{"title":"经典瑞利长度定义的解析解,包括在任意值处的截断。","authors":"Aufried Lenferink, Cees Otto","doi":"10.1111/jmi.70016","DOIUrl":null,"url":null,"abstract":"<p>We present the analytical solution to the diffraction integral that describes the Rayleigh length for a focused Gaussian beam with any value of a spherical truncating aperture. This exact solution is in precise agreement with numerical calculations for the light distribution in the near focal area. The solution arises under assumption of the paraxial approximation, which also provides the basis for the classical Rayleigh length definition. It will be shown that the non-paraxial regime can be included by adding an empirical term (<i>C</i><sub>np</sub>) to the solution of the diffraction integral. This extends the validity of the expression to high numerical apertures (<i>NA</i>) up to <i>n</i> times 0.95, with <i>n</i> being the refractive index of the immersion medium. Thus, the entire practical range of NA, encountered in optical microscopy, is covered with a calculated error of less than 0.4% in the non-paraxial limit. This theoretical result is important in the design of optical instrumentation, where overall light efficiency in excitation and detection and spatial resolution must be optimised together.</p>","PeriodicalId":16484,"journal":{"name":"Journal of microscopy","volume":"300 1","pages":"124-133"},"PeriodicalIF":1.9000,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jmi.70016","citationCount":"0","resultStr":"{\"title\":\"Analytical solution of the classical Rayleigh length definition, including truncation at arbitrary values\",\"authors\":\"Aufried Lenferink, Cees Otto\",\"doi\":\"10.1111/jmi.70016\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We present the analytical solution to the diffraction integral that describes the Rayleigh length for a focused Gaussian beam with any value of a spherical truncating aperture. This exact solution is in precise agreement with numerical calculations for the light distribution in the near focal area. The solution arises under assumption of the paraxial approximation, which also provides the basis for the classical Rayleigh length definition. It will be shown that the non-paraxial regime can be included by adding an empirical term (<i>C</i><sub>np</sub>) to the solution of the diffraction integral. This extends the validity of the expression to high numerical apertures (<i>NA</i>) up to <i>n</i> times 0.95, with <i>n</i> being the refractive index of the immersion medium. Thus, the entire practical range of NA, encountered in optical microscopy, is covered with a calculated error of less than 0.4% in the non-paraxial limit. This theoretical result is important in the design of optical instrumentation, where overall light efficiency in excitation and detection and spatial resolution must be optimised together.</p>\",\"PeriodicalId\":16484,\"journal\":{\"name\":\"Journal of microscopy\",\"volume\":\"300 1\",\"pages\":\"124-133\"},\"PeriodicalIF\":1.9000,\"publicationDate\":\"2025-07-31\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jmi.70016\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of microscopy\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/jmi.70016\",\"RegionNum\":4,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MICROSCOPY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of microscopy","FirstCategoryId":"5","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/jmi.70016","RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MICROSCOPY","Score":null,"Total":0}
Analytical solution of the classical Rayleigh length definition, including truncation at arbitrary values
We present the analytical solution to the diffraction integral that describes the Rayleigh length for a focused Gaussian beam with any value of a spherical truncating aperture. This exact solution is in precise agreement with numerical calculations for the light distribution in the near focal area. The solution arises under assumption of the paraxial approximation, which also provides the basis for the classical Rayleigh length definition. It will be shown that the non-paraxial regime can be included by adding an empirical term (Cnp) to the solution of the diffraction integral. This extends the validity of the expression to high numerical apertures (NA) up to n times 0.95, with n being the refractive index of the immersion medium. Thus, the entire practical range of NA, encountered in optical microscopy, is covered with a calculated error of less than 0.4% in the non-paraxial limit. This theoretical result is important in the design of optical instrumentation, where overall light efficiency in excitation and detection and spatial resolution must be optimised together.
期刊介绍:
The Journal of Microscopy is the oldest journal dedicated to the science of microscopy and the only peer-reviewed publication of the Royal Microscopical Society. It publishes papers that report on the very latest developments in microscopy such as advances in microscopy techniques or novel areas of application. The Journal does not seek to publish routine applications of microscopy or specimen preparation even though the submission may otherwise have a high scientific merit.
The scope covers research in the physical and biological sciences and covers imaging methods using light, electrons, X-rays and other radiations as well as atomic force and near field techniques. Interdisciplinary research is welcome. Papers pertaining to microscopy are also welcomed on optical theory, spectroscopy, novel specimen preparation and manipulation methods and image recording, processing and analysis including dynamic analysis of living specimens.
Publication types include full papers, hot topic fast tracked communications and review articles. Authors considering submitting a review article should contact the editorial office first.