光谱相干性,第一部分:无源谐振器线宽,基本激光线宽,和肖洛-汤斯近似

IF 7.4 1区 物理与天体物理 Q1 ENGINEERING, ELECTRICAL & ELECTRONIC
Markus Pollnau , Marc Eichhorn
{"title":"光谱相干性,第一部分:无源谐振器线宽,基本激光线宽,和肖洛-汤斯近似","authors":"Markus Pollnau ,&nbsp;Marc Eichhorn","doi":"10.1016/j.pquantelec.2020.100255","DOIUrl":null,"url":null,"abstract":"<div><p>The degree of spectral coherence characterizes the spectral purity of light. It can be equivalently expressed in the time domain by the decay time <em>τ</em> or the quality factor <em>Q</em><span> of the light-emitting oscillator, the coherence time </span><em>τ</em> <sup><em>coh</em></sup> or length <span><math><mi>ℓ</mi></math></span><sup><em>coh</em></sup><span> of emitted light or, via Fourier transformation to the frequency domain, the linewidth Δ</span><em>ν</em><span><span><span> of emitted light. We quantify these parameters for the reference situation of a passive Fabry-Pérot resonator. We investigate its </span>spectral line shapes, mode profiles, and Airy distributions and verify that the sum of all mode profiles generates the corresponding Airy distribution. The Fabry-Pérot 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-Pérot 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 </span>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 </span><em>τ</em><sub><em>c</em></sub> 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.</p></div>","PeriodicalId":414,"journal":{"name":"Progress in Quantum Electronics","volume":"72 ","pages":"Article 100255"},"PeriodicalIF":7.4000,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.pquantelec.2020.100255","citationCount":"30","resultStr":"{\"title\":\"Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation\",\"authors\":\"Markus Pollnau ,&nbsp;Marc Eichhorn\",\"doi\":\"10.1016/j.pquantelec.2020.100255\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>The degree of spectral coherence characterizes the spectral purity of light. It can be equivalently expressed in the time domain by the decay time <em>τ</em> or the quality factor <em>Q</em><span> of the light-emitting oscillator, the coherence time </span><em>τ</em> <sup><em>coh</em></sup> or length <span><math><mi>ℓ</mi></math></span><sup><em>coh</em></sup><span> of emitted light or, via Fourier transformation to the frequency domain, the linewidth Δ</span><em>ν</em><span><span><span> of emitted light. We quantify these parameters for the reference situation of a passive Fabry-Pérot resonator. We investigate its </span>spectral line shapes, mode profiles, and Airy distributions and verify that the sum of all mode profiles generates the corresponding Airy distribution. The Fabry-Pérot 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-Pérot 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 </span>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 </span><em>τ</em><sub><em>c</em></sub> 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.</p></div>\",\"PeriodicalId\":414,\"journal\":{\"name\":\"Progress in Quantum Electronics\",\"volume\":\"72 \",\"pages\":\"Article 100255\"},\"PeriodicalIF\":7.4000,\"publicationDate\":\"2020-08-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1016/j.pquantelec.2020.100255\",\"citationCount\":\"30\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Progress in Quantum Electronics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0079672720300094\",\"RegionNum\":1,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Progress in Quantum Electronics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0079672720300094","RegionNum":1,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 30

摘要

光谱的相干度表征了光的光谱纯度。它可以在时域中等效地表示为发光振荡器的衰减时间τ或质量因子Q,发射光的相干时间τ coh或长度r coh,或者通过在频域的傅里叶变换表示为发射光的线宽Δν。我们将这些参数量化为无源法布里-普氏谐振器的参考情况。我们研究了它的谱线形状、模式分布和Airy分布,并验证了所有模式分布的总和产生相应的Airy分布。法布里-帕姆罗特谐振器被描述为,一个振荡器,通过它的洛伦兹线宽和精细度,作为一个扫描光谱仪,通过它的艾里线宽和精细度。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。对法布里-帕姆罗特谐振腔内原子发射的研究、洛伦兹振子模型、Kramers-Kronig关系、幅相图和量子化电场的总和一致表明,受激光和自发光的发射发生在入射场的前导相位为90°。这些发现对量子光学图像提出了质疑,首先,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角上,其次,激光线宽是由自发发射引起的幅度和相位波动引起的。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:(i)它是一个真正的连续波(cw)激光器,(ii)它是一个理想的四能级激光器,(iii)它的谐振腔没有本征损耗,以及(iv)一个光子自发耦合到激光模式每个光子衰减时间τc谐振腔,独立于泵浦速率。在讨论了现有的半经典和量子光学描述激光线宽的不一致性之后,我们引入了光谱相干系数,它量化了主动模式下与底层被动模式下的光谱相干性,并基于增益延长光子衰减时间和收窄线宽的原理,推导出了半经典的单激光模式的基本线宽。该线宽适用于任意能级系统的激光器,在连续波或瞬态激光状态下工作,低于、等于或高于阈值,增益小于、等于或大于损耗。通过应用近似(i)-(iv),我们再现了原始的Schawlow-Townes方程。它提供了迄今为止在激光作为自发发射放大器的描述和肖洛-汤斯方程之间缺失的联系。自发发射要求在连续波激光模式下,增益小于损耗。我们还验证了基于密度算子主方程的激光线宽的量子光学方法中,增益小于损耗。最后,我们给出了坚果壳中激光线宽的推导。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Spectral coherence, Part I: Passive-resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation

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 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-Pérot 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-Pérot 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-Pérot 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.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
Progress in Quantum Electronics
Progress in Quantum Electronics 工程技术-工程:电子与电气
CiteScore
18.50
自引率
0.00%
发文量
23
审稿时长
150 days
期刊介绍: Progress in Quantum Electronics, established in 1969, is an esteemed international review journal dedicated to sharing cutting-edge topics in quantum electronics and its applications. The journal disseminates papers covering theoretical and experimental aspects of contemporary research, including advances in physics, technology, and engineering relevant to quantum electronics. It also encourages interdisciplinary research, welcoming papers that contribute new knowledge in areas such as bio and nano-related work.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信