The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of $n$ cloned photons, resulting in a $1/n$ reduction of the diffraction limit. In the limit of $n$th-order coherence, all photons within the $n$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.

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通过多光子干涉克服衍射极限：教程

通过比较基于场的统计光学方法和基于光子的量子光学方法，从衍射的角度审查了从光学到X射线范围的光的性质。通过比较基于波干扰的历史衍射概念，狄拉克的光子自干扰概念，费曼的时空光子概率振幅干扰以及格劳伯基于光子检测的相干函数公式，介绍了本主题。通过回顾半经典组合的方式来阐明这些概念不同的光子和波的概念已被用来描述光的产生，衍射和检测。然后在波和光子图中都讨论了基本衍射极限的起源。通过使用费曼的与独立光子相关的概率振幅的概念，我们证明了量子电动力学，即光的完整理论，以最低的顺序降级到传统的衍射波形式。作为多光子效应的简介，我们将回顾基本的单光子和双光子实验和检测方案，特别是开创性的Hanbury Brown-Twiss实验。本文的正式论述始于一阶相干理论的处理。在一阶中，相干的统计光学和量子光学公式显示为等效。通过讨论Zernike强大的部分相干传播定理（统计光学的基石），然后基于单光子概率振幅的干涉进行量子推导，可以阐明这一点。然后将处理扩展到二阶相干理论，其中波和粒子描述的等效性被分解了。这可以通过考虑两个光子来说明，这两个光子的时空振幅通过非线性出生过程相关，从而导致纠缠或克隆。在两种情况下，双光子衍射图样均显示出低于传统衍射极限的分辨率，该极限由单光子衍射图样定义。减少的起因被证明是由两光子概率振幅的干扰引起的。通过比较一阶和二阶衍射，可以看出传统的一阶概念 导致纠缠或克隆。在两种情况下，双光子衍射图样均显示出低于传统衍射极限的分辨率，该极限由单光子衍射图样定义。减少的起因被证明是由两光子概率振幅的干扰引起的。通过比较一阶和二阶衍射，可以看出传统的一阶概念 导致纠缠或克隆。在两种情况下，双光子衍射图样均显示出低于传统衍射极限的分辨率，该极限由单光子衍射图样定义。减少的起因被证明是由两光子概率振幅的干扰引起的。通过比较一阶和二阶衍射，可以看出传统的一阶概念具有相干性限制的部分相干和一阶相干具有部分纠缠的二阶类似物，其限制对应于两个纠缠光子（“纠缠双光子”）和两个克隆光子（“克隆双光子”），后者是二阶相干。克隆双光子的概念扩展到了$ñ$ 克隆的光子，导致 $\mathrm{1个}/ñ$降低衍射极限。在极限$ñ$三次相干，所有光子在 $ñ$如图所示，三阶集合态在粒子状轨迹上传播，从而再现了零阶光线图像。根据量子力学的线性和海森堡的空间-动量不确定性原理讨论了这些结果。发展了基于光子密度的相干性的一般概念，其一阶等效于常规的基于波的图片。