Abstract The degree of spectral coherence characterizes the spectral purity of light. It can be equivalently expressed in the time domain by the decay time τ or the quality factor Q of the light-emitting oscillator, the coherence time τ coh or length l coh of emitted light or, via Fourier transformation to the frequency domain, the linewidth Δν of emitted light. We quantify these parameters for the reference situation of a passive Fabry-Perot resonator. We investigate its spectral line shapes, mode profiles, and Airy distributions and verify that the sum of all mode profiles generates the corresponding Airy distribution. The Fabry-Perot resonator is described, as an oscillator, by its Lorentzian linewidth and finesse and, as a scanning spectrometer, by its Airy linewidth and finesse. Furthermore, stimulated and spontaneous emission are analyzed semi-classically by employing Maxwell′s equations and the law of energy conservation. Investigation of emission by atoms inside a Fabry-Perot resonator, the Lorentz oscillator model, the Kramers-Kronig relations, the amplitude-phase diagram, and the summation of quantized electric fields consistently suggests that stimulated and spontaneous emission of light occur with a phase 90° in lead of the incident field. These findings question the quantum-optical picture, which proposed, firstly, that stimulated emission occurred in phase, whereas spontaneous emission occurred at an arbitrary phase angle with respect to the incident field and, secondly, that the laser linewidth were due to amplitude and phase fluctuations induced by spontaneous emission. We emphasize that the first derivation of the Schawlow-Townes laser linewidth was entirely semi-classical but included the four approximations that (i) it is a truly continuous-wave (cw) laser, (ii) it is an ideal four-level laser, (iii) its resonator exhibits no intrinsic losses, and (iv) one photon is coupled spontaneously into the lasing mode per photon-decay time τc of the resonator, independent of the pump rate. After discussing the inconsistencies of existing semi-classical and quantum-optical descriptions of the laser linewidth, we introduce the spectral-coherence factor, which quantifies spectral coherence in an active compared to its underlying passive mode, and derive semi-classically, based on the principle that the gain elongates the photon-decay time and narrows the linewidth, the fundamental linewidth of a single lasing mode. This linewidth is valid for lasers with an arbitrary energy-level system, operating below, at, or above threshold and in a cw or a transient lasing regime, with the gain being smaller, equal, or larger compared to the losses. By applying approximations (i)-(iv) we reproduce the original Schawlow-Townes equation. It provides the hitherto missing link between the description of the laser as an amplifier of spontaneous emission and the Schawlow-Townes equation. Spontaneous emission entails that in a cw lasing mode the gain is smaller than the losses. We verify that also in the quantum-optical approaches to the laser linewidth, based on the density-operator master equation, the gain is smaller than the losses. We conclude this work by presenting the derivation of the laser linewidth in a nut shell.

中文翻译：

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光谱相干性，第 I 部分：无源谐振腔线宽、基本激光线宽和 Schawlow-Townes 近似

摘要 光谱相干程度表征光的光谱纯度。它可以在时域中用衰减时间 τ 或发光振荡器的品质因数 Q、发射光的相干时间 τ coh 或长度 l coh 或通过傅立叶变换到频域的线宽等价表示发射光的Δν。我们针对无源法布里-珀罗谐振器的参考情况量化这些参数。我们研究了它的谱线形状、模式分布和艾里分布，并验证所有模式分布的总和生成相应的艾里分布。法布里-珀罗谐振器被描述为振荡器，其洛伦兹线宽和精细度以及作为扫描光谱仪的艾里线宽和精细度。此外，利用麦克斯韦方程组和能量守恒定律，对受激辐射和自发辐射进行了半经典分析。对 Fabry-Perot 谐振器内原子发射的研究、洛伦兹振荡器模型、Kramers-Kronig 关系、幅相图和量子化电场的总和一致表明光的受激和自发发射发生在 90 相° 在入射场的领先。这些发现质疑量子光学图像，首先提出，受激发射发生在相位上，而自发发射发生在相对于入射场的任意相位角上，其次，激光线宽是由振幅和相位引起的自发辐射引起的波动。我们强调 Schawlow-Townes 激光线宽的第一个推导完全是半经典的，但包括四个近似值：(i) 它是真正的连续波 (cw) 激光器，(ii) 它是理想的四能级激光器，（iii）其谐振器没有固有损耗，并且（iv）一个光子自发耦合到谐振器的每个光子衰减时间 τc 中的激光模式，与泵浦速率无关。在讨论了激光线宽的现有半经典和量子光学描述的不一致之后，我们引入了光谱相干因子，该因子量化了主动模式与其潜在被动模式相比的光谱相干性，并基于以下半经典推导：增益会延长光子衰减时间并缩小线宽，即单激光模式的基本线宽。该线宽适用于具有任意能级系统的激光器，在阈值以下、等于或高于阈值以及在 cw 或瞬态激光状态下工作，与损耗相比增益更小、相等或更大。通过应用近似 (i)-(iv)，我们重现了原始的 Schawlow-Townes 方程。它提供了迄今为止将激光描述为自发辐射放大器与 Schawlow-Townes 方程之间缺失的联系。自发发射意味着在连续激光模式中，增益小于损耗。我们也验证了在激光线宽的量子光学方法中，基于密度算子主方程，增益小于损耗。我们通过简要介绍激光线宽的推导来结束这项工作。