atomic, chemical , and magnetic  specificity. In addition to polarization dependent x-ray absorption measurements, which provide bonding and magnetic information, one can also obtain structural information through diffraction experiments. They are carried out as illustrated in Fig. 4(c) and yield information on the nanoscale domain structure [81,85]. In diffraction experiments, one employs position sensitive CCD detectors to record the entire pattern, and with XFELs the diffraction pattern may even be obtained with a single shot .
Before the advent of XFELs, the incident beam, even at the most advanced SR facilities, had a degeneracy parameter below 1, as shown in Fig. 1(b). The probability of stimulating decays with the incident beam itself was therefore negligible. With XFELs, however, the large degeneracy parameter may readily lead to stimulated decays. If the excitation and emission process are both driven by the incident beam one speaks of impulsive stimulation. The resonant x-ray scattering process then combines the first absorption step with the second decay step, and with increasing incident intensity one may observe how the conventional diffraction pattern changes with the onset of stimulation [171,172]. This will be discussed in Section 4.5 after we have introduced the concepts of two-photon detection.
Stimulated resonant scattering is a second-order nonlinear process. The atom is modeled as a two-level electronic system consisting of a core and a valence state, separated for soft x rays by an energy of the order of 1 keV. At lower intensities the process may still be described in second-order Kramers–Heisenberg–Dirac perturbation theory [17,173], where any population changes of the atomic ground and excited states, averaged over all illuminated atoms, are neglected. At higher intensities, population changes in the coherently illuminated number of atoms need to be included by use of the optical Bloch equations [45,174]. In elastic resonant scattering, the temporal degree of second-order coherence is related to the relative upper state population of the collective atomic two-level system by [45,48,175]
Equilibrium is consistently reached only if the coherence time of the incident x rays is longer than the total spontaneous decay time of the upper state. In the soft x-ray region, is the total (Auger plus radiative) decay energy width [172,174], corresponding to excited state decay times . The strength and coherence time of the field ensure that it controls the electronic transitions for a sufficient time to equalize the upper and lower state populations with the process becoming quasi-stationary. Absorption is then compensated by stimulated emission and the sample becomes transparent . Stimulated resonant scattering scales with the square of the incident intensity and may be described semiclassically by a third-order nonlinear susceptibility .
The production of cloned biphotons in impulsive stimulated resonant forward scattering does not violate the famous “no-cloning theorem” [133,158–160] because the system evolves into an equilibrium state where stimulated emission completely dominates. The situation is similar to transportation and amplification of telecom signals in optical fibers, where stimulated emission dominates over spontaneous emission. The no-cloning theorem then does not apply . The inapplicability of the theorem in these cases may also be explained by the violation of the basic premise of an arbitrary input state. Since all incident photons are in the same mode, prior information about a state to be cloned is available and it can be cloned with higher fidelity .
Stimulated elastic resonant scattering entails the interaction of two independent photons in the same mode (first-order coherent) with an atom. It requires only a single beam of sufficient degeneracy parameter tuned to an atomic resonance to control the stimulated decay. In the atomic interaction, two photons are converted into indistinguishable clones, as illustrated in Fig. 8(b). This process needs to be distinguished from other stimulated processes. Examples are amplified spontaneous emission (ASE) utilized in a conventional laser, where the process is started by a spontaneous atomic decay that then self-amplifies. This process has also been observed in the x-ray range [178–181]. Another process is the impulsive stimulation of the inelastic channel by the incident beam, referred to as stimulated x-ray Raman scattering . It has been observed by use of broadband incident radiation from a SASE XFEL . The broad incident spectrum contained photons above the ionization energy to create core holes and below the ionization energy to stimulate inelastic decays into the created core holes.
4. Fundamental One- and Two-Photon Experiments
For many quantum optics experiments one would like the detection time uncertainty to be the same as the coherence time given by the length of an uninterrupted wavetrain (wavepacket). For flat-top distributions, the coherence time is given by , which may be lengthened through a reduction of the wavelength bandwidth or energy bandwidth by a filter or monochromator inserted into the beam. As shown in Fig. 1(e), one may achieve in the optical regime, but this is impossible at x-ray energies where .
Some of the most fundamental experiments in quantum optics have dealt with the spatial or temporal correlations between two photons. In such experiments, the existence of photon–photon correlations is typically revealed by detecting the two-photon arrival probability at the same position as a function of the difference in their arrival time , or detecting the probability of two photons arriving at the same time as a function of the difference in their position . The two types of measurements yield information of their temporal and spatial correlations, respectively. Below we focus on spatial correlation, interference, and diffraction.
4.1. Hanbury Brown–Twiss Experiment
Together with the development of the maser and laser, it was the Hanbury Brown–Twiss (HBT) experiment conducted with chaotic sources in the mid1950s that played an important role in the development of statistical optics as outlined in a recent book by Wolf  and especially quantum optics, as recalled by Glauber in his Nobel lecture .
HBT performed two experiments, illustrated in Fig. 9, one for the star Sirius [39,42] and another with a laboratory mercury arc lamp [38,41]. In both cases, the thermal sources were quasi-monochromatic, with the photons emitted by chaotic processes. The coincident arrival of two photons was detected in coincidence at two spatial points as a function of their separation. The correlated arrival of two photons was detected with one-photon detectors whose response time was much longer than the coherence time of the photons, determined by their spectral bandwidth. Since the detector response time significantly exceeded the coherence time, the identification of the true coincidence signal required excellent signal-to-noise data as reviewed by Wolf .
For their experiment with light emitted by a star, HBT used the schematic arrangement shown in Fig. 9(a). The measurement allowed the determination of the angular size of Sirius. The experiment with a mercury arc lamp used the scheme in Fig. 9(b), where the two single-photon detectors, joined by a coincidence circuit, are placed at the exit ports of a beam splitter. For convenience, only one of the detectors was scanned perpendicular to its incident beam to determine the relative spatial correlation of the two photons. The two-photon detection schemes introduced by HBT have provided the foundation for many modern quantum optics experiments, as will become apparent below.
The results of the HBT experiment are sketched in Fig. 9(c), and the two contributions to the coincidence signal are indicated. The blue curve represents true correlations between photons at small detector distances, referred to as “photon bunching.” Its shape depends on the macroscopic shape of the source. It is taken here to be a Gaussian, corresponding to a source with a Gaussian intensity distribution. For a chaotic circular flat-top source, the blue curve would have the shape of an Airy function of the form  [see Eq. (24)]. For chaotic sources, the random emission of photons also leads to a background, shown in red, that arises from accidental coincident arrival of two photons that were born at different points and different times in a completely uncorrelated way, but their path lengths to the detectors just happen to compensate for their different birth times. This contribution does not depend on the detector separation and is constant.
As indicated by the label of the ordinate in Fig. 9(c), the measured normalized coincidence rate reflects the degree of second-order coherence, which as discussed below is given by Eq. (66). Under the condition of spatio-temporal separability, it is given by
For the measurement of the temporal correlation , the light is made spatially coherent, , by a small aperture. However, the temporal coherence measurement is more difficult because it requires a time resolution of the order of the coherence time . Such a measurement was only accomplished about 10 years after the original HBT spatial correlation measurement by Morgan and Mandel , using the blue line of a mercury discharge lamp with a 5 ns temporal coherence time.
In the x-ray region, a temporal correlation measurement is much more difficult due to the much shorter coherence time, which even with a monochromator typically does not exceed (see Fig. 1). Therefore, only the spatial coherence has been measured so far [186,187]. For these measurements, one can overcome the slow detector response time by gating the detector to the arrival time of the x-ray pulses, so that the counting window is effectively determined by the shorter x-ray pulse length. Since the pulse repetition time is typically 2 ns for synchrotron sources and for XFELs (see Figs. 2 and 3), the detector may be read out pulse-by-pulse and a statistical average is obtained by averaging over many pulses .
4.2. Quantum Derivation of the HBT Effect
The HBT result may be accounted for by intensity fluctuations within the framework of statistical optics in conjunction with a semiclassical model for photon detection [44,61,188]. Since intensity fluctuations are nothing but fluctuations in the number of emitted photons, we might as well derive the effect in the more fundamental QED formulation. In QED, the HBT effect originates from the interference of the probability amplitudes associated with two photons. This was first pointed out by Fano  and supported by Glauber’s treatment of two-photon coherence based on their detection [28–30].
The original HBT effect was observed with sources of finite extent that obviously contain a large number of independent emitters. The shape of the bunching peak in Fig. 9(c) is then defined by a Fourier transform of the shape of the source, as will become apparent later. Theoretical studies of the effect, particularly of its quantum mechanical origin, have typically used a simple two-point source model, first considered by Fano  and also treated in the book by Scully and Zubairy . In the following section we shall first explore this model system using Feynman’s perturbation treatment of QED and then treat the more complicated realistic source case for the example of a uniform circular source.
4.2a. Fano’s Simple Four-Atom Model
For independent single photons, probability amplitudes for alternative paths are added, similar to the superposition of fields, before the detection probability is calculated as the absolute value squared. For two or more photons, the multi-photon probability amplitudes are constructed from those of single photons by multiplication and addition . This extension consists of remarkably simple rules that we shall illustrate in the following through treatment of the HBT effect.
To illustrate the quantum mechanical origin of the HBT effect, we start with the four atom case shown in Fig. 10 following Fano , where photons are created at two source atoms, separated by a distance and detected by two atoms separated by .
In Fig. 10 we depict one-photon probability amplitudes with the birth and propagation of a photon from source point I to detection point J. We note that the atomic model assumes that only single photons can originate from a given source atom. It therefore does not include the case where biphotons are birthed at the same point or within an indistinguishably small source area. However, as will be shown in Section 6.5, the atomic model remains valid in a more rigorous treatment since all other possible interference processes average to zero. This detailed discussion is postponed until we have properly introduced the concept of second-order coherence in Section 6.1.
According to Feynman’s rules, concomitant two-photon amplitudes, shown in the same color in Fig. 10 are formed by multiplication of one-photon amplitudes. The resulting two-photon amplitudes are then (green) and (orange). Alternatives of the concomitant amplitudes, shown in different colors, are then added to form the total two-photon amplitude , which is given by6). The two-photon detection probability is calculated from the amplitudes according to .18), keeping only the cross terms that lead to interference. This gives the Fraunhofer approximation expression:
When the two detectors are scanned in opposite directions parallel to the line defined by the two source points (), then the dot product in Eq. (20) is simply the product of the absolute values and (see Fig. 10), and we have20) and .
4.2b. Finite Size Sources
For the considered case of two source points of infinitesimal area, the cosine modulation has a constant unit amplitude and does not resemble the peak-shaped blue bunching curve in Fig. 9(c). If the source has a finite size, however, a bunching peak appears since the interference-modulation averages to zero with increasing . For example, the simplest case is a continuous line of source points, so that and lie within an infinitesimally thin slit of length . For such a chaotic line source, we can assume that there are only microscopic coherence regions of length , where photons are born without a relative phase shift, corresponding to a 1D version of the 2D case treated in Section 2.4. If the detectors are scanned so that , then Eq. (21) changes to9(c). Another example is a circular source of area with a flat-top intensity distribution. As discussed in Section 2.4, in this case there are only microscopic coherence areas where the photons can be assumed to be born without a relative phase shift. Taking this into account, the coincidence detection probability is derived as the expectation value: 9(c) with the Airy distribution replaced by a Gaussian. The association of the expression with the Airy function in Eq. (24) follows from Eq. (61) below, and confirms the Siegert relation [Eq. (16)].
4.2c. The Physics behind the HBT Effect
Our quantum mechanical derivation of the HBT effect reveals its origin within the fundamental QED-based photon picture of light. While it can also be accounted for in the framework of statistical optics, there is really no argument that the quantum formalism is more fundamental. It just turns out that for a chaotic source the treatments of second-order correlations give the same answer in both cases.
In Feynman’s space–time probability amplitude formulation of QED, the HBT bunching peak arises in second-order perturbation theory from the interference between alternatives of concomitant two-photon probability amplitudes, formed by products of one-photon amplitudes. The constant background is due to the arrival of independent single photons at the two detectors at the same time. These photons were born independently at different points and different times, but the distances covered from birth to detection just accidentally compensate for their different birth times.
As will be formally shown in Section 6.5, the birth of biphotons at the same point, not considered in the above derivations based on Fig. 10, make no contribution to the HBT effect. This is expected from the fact that in thermal sources the births of cloned biphotons by stimulated emission and entangled biphotons by spontaneous parametric downconversion are considerably less probable than the birth of single photons.
4.3. Single Photons, Detection, and Self-Interference
The quantum theory of light is based on the existence of photons, which is often said to be proven by the photoelectric and Compton effects. However, both effects can be explained by assuming classical EM waves with quantum properties associated only with the atomic and electronic structure of matter in the photoelectron generation process . The experiment that convincingly proved the existence of photons as elementary single units was performed in 1986 by Grangier, Roger, and Aspect  and is illustrated in Fig. 11.
The experiment utilized two photons emitted consecutively in an atomic cascade decay, with the first photon serving as a trigger to mark the arrival of the second photon, emitted about 4.7 ns later. The second photon entered the shown HBT-like detection scheme, where it could choose between two paths through a beam splitter, leading to two single-photon detectors, gated to the arrival of the first trigger photon. The second photon was found to register in only one of the two detectors with zero cross-correlation coincident counts. This confirmed the quantum mechanical prediction of indivisible single photons.
In a wave picture, it takes the constructive interference of two fields, represented by the product , to account for photon creation and destruction (detection). For a single photon, only detector counts 0 and 1 are possible, corresponding to the classical picture of destructive or constructive interference between two fields. From a quantum point of view, a detector count of 1 corresponds to interference of a single photon with itself, as stated by Dirac.
4.4. Observation of Two-Photon Interference
One of the most important experiments in quantum optics was carried out in 1987 by Hong, Ou, and Mandel (HOM)  using entangled biphotons. It is the basis for what we today refer to as two-photon interference. In the following we will discuss two versions of the experiment that illustrate its complementary aspects.
The original HOM experiment, shown schematically in Fig. 12(a), utilized a Type I SPDC source whose two beams of the same polarization were inserted into two input ports of a loss-less beam splitter, and the arrival of the two photons at the two output ports was measured with single-photon detectors as a function of their relative arrival time.
Femtosecond time resolution was achieved through precision movement of the beam splitter to change the relative path length to the two detectors. With decreasing delay between the arrival of the two photons, a “dip” in the coincidence counts was observed, with zero counts or a coincident arrival probability for no delay, as indicated in the figure. The situation encountered in the experiment was independently shown in 1987 to exist by Fearn and Loudon, who treated the effect of a loss-less beam splitter quantum mechanically .
The missing one-photon cross-coincidence counts, labeled , cannot be explained classically. On the right of Fig. 12(a), we show the four different possibilities of how photons, viewed as distinguishable classical particles, can propagate through the beam splitter. To indicate their classical distinguishability, we have colored their respective paths black and red and used a notation to denote their arrival probabilities, where due to photon conservation, and the sum of all probabilities is unity. Two photons arriving at the same output port can happen two ways—either both are reflected with a probability or both are transmitted . Two photons exiting different output ports can also happen two ways—either one photon is reflected and the other is transmitted and vice versa . Hence, from a classical point of view, cross-coincidence counts are expected half of the time, i.e., a total of , in contrast to the measured result.
From a quantum perspective, the situation is different. The quantum theory of a beam splitter is presented in Refs. [33,193–195], and here we only need to utilize the fact that there is no phase shift for the transmitted beam but the reflected beam is phase shifted by . The probability amplitudes of the reflected beams (see right side of Fig. 12) are therefore both changed by a phase factor of . The two-photon probability amplitudes follow the rules established in Section 4.2. The amplitudes of concomitant paths are multiplied, and those of alternative concomitant paths are added. We therefore have the detection probabilities and . One may also say that the result is due to destructive interference of biphoton probability amplitudes at the two detectors caused by their relative phase shift of .
A more complete version of the HOM experiment carried out in 2003 by Di Giuseppe et al. , which reveals the complementarity of the two photons exiting the beam splitter together versus separately, is schematically shown in Fig. 12(b). The real novelty of the experiment was the use of special detectors  that could discriminate one (gray) from two (blue) photon arrivals within the detector response time . The experiment also differed from that of HOM by utilizing collinear biphotons with orthogonal polarizations created in a Type II SPDC process, following an earlier scheme introduced by Shih and Sergienko .
The collinear biphotons were sent into the same beam-splitter input port, and the combination of a non-polarizing beam splitter and two 45° polarizers in each detector arm reproduced the HOM dip. The two polarizers behind the beam splitter served to establish polarization indistinguishability and also added to the beam-splitter phase shift between reflected and transmitted photons, so that the total phase shift between the wavepackets arriving at the two one-photon coincidence detectors was again , as in the HOM experiment. The coherence time of the biphotons was not increased by use of energy filters, but accidental coincidences were avoided by using a low-power pump beam so that the time between biphoton arrivals exceeded the coincidence detection window of .
The experiment thus allowed simultaneous measurement of the coincident cross correlation of single-photon arrivals at two exit ports and the complementary arrival of two photons at a single exit port, which was not measured by HOM. The results are schematically indicated in gray and blue in Fig. 12(b). The three arrival probabilities were found to be complementary, as required by the conservation of photons, adding the missing piece of the original HOM experiment.
If two photons (2 counts) are detected at one beam-splitter output, there are zero photons (0 counts) at the other beam-splitter output, as required by conservation of photons. This may be interpreted as constructive interference of the two-photon probability amplitudes at one port and destructive interference at the other. The other possibility of single photons exiting simultaneously at the two exit ports is not observed. It has zero probability because of the opposite phase of their wavepackets at the two exit ports. The implied destructive interference of coincident two probability amplitudes at two positions is a pure quantum phenomenon.
Comparison of the entangled biphoton results in Fig. 12(b) with the single photon results in Fig. 11 reveals a striking similarity. The single-photon counts are simply replaced by two-photon counts. In Dirac’s formulation, one might say that similar to individual photons, entangled biphotons form a collective new quanton state that may interfere only with itself. Then constructive biphoton self-interference corresponds to 2 counts and destructive biphoton self-interference to 0 counts.
4.5. Two-Photon Diffraction Experiments
Two-photon diffraction experiments are based on the detection of two rather than single photons. This is typically accomplished in the optical regime by use of a “two-photon detector” consisting of a beam splitter and two single-photon detectors, as in the experiments discussed above. In principle, a true two-photon detector could also be used as done by Di Giuseppe et al. , but the use of two single-photon APD detectors is more convenient. The HOM experiment, however, reveals that care has to be taken that the single-photon cross-coincidence probability does not vanish. This can always be achieved by suitable choices of beam splitters and phase shifters.
We note that the manipulation of photon beams by beam splitters and polarizers is significantly more difficult in the x-ray regime . The following discussion of optical concepts and experiments is envisioned to be fully extendable in the future to the x-ray regime through the development of a complete “x-ray optics toolbox.”
Two-photon diffraction experiments have typically been carried out by introducing a slit or transmission gratings right after the entangled biphoton source, essentially repeating Young’s experiment with biphotons. In this case the diffraction pattern is one dimensional, which facilitates its interpretation. Below we first illustrate typical results through two examples obtained with Type I and Type II entangled biphotons. We then discuss a soft x-ray experiment with cloned biphotons where the slit was replaced by a circular aperture. This experiment used the geometry of Fig. 6 and thus directly revealed how the diffraction limit can be overcome.
4.5a. Diffraction of Entangled Biphotons with Orthogonal Polarizations
Our first example of biphoton diffraction is shown in Fig. 13. The experiment was performed by D’Angelo et al.  utilizing collinearly Type II biphotons of wavelength with orthogonal polarizations produced by shining a 458 nm Ar ion laser beam onto a (BBO) crystal. The diffracting object was a double slit positioned immediately behind the biphoton source, as shown in Fig. 13(a).
The two-photon diffraction pattern was recorded as a function of the angle from the optical axis by use of a two-photon detector, represented by the entire blue box, consisting of a polarizing beam splitter and two one-photon coincidence detectors. The polarizing beam splitter optimized the coincident one-photon counting rate at the expense of the two-photon counting rate behind each output port. For experimental convenience, the detector was kept fixed in position, and rotation of a mirror was used to sweep the beam across the entrance pinhole of the detector. The experiment was also repeated by use of a conventional first-order coherent source of the same 916 nm wavelength using one-photon detection. This pattern constitutes the conventional double-slit diffraction pattern for reference.
The diffraction patterns for the two cases are shown in Figs. 13(b) and 13(c). Remarkably, the pattern for the biphotons of 916 nm wavelength corresponds to the conventional one-photon pattern of photons with half the wavelength, i.e., 458 nm. It thus beats the conventional diffraction limit by a factor of 2. From a diffraction point of view, the biphotons act as if they were a collective quanton that still has the wavelength of the 458 nm Ar ion pump beam that created them through a fission process. This remarkable observation cannot be explained classically, and arises as a consequence of quantum mechanical entanglement of the two photons.
4.5b. Diffraction of Entangled Biphotons with Identical Polarizations
A similar biphoton diffraction experiment performed with collinear Type I entangled biphotons with the same polarization by Shimizu et al.  is shown in Fig. 14(a). The biphotons of wavelength were created in a potassium niobate () crystal by a Ti:sapphire pump laser (862 nm) that was frequency doubled to . The biphotons were diffracted by a transmission grating placed immediately behind the crystal. The pump beam of opposite polarization was separated from the biphoton beam by a polarizing beam splitter. The experiment used the same scanning scheme as that by D’Angelo et al. and a similar two-photon detector, but a simple rather than polarizing beam splitter provided the required non-vanishing cross-coincidence two-photon detection probability .
For comparison, the one-photon counting rate was recorded concomitantly with the two-photon counting rate. Another experiment was performed after removal of the crystal with the grating directly illuminated by the 862 nm light from the Ti:sapphire laser. In this case, the light incident of the grating was coherent and not entangled in contrast to the biphoton case. The diffraction pattern was then recorded in the same experimental configuration by using either one of the green one-photon detectors or the entire blue-box two-photon detector. The results are shown in Figs. 14(b)–14(e).
The constant one-photon intensity in the detector plane in Fig. 14(b) is due to the fact that the biphoton source is mutually incoherent so the one-photon intensity is constant similar to that of a chaotic source. A similar one-photon measurement was not performed by D’Angelo et al.  but it would have yielded the same result. The diffraction pattern of the entangled biphotons of wavelength 862 nm, shown in Fig. 14(c), is again representative of a conventional pattern of single photons of 431 nm wavelength, i.e., half of the biphoton wavelength. This is revealed by comparison of the one-photon reference pattern in Fig. 14(d) recorded with light from a conventional source.
Comparison of the patterns in Fig. 14(d) recorded by one-photon detection and in Fig. 14(e) by two photon detection contains some altogether new information. Note that the two cases correspond to photon detection at the same point determined by the entrance slit of the detector, as a function of momentum transfer changed by rotation of the mirror. The peaks of both patterns are in the same positions but their relative intensities are different. The solid line fit through the data points reveals that the shape of the two-photon pattern in Fig. 14(e) is simply the square of the shape of the one-photon pattern in Fig. 14(d). Remarkably, the conventional one-photon diffraction pattern of the coherently illuminated grating can be changed to its square by picking out those events where two photons arrive simultaneously at a given position in the detector plane!
Closer inspection also shows that the widths of the peaks in the two-photon pattern in Fig. 14(e) are narrower than those in the one-photon pattern in Fig. 14(d). This is an indication that two-photon diffraction may reduce the diffraction limit. The reason for this behavior will become clear through the formal derivation of the one- and two-photon diffraction patterns below. It is a manifestation of the evolution of coherence in quantum optics from first to second order.
4.5c. Diffraction of Cloned Biphotons
Our last example is a cloned biphoton experiment performed by Wu et al. , illustrated in Fig. 15. The experiment was conducted at the Linac Coherent Light Source (LCLS) XFEL using 50 fs pulses of linearly polarized laterally coherent SASE pulses.
The x rays were transmitted through a monochromator, which defined the photon energy to , matching the resonant energy of the transition from the core to the valence shell. The sample was a Co/Pd multilayer with a nanoscale magnetic worm domain structure as shown. The Co/Pd film plus a circular gold mask of 1.45 μm diameter, deposited directly onto the film on the side of the incident beam, constituted the diffracting object. The diffraction pattern was recorded with a position sensitive CCD detector, with the total accumulated charge per pixel forming the diffraction pattern. No temporal gating of the detector and no electronic coincidence circuit was employed.
As indicated in Fig. 15(a), the diffracted signal consists of a magnetic part, determined by the domains, and a charge part due to the aperture and the homogeneous charge distribution in the film, which according to Fig. 6 only attenuates the intensity of the first-order diffraction pattern. For the employed linear polarization, the magnetic and charge patterns are separable, as indicated in the figure .
When the incident intensity was increased, a change of the diffracted intensity from both the magnetic domains and the circular aperture was observed . This is shown by comparison of the low- and high-intensity patterns shown in Figs. 15(b) and 15(c). The decrease of the diffracted intensity was attributed to intrinsic interference of the two cloned photons in the forward direction [171,174], enhancing the “in-beam” transmitted at the expense of the “out-of-beam” diffracted intensity.
The decrease of the outer Airy rings, in particular, is indicative of a stimulation effect that redirects the outer ring intensity into the central part of the Airy disk . The intensity enhancement in the forward direction, the hallmark of stimulation, could not be observed in the original experiment  due to a beamstop, but it was directly observed in a recent experiment .
Figures 15(d) and 15(e) compare the experimentally observed change in the outer Airy rings with that predicted by the theory of Ref. . It predicts that the conventional one-photon diffraction pattern changes to its square when the stimulation effect saturates. This behavior parallels the change of the one- to two-photon pattern in Figs. 14(d) and 14(e) recorded with the same coherent illumination of the grating but changing one- to two-photon detection. The two observations are indeed related since the roles of two-photon creation and detection are reversed.
While the two-photon pattern in Fig. 14(e) depends on coincidence detection of the two photons at the same point in the detector plane, the cloned biphoton pattern does not require coincidence detection. Rather, the correlation of the cloned biphotons is intrinsically created in the source and is maintained upon propagation to the detector plane where the two photons are naturally deposited at the same time. There is no need of inducing two-photon interference extrinsically by multiplication of the single-photon patterns by a coincidence circuit. Comparison of the two cases demonstrates the conjugate nature of two-photon creation in the source and destruction in the detector plane.
In the following sections, we shall formally treat one- and two-photon diffraction, comparing the predictions based on the wave and particle nature of light. We shall see that the experimental observations discussed above naturally follow from the quantum theory of light, which emphasizes the photon behavior. In contrast, statistical optics, which emphasizes the wave nature, will be seen to fail in some cases.
5. First-Order Coherence
Our starting point for the formal treatment of coherence is the description of light propagation from two points in a source to two others in a distant detector plane, as illustrated in Fig. 16. It was first treated by van Cittert in 1934  and Zernike in 1938 . What we are interested in is how the fields created within the source area propagate to a distant “detector” plane, where the quotation marks indicate that, in practice, the fields cannot be detected directly. In an experiment one would place mirrors at the points and and reflect the fields to a detector that measures the interference intensity.
Classically, we distinguish the two-point correlation of fields and intensities. Field propagation is formally expressed by a correlation function between two fields and is referred to as a “second-order” correlation in statistical optics [43,44,61]. In contrast, quantum optics is built on the notion of indivisible photons, and following Glauber, it is customary to refer to field–field or one-photon probability amplitude correlations (see Section 2.10) as first-order coherence [45,46,193]. In Section 4.2 we have already introduced the correlation between two-photon probability amplitudes corresponding to the classical correlation of two intensities or four fields. This case, which is the main topic of Section 6 below, constitutes second-order coherence in quantum optics and fourth-order coherence in statistical optics. We shall use the quantum optics terminology in this paper.
5.1. Field Quantization and Definition of First-Order Coherence Function
In the quantum theory of light, the electric field is an operator that contains both creation and annihilation operators associated with the positive and negative frequency parts of the field according to 
To briefly outline the quantum formalism, we use the general coordinate label to designate either the coordinates in the source plane or in the detector plane, as shown in Fig. 16. The quantum-optical spatio-temporal first-order correlation function is defined as 28,29]. The normal ordering also assures that no zero-point fluctuations of the EM field contribute to photon detection, since it would require the ordering , where the photon is first created from the zero-point field.
The statistical optics notation is similarly given by replacing and , with the triangular brackets in Eq. (27) now referring to a statistical ensemble average. However, in principle, one needs to distinguish the quantum and statistical optics formulations. In statistical optics, and correspond to complex numbers that convey equivalent information, with defining the detected intensity, given by in SI units. In contrast, the quantum operators and are intimately linked with the concept of the photon and describe the different processes of creation and destruction. We shall see that within first order, the statistical and quantum approaches give the same results but that this is not necessarily true in second order.
In the following we shall assume spatio-temporal separability. The “same time” complex degree of first-order spatial coherence is defined as
For a lateral distribution that is quasi-homogeneous, the lateral coherence depends only on the separation between the two points. Similarly, for a temporal distribution that is quasi-stationary, the temporal coherence depends only on the time difference [43,44].
5.2. First-Order Coherence Propagation and Diffraction: Zernike’s Theorem
We begin our formal treatment of coherence propagation and diffraction with Zernike’s formula for the propagation of the correlation of fields from a source to a distant detector plane . Zernike’s formula describes the general case of partial optical coherence and constitutes the central theorem of “first-order” optical coherence. We note again that our nomenclature of “first-order” coherence is that used in quantum optics [45,46,193], while it is referred to as “second-order” coherence in statistical optics [43,44,61]. The different labeling simply reflects the fact, that in statistical optics, fluctuating “fields” are the basic ingredients of light, while in quantum optics it is the photons, which correspond to an intensity or product of two fields. The importance of Zernike’s theorem is that it describes the complete range of first-order coherence, from complete incoherence or chaotic behavior to complete coherence. We will see that it naturally reduces to the two coherence limits, expressed by the Fraunhofer case of coherence and the van Cittert–Zernike case of incoherence.
Below we shall simply state Zernike’s theorem in its modern form, noting that its statistic optics derivation may be found in the books by Goodman  and Mandel and Wolf [43,61]. The most important ingredient in its conventional derivation is the acceptance of the Huygens–Fresnel principle, which follows from a lowest-order space–time treatment of quantum electrodynamics as discussed in Sections 2.7 and 2.10. For this reason, Zernike’s theorem also holds within the framework of quantum optics, as we shall discuss below in Section 5.4.
Following Fig. 16, we denote the locations of the source and detector planes as and . The two-field lateral correlation for a quasi-monochromatic source of arbitrary first-order lateral coherence then propagates according to the following Fourier transformation, which constitutes Zernike’s theorem:30) we have made the far-field approximation by pulling the position averaged distance out of the integral. This expression is referred to in statistical optics as the “far-field propagation of the mutual intensity” . The triangular brackets indicate that we need to take a statistical average over many “same time” measurements. The phase factor is given by 16.
The real space correlation law [Eq. (30)] can also be written as a Fourier transform between real and reciprocal space variables, and with the substitution we obtain30) is when the two points in the detector plane fall together. The single point intensities then constitute the conventional first-order diffraction pattern. It is composed by interference of the fields at different points in Fig. 16. Since in this case , we obtain from Eq. (30) 30) and the associated diffraction pattern of Eq. (33) describe the general case of partial coherence. The solution for this general case is quite complicated because it covers all field distributions between the chaotic and coherent limits. Only in these limits can an analytical solution be obtained. If the source is completely first-order coherent, one obtains the Fraunhofer diffraction formula, and the other limit of incoherence or chaoticity yields the van Cittert–Zernike theorem [44,202].
5.3. Quantum Approach to Coherence: Single-Photon Wavefunction and
The quantum theory of light pays particular attention to the processes of light creation and destruction, since they are local processes where light is created by an electronic down transition in an atom and is similarly destroyed by an atomic up transition that creates the detected photoelectron. The birth and destruction of photons is mathematically described by the action of operators that either create light from or destroy it back into the zero-point background. Since QED is the fundamental theory of light, Zernike’s theorem [Eq. (30)] must follow from a lowest order perturbation treatment within its framework, as will be shown in Section 5.4.
Following Newton and Wigner , the association of a photon with a wavefunction was long thought to be problematic , since owing to the lack of a photon rest mass, an unambiguous position-based wave function cannot be introduced. More recently, a wavefunction description of single photons has been introduced by Bialynicki-Birula  and Sipe , based on the notion that its modulus squared, , represents the photon’s energy density and the photon energy. It is supported by the fact that photon detection consists of local energy extraction from the total field or collapse of the photon wavefunction, which is also the essence of our previous expression [Eq. (1)].
In Section 2.10 we used Feynman’s concept of a single-photon probability amplitude associated with two points and . To link these probability amplitudes to quantum field operators, let us consider the action of the operators and , defined in Eq. (25), at the same point. They obey the conjugate property . In practice, the photon destruction probability is also its detection probability. Thus if we act with the destruction operator on the photon wavefunction, we demand that it is destroyed, leaving only the zero-point field. The photon destruction process involves taking an atomic electron from an initial state to a final state . The total initial states and final states are products of electronic and photon states according to and . The two parts are separable with the photon part of the transition amplitude being proportional to . The transition probability may therefore be written as45)].
Since the zero-point field contains all modes, it forms a complete set, and using the short form for the closure relation, we can rewrite Eq. (34) by use of to obtain the first-order spatial correlation function at position as30)] into that of Feynman’s space–time probability amplitudes.
5.4. Quantum Derivation of Zernike’s Theorem
For the derivation of Zernike’s theorem we follow Fig. 17, where we illustrate coherence propagation in terms of probability amplitudes associated with the propagation of independent photons emitted at two source points. For the single-photon case illustrated in Fig. 17, there are in principle three ways a photon can be detected. We may either locate single-photon detectors at point X or point Y (not shown). This case was treated in Section 2.10 for single source and detection points (see Fig. 7). We are here interested in the case illustrated in Fig. 17, where a single photon launched at either source points A and B may reach the one-photon detector located at point D by passing either through points X and Y in a “detection plane.” This corresponds to Zernike’s field-based scenario shown in Fig. 16 and expressed by Eq. (30).
In contrast to a field-based description, the quantum formulation always needs to be linked to the detection process. To do so we simply insert the shown mirrors to allow for one-photon detection at point D. Additional differential path-lengths from X and Y to D may be avoided by choosing . The scenarios A to D via X and via Y are alternatives, and they have the same birth phase [see Eq. (6)]. Similarly, the alternatives B to D via X and via Y have the same birth phase . In both cases the two alternative one-photon amplitudes may therefore interfere at the detector position D.
Our goal is to derive for a pseudo-monochromatic source of arbitrary first-order lateral coherence in a quantum formulation. We start with Eq. (35) in the two-point form:17 we write this expression in terms of dimensionless single-photon probability amplitudes (see Sections 2.10 and 4.2) as 45)].
The total one-photon amplitudes at the detector position via X and via Y are related to the amplitudes at X, , and at Y, , according to and , respectively. For , the additional differential phase factors fall out when the one-photon detection probability at point D is calculated according to39) by an integral. In doing so we have to consider that the integration of will also include the case where takes the position of in Fig. 17. To avoid such “double counting,” we choose the cases in which one of the one-photon amplitudes links points A and X and the other B and Y. We then obtain the total one-photon amplitude associated with point X with coordinate by use of the paraxial approximation in Eq. (9), and neglect of the term, as 43), is the quantum theoretical equivalent of Zernike’s field propagation law. To see the equivalence, we rewrite Zernike’s theorem [Eq. (30)] by use of as
In the following, we illustrate the entire transition between the chaotic and coherent limits by use of a particular approximation called the Schell model .
5.5. Schell Model Source
We consider the general case of a partially coherent source with cylindrical symmetry around the optical axis in the convenient form of a Gaussian Schell model [43,44,61]. The Schell model approximation allows us to separate the source distribution into separate intensity and coherence functions. If both functions are Gaussians, the first-order correlation function can be expressed analytically as 
As expected, is entirely determined by the coherence part of Eq. (46), and has the following chaotic and first-order coherent limits:49), is the 2D Dirac -function defined through its dimensionless unit integrated area according to Eq. (3).
For the general case of partial coherence, the integrations in Eqs. (30) and (33) need to be performed numerically for an assumed source coherence function . Below we shall consider the case of a partially coherent source with a symmetrical distribution function around the optical axis. In particular, we consider the case of a planar circular source with a flat-top (constant) intensity profile, owing to its practical importance in establishing the fundamental diffraction limit . For a circular flat-top source, the “intensity” part of Eq. (46) is replaced by46) becomes 61,77]. The approximation of quasi-homogeneity assumes that the intensity distribution is approximately constant across the source, while the coherence function is allowed to change more rapidly.
Before we consider the full coherence function of Eq. (51), we shall evaluate the limiting cases of complete coherence and chaoticity described by Eqs. (48) and (49). This yields the famous Fraunhofer and van Cittert–Zernike diffraction formulas, respectively.
5.6. Van Cittert–Zernike and Fraunhofer Diffraction Formulas
For a source that emits its radiation symmetrically about its optical axis, the geometry in Fig. 16 can be simplified, since all directions from the optical axis in the detector plane are equivalent. We can therefore choose two positions in the detector plane and along a line, with their position vectors forming angles and with the position vectors and in the source plane, respectively, as shown in Fig. 18.
The correlation of the fields at the points and in the detector plane are then given by the first-order correlation function [Eq. (30)] in terms of a Fourier transform integration over the source coordinates and . For a circular source with a flat-top intensity distribution given by Eq. (50), we can simply change our integration in Eq. (30) up to the source radius , and by use of Eq. (51) the detector plane correlation may be written in terms of the degree of first-order coherence of the source. In cylindrical coordinates we obtain47) with the coherent and chaotic limits given by Eqs. (48) and (49).
5.6a. Coherent Source: Fraunhofer Formula
In the coherent limit we have , and the integrations in Eq. (52) can be performed separately to yield202], already encountered in Eq. (5). The degree of first-order coherence is given by . The correlation function in the detector plane is seen to factor, except for the phase factor , which still depends on the coordinate difference in the detector plane according to Eq. (31). The expectation value is determined by averaging over many measurements.
The corresponding diffraction pattern is obtained by setting , which eliminates the phase factor. The one-photon intensity is related to the correlation function according to53) the square of the so-called Fraunhofer diffraction formula for a first-order coherent source : 19. In practice, one may produce a uniform coherent source indirectly as a secondary source by illuminating the circular aperture by a chaotic source that is sufficiently far away. Then all path lengths from the chaotic source to points within the plane of the aperture can be made to be smaller than the wavelength.
In Fig. 19 we also show in black a flat-top distribution that contains the same power as the Airy distribution shown in green. It is convenient to define the effective area of any peaked symmetric distribution as202].
5.6b. Chaotic Case: Van Cittert–Zernike Formula200,201]. In contrast to the coherent source Fraunhofer expression [Eq. (53)], it does not factor into functions of and but depends on the separation between two points in the detector plane, as shown in Fig. 18.
The measurement of the correlation function is illustrated in Fig. 20(a), where the shown symmetrical mirror arrangement about the optical axis is a simplified version of a Michelson interferometer.
It can be shown that44]. This area corresponds to the maximum resonant atomic interaction cross section with emission of a spherical wave, derived by Breit and Wigner [86,207]. The Breit–Wigner cross section describes the case where the atomic absorption and scattering cross sections become the same, and its value remarkably depends only on the wavelength and is independent of all other atomic properties!
The green curve represents the total intensity distribution, measured as for the case of a coherent source by scanning a single detector as a function of its separation from the optical axis or simply using a position sensitive CCD pixel detector (shaded green). We would obtain the same result if we eliminated one of the mirrors. The intensity distribution shown in green is given by20(b).
The dashed pink curve in Fig. 20(b) represents the coherent fraction of the intensity distribution, which is determined by the VCZ theorem as the difference of the coordinates of the two points corresponding to the mirror positions in Fig. 20(a). The coincident interference intensity is given by43,44,61] from two points and separated by .
If we define as the distance from the optical axis as in the Fraunhofer case, we obtain the red curve. It has a width below the diffraction limit, which is represented by the thin gray curve taken from Fig. 19.
The detector plane intensity distributions for a coherent and chaotic circular flat-top source given by Eqs. (55) and (60) differ by the value of the central on-axis intensity. It is lower for an incoherent source by a factor , which is simply the ratio of the coherence areas of the source.
The change of with , when normalized to its value at , is a direct measure of the degree of first-order coherence given by208].
5.7. First-Order Coherence Matrix
For a circular source, the cylindrical symmetry about the optical axis allows a particularly simple choice of the coordinates and in the general expression for the mutual intensity [Eq. (30)]. Owing to symmetry, the two points in the detector plane may be chosen along any line through the optical axis since all such lines are equivalent, as illustrated in Fig. 21.
Following Fig. 20, we can conveniently identify the two points and in Fig. 21 with different mirror positions so that the mutual intensity is due to interference of two fields reflected by the two mirrors onto a detector. As shown in Fig. 21, the mutual intensity for different position pairs and along a 1D line through the optical axis may be expressed as a 2D matrix.
The coherence matrix approach allows us to directly compare the detector plane intensity distributions for the extreme cases of a coherent and chaotic circular flat-top source, as shown in Fig. 22.
In the figure we have plotted the normalized coherence functions21. We have used the reciprocal space notation with the wavevector related to the real space position in Fig. 21 by . This allows a more compact notation of the framed Airy functions representing the shown distributions. In comparing the two figures one may simply substitute .
Figure 22(a) represents the coherent case and Fig. 22(b) the chaotic case. On the left we illustrate the intensity distributions in 3D and underneath as the 2D projection onto the detector plane. The plotted intensity distributions originate from field interference from two points according to Fig. 21, which in practice are defined by the position of mirrors and their scanning along a line, as in Fig. 20.
For the coherent source case, shown in Fig. 22(a), the interference intensity measured with a single mirror scanned according to is the same as if a detector was directly placed at the mirror position. In this case, the intensity distribution is also the same as that measured by use of two mirrors that are arranged and scanned symmetrically with respect to the optical axis according to . This is seen from the equivalence of the distributions (green) and (red), which have the form of the boxed Airy function. It is given by Eq. (53) previously plotted on a linear scale in Fig. 19.
For the chaotic case, shown in Fig. 22(b), the distribution function is highly asymmetrical, with the single point measurement yielding a constant (neglecting the flatness of the detector plane) intensity distribution, characteristic of a chaotic source. In contrast, picks out the coherent fraction according to the VCZ theorem and is given by Eq. (57). It is equivalent to the red distribution shown on a linear scale in Fig. 20.
Finally, Fig. 22(c) shows the intensity distribution resulting from interference of the on-axis field () with a field from point , corresponding to the light blue and pink scanning cases in Fig. 21. These distributions are the same for the coherent and chaotic cases and correspond to the absolute value of the degree of first-order coherence [Eq. (61)] but with instead of (or instead of ).
Up to now, our discussion of first-order coherence was based on the classical correlations of fields, as utilized in Fourier (wave) optics and statistical optics. We will now cast this treatment into the language of quantum optics by linking the concept of coherence to that of a photon wavefunction. We shall see that the two approaches are equivalent in the first-order treatment of light.
5.8. Partial Coherence: Transition from Chaoticity to First-Order Coherence
Zernike’s mutual intensity expression [Eq. (30)] is valid for the general case of partial lateral coherence under the assumption of first-order temporal coherence. We can utilize it to illustrate the evolution from chaoticity to first-order lateral coherence. The general case of partial coherence propagation for our case of a circular source with a Gaussian Schell model coherence distribution is given by inserting Eq. (47) into Eq. (52), so that48) and (49), needs to be evaluated numerically in the general case. The cylindrical symmetry allows us, however, to choose the coordinates and associated with the mirror positions in Fig. 20 along a line through the optical axis, as in Fig. 21. We can then evaluate Eq. (63) for different position pairs and along the line and plot the normalized mutual intensity as a 2D array, as previously shown in Fig. 22.
The results of a calculation for different values of the relative coherence length in the source plane are shown in Fig. 23.
We show an array plot in the middle framed by the most important line plots of the normalized intensities, where in all cases we have plotted to enhance the weak outer interference rings. The detector plane distances are given according to , where is the first zero of the Airy pattern. All patterns in Fig. 23 were calculated numerically by use of Eq. (63). For the shown top and bottom distributions, labeled “coherent limit” and “chaotic limit,” we replaced the Gaussian distribution function by and according to Eqs. (48) and (49).
The Schell model allows us to visualize the evolution of the intensity in the detector plane measured by scanning the mirrors in different ways. Again the case , reflecting use of only one mirror, is equivalent to placing a detector at the position , while the case reflects the mirror positions according to the VCZ theorem.
The change of the normalized mutual intensity in the detector plane, indicated by a vertical dashed green line in Fig. 23, directly reflects the transition from chaotic and coherent behavior. It changes from unity for a chaotic circular source to zero for the coherent case. One may therefore associate the detector intensity with the degree of chaoticity of the source. It is plotted in Fig. 24 versus the relative coherence length in the circular source of radius . It nicely reveals the reduction of the chaotic source behavior with increasing coherence length .
6. Second-Order Coherence
The concept of second-order coherence emerged in the wake of the HBT experiment [39–41] and can be formulated within the framework of statistical optics pioneered by Mandel and Wolf  and Glauber’s and Feynman’s formulation of quantum optics [27,30,115]. The fact that the HBT effect can be explained within either framework has resulted in many debates over its physical interpretation. We have already discussed the quantum interpretation of the HBT experiment in Section 4.2. Owing to the fundamental granular nature of light based on QED, and its first-order reduction to a wavefield-based description (see Section 2.7), there is doubt that the quantum optical description of the HBT effect reveals its true essence. This will become even more apparent in the general formulation of second-order coherence in this section, which will reveal the shortcomings of the statistical optics description.
As expressed by Eq. (45), the concept of first-order coherence may be equally expressed by the correlation of fields in statistical optics or the correlation of one-photon probability amplitudes in quantum optics. The two approaches, however, are not equivalent in second order, although there is a large overlap. It is the purpose of the present section to elucidate the difference.
By extension of the concept of first-order coherence, second-order coherence is based on the classical correlation of two intensities or four fields, which in quantum theory take the form of operators. The second-order quantum correlation function is defined as64) and the corresponding first-order expression [Eq. (27)], field correlations or “interference” are expressed by products of fields (not sums).
6.1. Second-Order Coherence Functions
For two points denoted and , the second-order lateral coherence is expressed through the same-time correlation function:45,193]. This arises from the fact that the photon character of light and its detection is properly accounted for only in the quantum approach through the concepts of creation and destruction operators and their ordering.
6.2. Link of First- and Second-Order Coherence
The key to the propagation of the degree of second-order coherence is its relationship with the degree of first-order coherence . The best known link between first- and second-order coherence is for a chaotic source whose emission is a random Gaussian process. Interestingly, a general link can be established only for the chaotic case and for second-order coherent light. For first-order coherent light there is no general link. This is readily revealed by comparing the first- and second-order coherence properties of different single-mode quantum states . The number, chaotic, and coherent states are all first-order coherent, but only the coherent states are second-order coherent. Hence, by knowledge of the first-order coherence alone, one cannot say anything about the degree of second-order coherence!
For chaotic light, the second-order coherence properties can be expressed in terms of the first-order properties through the so-called Reed theorem  or complex Gaussian moment theorem [44,47]. Under the assumption that the light is quasi-homogeneous, the theorem states67), often referred to as the Siegert relation : 68) famously accounts for the bunching enhancement measured in HBT interferometry, as shown in Fig. 9(c). The relation was previously derived from a quantum treatment and for a circular source was evaluated as Eq. (24).
The complex Gaussian moment theorem [Eq. (67)] holds for both the statistical and quantum optics formulations of second-order coherence, and the HBT effect can therefore be accounted for by either description. In general, however, the statistical optics and quantum description of second-order coherence are not the same, and there are cases, first demonstrated by Mandel and co-workers [37,132,134], that can be accounted for only by quantum optics.
The other case where and are linked is for second-order coherent light. While first-order coherent light is not necessary second-order coherent, the reverse is true. From Eq. (66) we then obtain the following simple relations:9(c), while .
In the following we shall discuss the quantum approach to second-order coherence, which properly accounts for the photon character of light and its detection. We start by considering the quantum description of the fundamental case of two photons.
6.3. Photon–Photon Interactions or “Boson Exchange”
Photons are bosons and sometimes the interactions between photons are termed “boson exchange” in analogy with fermion exchange giving rise to the magnetic exchange interaction . Exchange is not a force, but it reflects an interference effect due to the symmetry properties under exchange .
An exchange formulation requires establishment of a photon wavefunction. The wavefunction description of single photons discussed in Section 5.3 can be extended to multi-photon states as reviewed by Smith and Raymer . For example, a two-photon wavefunction can be written in the general form
The modulus squared of the two-photon wavefunction in Eq. (71) contains self-interference and cross-interference terms. The positive sign in the sum reflects the bosonic nature of the photons, leading to the bunching behavior of bosons, also observed in nuclear collisions with pions . The corresponding negative sign in the wavefunction for fermions  leads to anti-bunching. The different behavior has been elegantly demonstrated by experiments with bosons and fermions . The bunching versus anti-bunching behavior of bosons versus fermions has also been discussed and experimentally demonstrated recently by Liu et al. .
In the optical region, photon–photon correlation experiments may be performed in the space-momentum and time-frequency domains. While optical coherence times may be of the same order as detector response times of about 1 ns, temporal correlation experiments are much more difficult in the x-ray regime because of the femtosecond x-ray coherence times. In practice, x-ray correlation experiments therefore typically investigate the lateral x-ray coherence properties. This is aided by the expansion of the coherence areas with distance, as shown in Fig. 4. With today’s position sensitive detectors containing pixels of about 10 μm size and use of source–detector distances around 10 m, spatial correlation experiments are straightforward, provided that the coincidence widow is defined by the pulse length rather the longer detector response time .
6.4. Two-Photon Wavefunction and
For two photons we can construct a two-photon probability amplitude similar to the one-photon case treated in Section 5.3. We again consider the normal ordering and action of the field operators and , which create and destroy single photons at position , respectively, and rewrite Eq. (65) as2]. Closer inspection reveals the peculiarity of the quantum theory of coherence in that it allows the interference between the same-time destruction of photons at two different positions and . In practice, this is measured by single-photon detectors at two points whose signal is multiplied by a coincidence circuit.
6.5. General Two-Photon Propagation Law
In the discussion of the HBT case in Section 4.2 we considered the birth of independent single photons at two source points and calculated the coincident one-photon detection probability at two points in the detector plane. In that case, the correlation function had the special form given by Eq. (67). More generally, the two-photon case also includes the production of biphotons at a single point, which was neglected in our HBT treatment. The general case consists of the four scenarios illustrated in Fig. 25 and we shall explore it now by use Eq. (73) and Feynman’s probability amplitude formulation.
As for the discussion of the HBT effect in Section 4.2, we associate one-photon probability amplitudes with the birth and propagation of photons from source point I to detection point J. Concomitant propagation of two single photons is again shown in the same color as for the HBT case in Fig. 10. Figure 25 shows both the two HBT concomitant scenarios and two additional scenarios associated with the birth of two photons at the same point. The case where X = Y, i.e., two-photons are detected at a single point, is not shown but is included in our general formalism below as a special case.
We start with the second-order correlation function in Eq. (72), which we write in terms of detector plane coordinates:
We follow Fig. 25 and the notation of Sections 2.10 and 4.2 by denoting dimensionless single-photon amplitudes associated with paths from point to point as [defined in Eq. (6)], summed single-photon amplitudes associated with a detection point as , and two-photon amplitudes associated with points and as . We write the two-photon amplitudes in Eq. (74) of dimension in terms of our dimensionless ones as where is given by Eq. (45). We can decompose the two-photon amplitudes in Eq. (74) as follows by reading from the bottom up:25 and has the form [115,215] 46].
As indicated in Fig. 25, in the top two (green and orange) scenarios, single photons are born at different points, and the two alternatives are described by the same correlation of the birth phases in Eq. (77). In the bottom two (pink and blue) scenarios, two photons are born at the same point and the general formulation allows them to be born with different phases that also differ from the green/orange scenario.
In general, the 16 possible two-photon interference phenomena may lead to different behavior for different sources and different detection schemes. Over the years, different scenarios have indeed been observed experimentally, as reviewed by Liu and Zhang . Similar to the simple two-point sources, these studies conveniently utilized two slits, as discussed in Section 4.5. The detectors are scanned in the plane perpendicular to the two slits to observe interference effects. The finite width of slits is accounted for by integration of Eq. (77) over the slit widths . Different types of slit illuminations by thermal, laser, or biphoton sources then conveniently allowed the adjustment of the relative birth phases of the photons created at the slit positions. As pointed out by Liu and Zhang , all observations can be explained by the general two-photon amplitude [Eq. (77)], due to preferential contributions of some of the 16 interference terms in . A particularly beautiful experiment that explored the bosonic and fermionic second-order interference cases by use of independent photons and application of Eq. (77) has recently been reported .
In the following we shall not discuss the rich landscape associated with all 16 terms but will focus on the difference between the HBT (green/orange) and biphoton (pink/blue) scenarios.
6.6. Biphoton Propagation Law
As done in Section 4.2 for the green/orange HBT scenario in Fig. 25, we now consider the pink/blue biphoton scenario corresponding to the last two terms in Eq. (77). By assuming that the phases for two photons born at the same point are the same, i.e., , we obtain9)] and neglecting both terms and , this becomes 45) the biphoton correlation function becomes 20), the cosine term depends on the phase difference at the two source points , and the detector positions are linked by a plus sign, , instead of a minus sign, . We shall come back to this important point in the next section.
For the HBT one-photon coincident detection scheme, the detectors are positioned symmetrically, as shown in Fig. 9(a), so that and the second argument in the cosine function in Eq. (81) vanishes. For a chaotic source, the relative birth phases furthermore change randomly so that the quantum mechanical expectation value becomes . The cosine term in Eq. (81) therefore does not contribute to the HBT case, as assumed in its quantum derivation in Section 4.2.
For a finite-size source that emits biphotons from different points, the two-point case given by Eq. (78) needs to be integrated over all source points and . As in the quantum derivation of Zernike’s one-photon case discussed in Section 5.4, however, we need to avoid “double counting” of source points in the integration. We again use the paraxial approximation of Eq. (9) and neglect the term , but as in the derivation of Zernike’s law, we keep the term (which can be neglected in practice) to obtain45), we obtain the desired biphoton propagation law: 43) and indicated that it corresponds to .
For biphotons, the second-order coherence function differs from the Reed theorem expression [Eq. (67)] for the chaotic case in that there is no constant or accidental coincidence term. This is explained by the fact that for biphotons that are born at the same time, there are no accidental coincidences for the case of ideal one-photon detectors. In addition, given by Eq. (83) differs from the second term in Eq. (67) given by , which reflects true coincidences or bunching. To appreciate the difference between the two cases, we shall now compare them phrased in the language of Glauber’s field operators.
6.7. Discussion of First- and Second-Order Propagation Laws
To more clearly reveal the difference between the propagation laws for first- and second-order coherence, we recall Zernike’s one-photon propagation law written in terms of field operators, given by Eq. (30):84) expresses the correlation between single-photon destruction () and photon creation (), the quantum expression for given by Eq. (73) reflects the correlation between two-photon destruction () and two-photon creation (). For biphotons we may therefore write, according to Eq. (73), 84) given by
The biphoton version of Zernike’s law given by Eq. (83) corresponds to taking the magnitude squared of Eq. (84) and replacing in both the source and detector planes by the corresponding biphoton expression . This gives us Eq. (83) in terms of field operators as84) and (88) reveals that the difference is a consequence of the adjoint property of the quantum operators . The impulse response functions, expressed by the exponentials, therefore have different signs, as emphasized by others [50,51,198,217]. The important sign difference is also revealed by comparison of the HBT expression [Eq. (20)], where the detector positions are related by instead of in the corresponding biphoton expression [Eq. (81)].
As an illustrative example of two-photon propagation, we now discuss the case of a Schell model source, which will also allow a direct comparison with the one-photon case in Section 5.5.
6.8. Biphoton Schell Model Source
Assuming again a symmetrical distribution function around the optical axis in the source, the biphoton probability (wavefunction squared) in the source may be written in form of a generalized Einstein–Podolsky–Rosen (EPR) state as [218–220]
6.9. Correspondence of Partial Coherence and Partial Entanglement
As discussed in Section 3.2, in practice, one uses incident beams that are first-order coherent. For the case of a secondary circular biphoton source of radius that is created through uniform illumination by a first-order coherent beam, we can therefore assume , where is the radius of the source. The limiting case of a completely entangled biphoton source is represented by the case , while the opposite limit is represented by . This limit corresponds to a cloned biphoton source, where the biphotons are not entangled but completely indistinguishable and second-order coherent.
For the cloned biphoton case, Eq. (90) becomes, with ,
The limit of complete entanglement is given by , which in analogy to the treatment of the chaotic one-photon case [Eq. (49)], may again be represented by a 2D Dirac- function:49). The correspondence is illustrated in Fig. 26.
The derived correspondence between the first-order one-photon and second-order biphoton cases reveals more insight into the nature of light. In first order, the treatment of partial coherence has two limits, chaoticity and first-order coherence, expressed by Eqs. (49) and (48), respectively. Similarly, in second order, partial entanglement of two photons has the limits of entanglement and second-order coherence, given by Eqs. (91) and (92). Hence, the concept of incoherence in first order corresponds to the concept of entanglement in second order.
The quantum behavior in first- and second-order coherence theory have important photon counting implications. When the light is first- or second-order coherent, interference occurs intrinsically between single- or two-photon amplitudes, respectively. The interference pattern is simply recorded with a position sensitive detector that in each pixel registers the total charge created by the impinging single or two photons.
In contrast, when the light is chaotic in first order or entangled in second order, a coincidence measurement is required to reveal interference. The coincidence technique in first order measures the intensity due to overlapping fields (see Fig. 20) or self-interference of the single-photon wavefunction. In second order, the coincidence is between the arrival of two photons, or self-interference of the biphoton wavefunction.
6.10. Biphoton Diffraction for a Circular Source
To illustrate the correspondence between partial coherence in first order and partial entanglement in second order quantitatively, we now calculate the biphoton intensity distribution in the detector plane for a circular source, in complete analogy to the one-photon case discussed in Sections 5.6 and 5.8. To better reveal the difference between the cloned and entangled cases in the diffraction patterns in the detector plane, we use the shorter reciprocal space notation, where the momentum transfer is given by .
The general case of second-order partial entanglement propagation is given by Eq. (83) or Eq. (88). By inserting the biphoton wavefunction given by Eq. (89) under the assumption , the intensity distribution resembles the flat-top case given by Eq. (50), and the integration extends to the source radius . We obtain26.221]. We shall later in Section 7.4 discuss the first- and second-order diffraction patterns of a circular source of the same area. In this case, the behavior in the detector plane is exactly opposite, because for a fixed source size the first- and second-order coherence cones scale as [see Eq. (121)].
6.11. for Cloned and Entangled Biphotons
The partially entangled case described by Eq. (94) for a circular source can be solved analytically in the limits of complete entanglement (entangled biphotons) and second-order coherence (cloned biphotons) . This is in complete analogy to the one-photon case in which the Fraunhofer and VCZ expressions reflect the limiting cases of first-order coherence and chaoticity.
For the cloned biphoton case, we evaluate Eq. (94) with as94), which is a quantum mechanical signature, may also be replaced by a minus sign, i.e., by squaring the single-photon expression in Eq. (93). This reflects the fact that stimulated emission, resulting in cloned biphotons, may also be explained classically. In fact, Einstein’s introduction of stimulated emission  was based on treating absorption and stimulated emission as both driven by an classical field, with the two processes corresponding to classical destructive or constructive interference of the incident field with the scattered field in the forward direction.
In contrast, the entangled biphoton case is truly quantum mechanical in nature and depends on the positive sign in Eq. (94). It is evaluated with to give27). It resembles the square of the chaotic source expression in Eq. (57) but with inverted signs, .
6.12. Power Conservation: for Cloned and Entangled Biphotons
For the one-photon case, the first-order coherence function and the intensity are related according to the simple relation , since has the dimension of and corresponds to the number of photons (× photon energy / (area × time)). The second-order coherence function , however, has the dimension of . The one- and two-photon intensities linked by have the same dimension, and hence one needs to consider how is related to . This is accomplished by considering power or energy conservation between the source and detector planes, so that the relevant intensities integrated over the finite source area and detector plane are the same.
The power associated with the simultaneous emission of two photons by a source with a symmetrical distribution about the optical axis may be written as , where is the two-photon central peak intensity and the effective source area, defined as
The two-photon intensity in the detector plane is measured either with two one-photon detectors or one two-photon detector. As for the source, we may write the power in the detector plane as , where is the central intensity, and is the effective area of the detector-plane intensity distribution, given by99) for , we obtain our final result, which we can write by dropping the detector plane label “d” in the real-space vector notation as
The cloned biphoton intensity distribution is obtained from Eq. (96) as
For the entangled biphoton case we obtain from Eq. (97) with145]. The entangled photon pattern has a peak intensity that is about a factor 2 higher than for the cloned biphoton case, and the second-order coherence area is a factor of 2 smaller. For we obtain from Eq. (97) the expression in Eq. (104) without the Airy function, which is unity for . The intensity is thus constant and there is no diffraction structure.
We can now directly compare the two cases through their coherence matrix, in complete analogy to the one-photon (two field) case illustrated in Figs. 21 and 22. The extreme cases of entanglement and second-order coherence are illustrated in Fig. 27.
In the figure we have plotted the normalized two-photon coherence function . It directly corresponds to the extreme cases of chaoticity and first-order coherence shown in Fig. 22. In the one-photon case, the detector plane positions were in practice given by the locations of mirrors, which created the interference intensity of two fields at the detector. In the two-photon case, the mirrors are replaced by two one-photon detectors. They measure the interference intensity of two photons in complete analogy to the two fields that determine .
However, there is an important reversal of the coordinates from the one-photon case in Fig. 22 to the two-photon case in Fig. 27, according to ). It is due to the quantum mechanical treatment of the two-photon detection process, as reflected by the opposite signs in the impulse response functions in Eqs. (93) and (94).
6.13. Comparison of One-Photon and Biphoton Patterns and Detection Schemes
The calculated one-photon and biphoton diffraction patterns are summarized in Fig. 28. In the figure we also depict the assumed experimental geometry and detector positions for their measurement.
In all cases, the intensity distribution is taken to be uniform across the circular aperture of area that defines the spatial extent of the secondary source. The average intensity emitted by the source is given by the total number of photons of energy that are emitted over a time , according to
For the chaotic and first-order coherent cases, we can simply assume that the aperture contains no material. The diffraction patterns are those previously shown in Figs. 19, 20, and 22. They are shown again in Figs. 28(a) and 28(c), and are compared to the biphoton patterns in Figs. 28(b) and 28(d). In all cases we have also indicated how they may be measured in practice.
For the cloned biphoton case, shown in Fig. 28(b), we have assumed that the incident monochromatic radiation is tuned to an atomic resonance (two-level system) and that the intensity is sufficiently high that during the pulse the two-level system equilibrates . In this steady state, absorption is completely compensated for by stimulated emission . As shown on the right side, the diffraction pattern given by Eq. (103) may be recorded either by use of a single two-photon detector (red) at position (or ) or by use of two single-photon coincidence detectors at positions and (blue). According to Fig. 27, the two measurements give the same result. In practice, one may simply use a position sensitive detector like a CCD that gives a charge integrated signal per pixel that is directly proportional to the two coincident cloned biphotons incident on the pixel.
For the entangled biphoton case depicted in Fig. 28(d), we have assumed that the aperture contains a nonlinear material that creates biphotons in a SPDC process. In practice, the nonlinear material is placed right in front of the aperture [145,147,198]. If the nonlinear material creates collinear biphotons of opposite polarization, one would insert a suitable polarizer that makes the transmitted photons indistinguishable in their polarization. The red diffraction pattern is given by Eq. (104) and the blue one by the same equation, but with replacement of the Airy function by unity.
6.14. Partial Entanglement Patterns
Similar to the treatment of partial coherence in Section 5.8, we can now treat the case of partial entanglement for the case of a circular Schell model source by numerically evaluating Eq. (94). The results are shown in Fig. 29.
The two-photon diffraction patterns as a function of the normalized second-order coherence length are seen to be the square of the corresponding single-photon ones with reversal of the coordinates due to the opposite signs in the impulse response functions in Eqs. (93) and (94). As for the single-photon case, the intensity distributions shown in blue change along the dashed blue line corresponding to from zero for the second-order coherent case to unity for the completely entangled case.
In practice, the degree of entanglement can be changed, as demonstrated by Shimizu et al. , by changing the focus of the pump beam on the nonlinear crystal used to generate the biphotons, so that part of the transition from entangled to cloned behavior has indeed been observed!
Previously, we associated the change of in Fig. 23 with the degree of chaoticity. Figure 29 shows that we may similarly associate with the degree of entanglement. The correspondence is underscored by Fig. 30, where we directly compare the one- and two-photon cases, by plotting versus (green, taken from Fig. 24) and versus (dashed blue). In the figure we have taken into account the relationship between the first- and second-order coherence lengths according to Eq. (95). We again see that the second-order description is the square of the first-order one, with the degree of entanglement corresponding to the square of the degree of chaoticity.
6.15. Summary of the Forms of Diffraction Patterns for Different Cases
The summary of the diffraction structure for circular flat-top sources of different nature shown in Fig. 28 reveals an interesting correlation between the diffraction structure of chaotic and entangled sources on the one hand and that of first- and second-order coherent sources on the other.
For a chaotic source, the first-order coherent part of the constant total intensity, represented by , is picked out according to the VCZ theorem with two mirrors at positions and , as shown in Fig. 28(c). We have the relations28(c) is replaced by the HBT detector scheme in Fig. 28(d). The corresponding second-orderexpressions, given by 24), or 28(a) and 28(b), we have
The functional form of the Airy diffraction patterns thus have the same dependence for the chaotic and entangled sources. Similarly, the patterns have the same (half the periodicity!) dependence for the first- and second-order coherent sources. In both cases the patterns become squared in going from first to second order. This behavior reflects the duality of chaoticity and first-order coherence and of entanglement and second-order coherence, respectively, that emerged in Section 6.9.
7. Beyond the Diffraction Limit
7.1. Biphoton Case
The conventional diffraction limit is defined by the 2D area of the diffraction pattern of a coherent circular source, given by the Airy distribution function in Eq. (5) or Eq. (55). Hence, we can conveniently use this pattern to define the diffraction-limited area. For our case of a symmetrical intensity distribution function around the optical axis, the area of the central peak in the detector plane for the first and second order distributions is given by19.
For the first-order coherent case, is given by Eq. (55) and the width of this distribution, identified in Fig. 19, serves as our standard for the diffraction-limited case. The biphoton cases are described by , given for the cloned case by Eq. (103) and the entangled case by Eq. (104). In Fig. 31 we directly compare the biphoton cases with our standard for the diffraction limit, defined by the first-order area .
Figure 31(a) compares all three diffraction patterns plotted on a logarithmic intensity scale to emphasize the diffraction rings. The same patterns are shown on a linear scale in Fig. 31(b). This emphasizes the increase of the central peak intensity, while the normalized intensity distributions shown in Fig. 31(c) reveal the concomitant reduction of the central image area.
The conventional diffraction limit, corresponding to the area , is reduced by factors of 2.18 and 4 for the cloned and entangled biphoton cases, respectively. This reveals the important fact that the conventional diffraction limit may be reduced by the interference of multi-photon probability amplitudes! We shall come back to the discussion of our results in terms of the link between the diffraction limit and the uncertainty principle in Section 7.4.
7.2. Simple Biphoton Model
We may picture the origin of the biphoton diffraction patterns in terms of the simple biphoton models shown in Fig. 32. A single photon may be characterized by a wavevector mode and amplitude , with defining both the propagation direction and the frequency or wave length .
Stimulated emission, like absorption, may be viewed classically as an atomic transition driven by an incident field  (also see Section 6.11). This allows us to picture the cloning process in terms of the classical amplitude superposition picture shown in Fig. 32(a). Since classically, intensity is given by the absolute value squared of the field, the cloned biphoton has an associated intensity through constructive interference that is and not the value associated with two independent photons. The increased amplitude or intensity enhancement by a factor of 2 in the forward direction then requires that the area of the diffraction pattern reduces by the same factor of 2 due to power conservation.
As the name implies, the two photons forming a cloned biphoton are intrinsically indistinguishable in all of their properties: amplitude, polarization, wavevector, energy, and phase. They are both first- and second-order coherent, and , and naturally interfere because of their indistinguishability. Einstein , who first introduced stimulated emission, already conjectured that the two created photons were indistinguishable. In a remarkable paper, Ou et al. [35,222] have suggested that this picture may also be turned around, with stimulated emission being viewed as constructive interference due to photon indistinguishability. In a stimulated resonant scattering process, two incident first-order-coherent photons in the same mode but with a random relative phase hence become phase-locked through their resonant nonlinear interaction with an atom and become second-order coherent.
The incident first-order coherent photons are in single-mode number or Fock states with a second-order degree of coherence of . Stimulation causes the individual single photon wavepackets to lock in phase and become second-order coherent with . The cloned biphotons remain phase-locked upon propagation to the detector plane. When using a charge integrating detector, the one-photon diffraction pattern changes naturally to a two-photon pattern with the onset of stimulation. One may distinguish this intrinsic change of the diffraction pattern, whose origin lies in the source plane, from an extrinsically induced change of the diffraction pattern by two-photon coincidence destruction in the detector plane.
This intrinsic versus extrinsic behavior constitutes the fundamental difference between the change from the one- to the two-photon patterns in Figs. 14(d) and 14(e) and that in Figs. 15(d) and 15(e). The comparison of the two cases also reveals the conjugate behavior of photon creation in the source and photon destruction in the detector plane, which is employed in the quantum mechanical definition of the two-photon wavefunction in Eq. (72).
In this context, it is important to recall our discussion in Section 2.8 about the difference between a -photon number state and a coherent state of mean photon number . Number states of the same polarization are in the same monochromatic mode and their number is the photon degeneracy parameter. Following Dirac, it is usually stated that number states do not interfere with each other. As pointed out by Mandel , one needs to make a distinction, however, between a single observation and the measurement of an ensemble average over many observations (e.g., a temporal average).
A single observation may indeed exhibit interference, as first reported in 1955 by Forrester et al.  and discussed by Javanainen and Yoo  for Bose condensates in number states. Hence, a single measurement involving two photons in the same number state may exhibit interference since it depends only on the relative phase of the two photons. For example, it may happen to be the same, yielding constructive interference at a point or 2 counts, or opposite to reveal destructive interference and 0 counts. In contrast, when an ensemble average over many two-photon detections is formed, any interference is wiped out due to the random nature of the relative phases. On average the two photons thus give single-photon counts at different points and, in line with Dirac’s statement, the diffraction pattern is formed by one-photon detection events .
As a consequence, the conventional first-order diffraction pattern does not change with the photon degeneracy parameter, as mentioned in Section 2.8. From a conventional diffraction point of view, the improvement of x-ray sources has therefore only enabled faster measurements of diffraction patterns. From a multi-photon diffraction point of view, the degeneracy parameter leads to the simultaneous presence of photons in the sample and increases the probability of conversion of first-order into higher order coherent photons in the sample acting as a secondary source.
According to Fig. 32(b), the entangled biphoton may be envisioned to have a single-photon amplitude and wavevector of or twice that of the individual photons. Remarkably, the total biphoton wavevector is the same as that of the pump photon that is broken up into the pair, and the second-order biphoton diffraction pattern is the same as the first-order one of the pump photons. In Dirac’s language, entangled biphotons interfere only with themselves. In this case, we can attribute the reduction of the image area by a factor of 4 relative to that of photons of wavevector to the larger biphoton wavevector. Because of power conservation, the peak intensity needs to increase by the same factor of 4.
The entangled biphoton case cannot be described classically, since a wave-based picture cannot account for the effective reduction of the biphoton wavelength by a factor of 2. For this reason, entangled biphotons incorporate the essence of quantum behavior, which explains their popular use for the exploration of quantum phenomena, and their prominent role in quantum information science and technology.
In summary, cloned biphotons offer both an increased central intensity and reduced diffraction spot size. This may be viewed as arising from the interference of the probability amplitudes of the two photons with each other and the interference of the entire biphoton probability amplitude with itself. In contrast, the entangled biphoton diffraction pattern arises from interference of the biphoton probability amplitude only with itself. In practice, entangled biphotons offer no advantage for image formation (e.g., lithography) over using the pump photons directly, but rather the disadvantage of lower intensity due to the low SPDC conversion efficiency.
7.3. Extension to th-Order Coherent Source
The case of cloned biphotons may readily be extended to the case of clones emitted by an th-order coherent source, since factors into first-order correlation functions [31,113]. The associated th-order diffraction pattern is given by99), 103) for by 113), is the effective coherence area in the detector plane of an th-order coherent source. The evolution diffraction pattern with increasing is illustrated in Fig. 33.
The figure extends the case of shown in Fig. 31 to higher order. The increase of the central peak intensity by a factor of , revealed most prominently by Fig. 33(b), is compensated by a reduction of the effective area , which is evident from Fig. 33(c). The inset in Fig. 33(c) reveals the relationship for a flat-top source distribution:33(c). For a Gaussian source distribution, the relation holds exactly since 33, is that it overcomes the general paradigm of wave optics or first-order diffraction theory outlined in Section 2.9. The propagation behavior of an th-order coherent beam resembles that of non-diffracting Bessel beams [224–226]. We find that the collective -photon state created by boson exchange is characterized by the alignment of the linear momenta of the photons. This reveals a striking correspondence of boson exchange, resulting in the collective alignment of the wavevector of indistinguishable photons, exemplified by the laser, and fermion exchange leading to the collective alignment of identical atomic spins, giving rise to ferromagnetism .
7.4. Diffraction Limit and Uncertainty Principle
The conventional diffraction limit of a source is determined by Heisenberg’s position-momentum uncertainty principle . Kennard  first proved the modern inequality, which links the standard deviation of position, , with that in momentum, . For this reason, Gaussian intensity distributions of rms width naturally define the minimum uncertainty distributions in position and momentum , which can also be written in terms of the wavevector, , and angle, , according to119) may also be derived from the Fourier relation [Eq. (33)] between the source and the detector plane distributions by assuming a coherent Gaussian source distribution according to Eq. (46), with and , as shown in Appendix A.1. The minimum of the product, expressed by the equal sign, defines the conventional diffraction limit.
For a flat-top source of area , of interest here, the corresponding expression is given by (see Appendix A.2)115) and as 119) and (120). Equation (121) is strictly valid for Gaussian and approximately for flat-top source distributions, with the approximation improving with as revealed by the inset in Fig. 33(c). In both cases, the th-order uncertainty expression is reduced relative to the conventional first-order () one due to the interference of the probability amplitudes of all indistinguishable photons and the self-interference of the collective -photon state.
If the diffraction patterns for a first- and th-order coherent source are compared, one might be tempted to associate the reduced diffraction limit with a violation of the uncertainty principle. A comparison of the patterns alone, however, would not take into consideration all information about how the pattern was recorded, which is required for a proper physical interpretation. On the other hand, it is well known that quantum mechanics is a linear theory and therefore of first order . In principle, the uncertainty principle is therefore also of first order and is expected to scale with increasing order simply as .
7.5. Coherence in a Photon Picture and Multi-Photon Diffraction Patterns
The above discussion of the hierarchical nature of light in terms of orders of coherence allows us to develop a photon-based picture of lateral coherence that generalizes the conventional picture based on the wave description of light, which is recovered for . It emerges when we express the solid angle emission cone of the macroscopic source area given at a distance as a function of according to Eq. (121). The total emission cone of the macroscopic source is comprised of micro-cones that originate from micro-coherence areas , as illustrated in Fig. 34.
Our model associates the microscopic first-order coherence area in a source with the atomic Breit–Wigner cross section and then considers how the solid angle of emission changes with the order of coherence . In particular, for microscopic flat-top coherence areas, the diffraction limit can be written by use of Eqs. (120) and (121) as34 leads to the following simple picture of the change in coherence propagation with order of coherence .
We may associate a chaotic source with . It emits into the entire forward solid angle, in accord with the fact that the Breit–Wigner cross section corresponds to the maximum resonant atomic interaction cross section with emission of a spherical wave [86,207]. The case is the conventional (first-order) coherence case, which is usually derived by requiring that the emitted spherical wave can be approximated within the coherence cone by a plane wave. In this case, the coherence cone is reduced by a factor of 2 to . The case expresses the solid angle of emission of a second-order coherent source, which is again reduced by a factor of 2 to . For an th-order coherent source we have .
With increasing , the micro-cones increasingly resemble particle-like trajectories from the source to the detector plane, and the ray optics limit of no diffraction is increasingly recovered, as illustrated in Fig. 34(b). The figure supports the link of the lateral brightness of a source with the first-order lateral coherence and the areal density of the emitted photons, according to Eq. (4). With increasing order of coherence , the lateral brightness and photon density also increase by a factor . In space-time, brightness, coherence, and photon density therefore constitute alternative descriptions of the nature of a light source.
The photon-density-based description of coherence together with the link of coherence and photon detection facilitates the understanding of multi-photon interference. Conventional first-order diffraction patterns recorded with a pixelated CCD detector consist of the accumulation of single-photon counts per pixel. Since the photons are independent, a large degeneracy parameter simply speeds up the statistical formation of the one-photon-at-a-time pattern through the deposition of single photons in different pixels, but it does not change the pattern relative to that recorded with a low-degeneracy-parameter source.
In second order, the two-photon pattern may be formed either intrinsically in the photon creation process in the source or extrinsically in the photon destruction process in the detector plane through coincidence detection. If the second-order coherence is intrinsic no extrinsic coincidence detection is required. If the source is not second-order coherent, extrinsic coincidence detection is required.
Examples of this behavior are the changes from one- to two-photon patterns in Figs. 14(d) and 14(e) and in Figs. 15(d) and 15(e), respectively. In the former case, the source is first-order coherent and the two-photon pattern is extrinsically created by a coincidence circuit in the detector plane. In the latter case, the source emits a large number of intrinsically linked photon pairs within a single pulse. The entire two-photon pattern forms with sufficient statistics in a single pulse. Each pair is naturally coincident and can be counted by a charge-integrating pixel detector. By extrapolation, the th-order pattern is formed by coincident detection of photons (-photons-at-a-time) per pixel and consists of the th power of the one-photon pattern.
7.6. Emission of Conventional Chaotic versus Synchrotron Sources
As discussed in conjunction with Fig. 2 a synchrotron source is chaotic like an x-ray tube or a thermal source [62,63]. In contrast to conventional chaotic sources, however, the solid angle of emission of a SR source is greatly reduced due to relativistic effects, and we need to briefly discuss its implications.
The minimum first-order coherence product for a synchrotron and conventional source are the same, which means that the effective (one electron) source area in a synchrotron source is significantly larger. This is illustrated for Gaussian distributions in Fig. 35.
In practice, the emitted SR originates from the total bunch of electrons, and the diffraction limit is reached only when the bunch size can be squeezed below the size of the single electron source area, . Owing to the large value of ( for a 3 GeV ring), the single electron coherence area is fortunately quite large. This allows the squeezing of the total bunch into an area of order for photon energies below about 1000 eV in “ultimate storage rings” [112,228]. In the general case, the source areas and solid angles of the single electron and electron bunch distributions add in quadrature [110,112].
8. Summary and Conclusions
The present paper explores the nature of light from a spatial coherence and diffraction point of view, comparing both classical and quantum concepts. The diffraction aspect of light is particularly fundamental in that it is related to Heisenberg’s space-momentum uncertainty principle. We note that the spectroscopy aspects of light, which link time and energy through the so-called transform limit or time–bandwidth product are less fundamental since time cannot be defined in terms of a quantum mechanical operator, but is a parameter .
From a historical point of view, diffraction has limited the resolution of optical imaging, while in the x-ray range it has been the leading tool for deciphering the atomic structure of matter. It is quite remarkable that, even today, x-ray diffraction is almost exclusively formulated in terms of the wave concept of light . In contrast to the optical regime, x-ray photon–photon correlation experiments have been performed within only the past 10 years or so, mostly to characterize synchrotron and XFEL radiation [53,63,186,229] and only within the past two years from a perspective of quantum imaging [54–56,142–144]. By discussing the fundamentals of diffraction from a wave and photon perspective, the present paper is hoped to also serve the broader x-ray community as an introduction to the powerful paradigm of photon–photon interference, which emerges from either Glauber’s generalized treatment of coherence or Feynman’s concept of photon probability amplitudes.
The key topic of the paper is the evolution from the conventional concept of coherence based on the wavefield, referred to here as first-order coherence, to the quantum concept of second-order coherence. To elucidate this transition, we first show the equivalence of the wave and photon description of coherence in first order. It originates from the fact that the classical fields simply correspond to the probability amplitudes of independent single photons. This leads to the important recognition that the wave description is a special case of the full QED description when limited to first order. When extending the quantum description to second order, the equivalence breaks down and in general only the quantum description should be used. We show for the most fundamental one-photon and two-photon cases that the quantum formulation of diffraction is facilitated by simple rules regarding the addition or multiplication of single-photon probability amplitudes.
The second-order concept is discussed for three cases. Independent photons emitted by a chaotic source are shown to lead to the HBT effect resulting from cross interference of concomitant two-photon probability amplitudes. This case of independent photons is contrasted to the case where two-photons are born with an intrinsic correlation in space-time, referred to as biphotons. We show that for this case, one can distinguish two extreme cases, corresponding to complete entanglement or complete indistinguishability, which are represented by entangled and cloned biphotons, respectively. The biphoton case exhibits a remarkable duality in second order to the single photon case in first order, with partial entanglement in second order corresponding to partial coherence in first order. The first-order limits of chaoticity and first-order coherence for single photons directly map onto the second-order limits of entanglement, represented by entangled biphotons, and second-order coherence, embodied by cloned biphotons. We also show that, in general, partial entanglement of photons can be utilized to obtain spatial resolution below the conventional diffraction limit, and describe how this arises in a simple model for cloned and entangled biphotons.
The case of indistinguishable clones is discussed by generalization of the cloned biphoton case using Glauber’s formulation of th-order coherence. Remarkably, the conventional diffraction limit is shown to be reduced by a factor of . When the diffraction limit is defined in terms of Heisenberg’s uncertainty principle as the product of the source area and solid angle of emission , the product is shown to decrease as . This may be attributed to the linearity of quantum mechanics and an uncertainty principle that, in principle, is only valid to first order.
The disappearance of diffraction effects with increasing order of coherence may be interpreted as increasing localization and particle-like behavior of light within the collective state. At large , the image of the source in a distant detector plane becomes that of the source itself, similar to the ray optics result. The presented results reveal that increasing degrees of coherence of a source can be pictured simply as increasing density of the emitted photons.
This paper emphasizes diffraction by a source defined by a circular aperture, since it has long served as the paradigm to define the conventional diffraction limit. At hard x-ray energies, conventional Bragg diffraction by periodic atomic arrangements arises from interference of independent single-photon probability amplitudes, just like in Young’s experiment. We envision extension of conventional one-photon diffraction to multi-photon diffraction with several coincidence detectors to increase the spatial resolution at x-ray energies below the atomic scale.
Appendix A: Derivation of Uncertainty Relations from Fourier Transforms
A.1. Coherent Gaussian Distribution
The uncertainty relation in Eq. (119) may be derived from the Fourier relation in Eq. (33) between the source and the detector plane. For a coherent Gaussian source, the intensity distribution is given by Eq. (46) with , and the effective source area is . Inserting into Eq. (33) we obtain119).
A.2. Coherent Flat-Top Distribution
The uncertainty relation in Eq. (120) is also derived from the Fourier relation of Eq. (33) between the source and the detector plane with the intensity distribution given by Eq. (51) with , and the effective source area is . Inserting into Eq. (33) we obtain55)] 19) and associated solid angle of emission given by 120).
I have benefitted from discussions with Joseph Goodman, Z. Y. (Jeff) Ou, Robin Santra, Yanhua Shih, Justin Wark, and Joachim von Zanthier.
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Joachim Stöhr received his Ph.D. from the Technical University in Munich in 1974. After a postdoctoral period and staff position at Lawrence Berkeley National Laboratory he became a staff scientist at the Stanford Synchrotron Radiation Laboratory (SSRL) in 1977. In 1981 he joined Exxon Corporate Research Laboratory as Senior Staff Scientist and in 1985 moved to the IBM Almaden Research Center, where he was a department manager and wrote his first book NEXAFS Spectroscopy (Springer, 1992). In 2000 he joined Stanford University as Professor of Photon Science and Deputy Director of SSRL, assuming its Directorship in 2005. In 2009 he became the founding Director of the Linac Coherent Light Source (LCLS), the world’s first x-ray free electron laser. While at Stanford, he wrote his second book with H. C. Siegmann, Magnetism–From Fundamentals to Nanoscale Dynamics (Springer, 2006). In 2011 he received the Davisson–Germer Prize in Surface Physics from the American Physical Society. He has been Professor Emeritus of Photon Science since 2015.