{"id":9651,"date":"2020-09-09T11:00:42","date_gmt":"2020-09-09T11:00:42","guid":{"rendered":"https:\/\/www.ophiropt.com\/blog\/laser-measurement\/?p=9651"},"modified":"2026-03-15T14:16:05","modified_gmt":"2026-03-15T14:16:05","slug":"the-focal-spot-size-calculator","status":"publish","type":"post","link":"https:\/\/www.ophiropt.com\/blog\/the-focal-spot-size-calculator\/","title":{"rendered":"The Focal Spot Size Calculator"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\"><strong>Why the Thin Lens Formula Fails for Gaussian Beams<\/strong><\/h2>\n\n\n\n<p>It is not uncommon in working with <a href=\"https:\/\/www.ophiropt.com\/en\/\">lasers<\/a> to need to focus the laser beam down. <\/p>\n\n\n\n<p>When using a lens, our first impulse is to fall back on the thin\nlens equation we learned in geometric optics. <\/p>\n\n\n\n<p>In fact, that equation is applicable to spherical waves in the paraxial\napproximation. It is not reliable for Gaussian beams, especially when working in\nthe near field (defined as distance from the waist, z &lt; z<sub>R<\/sub>, the\nRayleigh length, to be defined below). In fact, the location of the new waist\nwill be shifted relative to that expected from using the geometrical thin lens\nequation.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Key Parameters of a Gaussian Laser Beam<\/strong><\/h2>\n\n\n\n<p>A spherical TEM00 mode Gaussian beam is characterized by two\nparameters, e.g. the waist, w<sub>0<\/sub>, and the divergence, q<sub>FF<\/sub>. Alternately, the Rayleigh length, z<sub>R<\/sub>, (AKA Rayleigh\nrange) or the wavelength, l, can be substituted for one or both of the above. The Rayleigh\nlength is defined as <\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img decoding=\"async\" width=\"70\" height=\"54\" data-src=\"\/blog\/wp-content\/uploads\/2019\/12\/pic1.png\" alt=\"\" class=\"wp-image-9653 lazyload\" data-srcset=\"https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic1.png 70w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic1-24x19.png 24w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic1-36x28.png 36w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic1-48x37.png 48w\" data-sizes=\"(max-width: 70px) 100vw, 70px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 70px; --smush-placeholder-aspect-ratio: 70\/54;\" \/><\/figure>\n\n\n\n<p>where l is the wavelength and  w<sub>0<\/sub> is the beam waist. <\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Using the Focal Spot Size Calculator<\/strong><\/h2>\n\n\n\n<p>There are matrix methods that can be used for propagating the diameter and curvature of the beam, but if your main interest is simply the location of the new focus (i.e. beam waist) or the size of the focal spot, you can use the <a href=\"https:\/\/ophiropt.com\/en\/laser-focal-spot-calculator\">Focal Spot Size Calculator<\/a> tool on the Ophir website.<\/p>\n\n\n\n<p>This tool uses the equations derived by Sidney Self which take a form reminiscent of the geometric optics (Gaussian) thin lens equation. [SA Self, Applied Optics <strong>22<\/strong>(5), 1983, pg 658-661] In normalized form, the thin lens Self equation is<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter\"><img decoding=\"async\" width=\"278\" height=\"92\" data-src=\"\/blog\/wp-content\/uploads\/2019\/12\/pic2.png\" alt=\"\" class=\"wp-image-9654 lazyload\" data-srcset=\"https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic2.png 278w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic2-24x8.png 24w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic2-36x12.png 36w, https:\/\/www.ophiropt.com\/blog\/wp-content\/uploads\/2019\/12\/pic2-48x16.png 48w\" data-sizes=\"(max-width: 278px) 100vw, 278px\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" style=\"--smush-placeholder-width: 278px; --smush-placeholder-aspect-ratio: 278\/92;\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>Rayleigh Range and Beam Waist Explained<\/strong><\/h2>\n\n\n\n<p>where s is the object distance, s&#8217; is the image\ndistance, f is the focal length of the lens and z<sub>R<\/sub> is the Rayleigh\nrange of the laser beam. Object and image distances refer to the locations of\nthe beam waist before and after the lens.<\/p>\n\n\n\n<p>Several important insights can be gleaned by analyzing\nthis equation.<\/p>\n\n\n\n<p>The first is that it approaches the geometrical optics\nthin lens equation in the limit (z<sub>R<\/sub>\/f)\u00e00.<\/p>\n\n\n\n<p>On the other hand, a beam with a waist at the front\nfocal plane of the lens will reach a new waist at the back focal plane,\nindependent of the Rayleigh length. This is very different from the geometrical\noptics case in which an object at the front focal plane is imaged at infinity. Conversely,\nfor object distances that are not exactly at the front focal plane but near it,\nthe image distance is strongly dependent on the Rayleigh length.<\/p>\n\n\n\n<p>In fact, there is no object distance that produces a\nbeam waist at infinity. For a positive lens, the waist will always be imaged at\na finite distance beyond the lens (except for a small range of parameters in\nwhich the image distance is negative \u2013 but still finite). <\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><strong>What the Calculator Reveals About Beam Focusing<\/strong><\/h2>\n\n\n\n<p>For a positive lens and real object, the maximum image\ndistance, s&#8217; = &nbsp;f + f<sup>2<\/sup>\/2z<sub>R<\/sub>\nis achieved at an object distance of s = f + z<sub>R<\/sub>. The minimum image\ndistance s&#8217; =&nbsp; f &#8211; f<sup>2<\/sup>\/2z<sub>R<\/sub>\noccurs for s = f &#8211; z<sub>R<\/sub>.<\/p>\n\n\n\n<p>While a more detailed analysis, which can be found in Self&#8217;s beautiful 1983 article, is insightful and rewarding, practitioners needing to save time are invited to use Ophir&#8217;s <a href=\"https:\/\/www.ophiropt.com\/en\/laser-focal-spot-calculator\">Focal Spot Size Calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>There are matrix methods that can be used for propagating the diameter and curvature of the beam, but if your main interest is simply the location of the new focus (i.e. beam waist) or the size of the focal spot, you can use the Focal Spot Size Calculator tool on the Ophir website.  Dr. Ivker Our expert &#8211; will explain you how<\/p>\n","protected":false},"author":8,"featured_media":9357,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center 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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[380,5],"tags":[27],"ophirposttype":[349],"class_list":["post-9651","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-focal-spot-size-calculator","category-laser-power-energy-measurement","tag-focal-spot","ophirposttype-blogpost-en"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Laser Focal Spot Size Calculator | Ophir Photonics<\/title>\n<meta name=\"description\" content=\"Calculate the focal spot size and waist location of a Gaussian laser beam using Ophir\u2019s focal spot size calculator.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link 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