The nature of light, extending from the optical to the x-ray regime, is reviewed from a diffraction point of view by comparing field-based statistical optics and photon-based quantum optics approaches. The topic is introduced by comparing historical diffraction concepts based on wave interference, Dirac’s notion of photon self-interference, Feynman’s interference of space–time photon probability amplitudes, and Glauber’s formulation of coherence functions based on photon detection. The concepts are elucidated by a review of how the semiclassical combination of the disparate photon and wave concepts have been used to describe light creation, diffraction, and detection. The origin of the fundamental diffraction limit is then discussed in both wave and photon pictures. By use of Feynman’s concept of probability amplitudes associated with independent photons, we show that quantum electrodynamics, the complete theory of light, reduces in lowest order to the conventional wave formalism of diffraction. As an introduction to multi-photon effects, we then review fundamental one- and two-photon experiments and detection schemes, in particular the seminal Hanbury Brown–Twiss experiment. The formal discourse of the paper starts with a treatment of first-order coherence theory. In first order, the statistical optics and quantum optics formulations of coherence are shown to be equivalent. This is elucidated by a discussion of Zernike’s powerful theorem of partial coherence propagation, a cornerstone of statistical optics, followed by its quantum derivation based on the interference of single-photon probability amplitudes. The treatment is then extended to second-order coherence theory, where the equivalence of wave and particle descriptions is shown to break down. This is illustrated by considering two photons whose space–time probability amplitudes are correlated through nonlinear birth processes, resulting in entanglement or cloning. In both cases, the two-photon diffraction patterns are shown to exhibit resolution below the conventional diffraction limit, defined by the one-photon diffraction patterns. The origin of the reduction is shown to arise from the interference of two-photon probability amplitudes. By comparing first- and second-order diffraction, it is shown that the conventional first-order concept of partial coherence with its limits of chaoticity and first-order coherence has the second-order analogue of partial entanglement, with its limits corresponding to two entangled photons (“entangled biphotons”) and two cloned photons (“cloned biphotons”), the latter being second-order coherent. The concept of cloned biphotons is extended to the case of n cloned photons, resulting in a 1/n reduction of the diffraction limit. In the limit of nth-order coherence, all photons within the nth-order collective state are shown to propagate on particle like trajectories, reproducing the 0th-order ray-optics picture. These results are discussed in terms of the linearity of quantum mechanics and Heisenberg’s space–momentum uncertainty principle. 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和基于光子的量子光学方法，从衍射的角度回顾了从光学到 X 射线范围的光的性质。通过比较基于波干涉的历史衍射概念、狄拉克的光子自干涉概念、费曼的时空光子概率幅度干涉以及格劳伯基于光子检测的相干函数公式，介绍了该主题。通过回顾如何使用不同的光子和波概念的半经典组合来描述光的产生、衍射和检测来阐明这些概念。然后在波和光子图片中讨论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幅度概念，我们表明量子电动力学，光的完整理论，在最低阶上简化为传统的衍射波形式。作为多光子效应的介绍，我们随后回顾了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。论文的正式论述始于对一阶相干理论的处理。首先，相干的统计光学和量子光学公式被证明是等效的。通过讨论 Zernike 的部分相干传播的强大定理（统计光学的基石），以及其基于单光子概率幅度干涉的量子推导，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打破。这可以通过考虑两个光子来说明，它们的时空概率幅度通过非线性出生过程相关，导致纠缠或克隆。在这两种情况下，双光子衍射图案的分辨率都低于由单光子衍射图案定义的常规衍射极限。显示减少的起源是由双光子概率幅度的干扰引起的。通过比较一阶和二阶衍射，表明具有混沌和一阶相干极限的常规一阶部分相干概念具有部分纠缠的二阶类似物，其极限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”），后者是二阶相干的。克隆双光子的概念扩展到 n 个克隆光子的情况，导致衍射极限降低 1/n。在 n 阶相干的限制下，所有处于 n 阶集体状态的光子都表现出在粒子状轨迹上传播，再现了 0 阶射线光学图像。这些结果根据量子力学的线性和海森堡的空间-动量不确定性原理进行了讨论。发展了基于光子密度的相干性的一般概念，其一阶等效于传统的基于波的图片。后者是二阶相干的。克隆双光子的概念扩展到 n 个克隆光子的情况，导致衍射极限降低 1/n。在 n 阶相干的限制下，所有处于 n 阶集体状态的光子都表现出在粒子状轨迹上传播，再现了 0 阶射线光学图像。这些结果根据量子力学的线性和海森堡的空间-动量不确定性原理进行了讨论。发展了基于光子密度的相干性的一般概念，其一阶等效于传统的基于波的图片。后者是二阶相干的。克隆双光子的概念扩展到 n 个克隆光子的情况，导致衍射极限降低 1/n。在 n 阶相干的限制下，所有处于 n 阶集体状态的光子都表现出在粒子状轨迹上传播，再现了 0 阶射线光学图像。这些结果根据量子力学的线性和海森堡的空间-动量不确定性原理进行了讨论。发展了基于光子密度的相干性的一般概念，其一阶等效于传统的基于波的图片。n 阶集体状态内的所有光子都显示为在粒子状轨迹上传播，再现了 0 阶射线光学图片。这些结果根据量子力学的线性和海森堡的空间-动量不确定性原理进行了讨论。发展了基于光子密度的相干性的一般概念，其一阶等效于传统的基于波的图片。n 阶集体状态内的所有光子都显示为在粒子状轨迹上传播，再现了 0 阶射线光学图片。这些结果根据量子力学的线性和海森堡的空间-动量不确定性原理进行了讨论。发展了基于光子密度的相干性的一般概念，其一阶等效于传统的基于波的图片。