{"id":66977,"date":"2026-10-13T10:45:03","date_gmt":"2026-10-13T07:45:03","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=66977"},"modified":"2026-10-13T10:45:03","modified_gmt":"2026-10-13T07:45:03","slug":"maluss-law-calculator","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/physics\/maluss-law-calculator\/","title":{"rendered":"Malus's Law Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Find How Much Polarized Light Passes Through an Analyzer<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"66977\" data-favorite-title=\"Malus&#039;s Law\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/physics\/maluss-law-calculator\/\" data-favorite-excerpt=\"Calculate how much linearly polarized light passes through a polarizer using Malus&#039;s law. Enter the incident intensity and the angle between the polarization direction and the analyzer axis.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-spectrum\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/physics\/maluss-law-calculator\/\" method=\"POST\" class=\"calculator\" id=\"calculator-66977\" data-post=\"66977\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">Incident intensity (I\u2080)<\/label><div class=\"term\">I\u2080<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"text\" name=\"incident_intensity_i\" id=\"field-1\" aria-describedby=\"calc-info-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"e.g. 100\" min=\"0\" value=\"100\"\/><div class=\"grid grid-cols-1\"><select id=\"field-2\" name=\"unit\" aria-label=\"Unit\"><option value=\"wm2\" selected>W\/m\u00b2<\/option><option value=\"mwcm2\">mW\/cm\u00b2<\/option><option value=\"wcm2\">W\/cm\u00b2<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-1\">Intensity of the linearly polarized light before the analyzer. One sunny-day sky reflection is roughly 20\u2013100 W\/m\u00b2.<\/small><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-3\">Angle between axes (\u03b8)<\/label><div class=\"term\">\u03b8<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><div class=\"field-group\"><input type=\"number\" name=\"angle_between_axes_\" id=\"field-3\" aria-describedby=\"calc-info-3\" placeholder=\"e.g. 30\" step=\"any\" value=\"30\"\/><div class=\"grid grid-cols-1\"><select id=\"field-4\" name=\"unit\" aria-label=\"Unit\"><option value=\"deg\" selected>degrees<\/option><option value=\"rad\">radians<\/option><\/select><svg class=\"pointer-events-none col-start-1 row-start-1 mr-2 size-5 self-center justify-self-end text-gray-500 sm:size-4\" viewBox=\"0 0 16 16\" fill=\"currentColor\" aria-hidden=\"true\" data-slot=\"icon\">\n              <path fill-rule=\"evenodd\" d=\"M4.22 6.22a.75.75 0 0 1 1.06 0L8 8.94l2.72-2.72a.75.75 0 1 1 1.06 1.06l-3.25 3.25a.75.75 0 0 1-1.06 0L4.22 7.28a.75.75 0 0 1 0-1.06Z\" clip-rule=\"evenodd\" \/>\n            <\/svg><\/div><\/div><small id=\"calc-info-3\">Angle between the light&#039;s polarization direction and the analyzer&#039;s transmission axis. 0\u00b0 = fully aligned (all light passes), 90\u00b0 = crossed polarizers (no light passes).<\/small><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"66977\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Malus&#039;s Law Results<\/h3><div class=\"result-box\"><div class=\"field-result is-column\" id=\"output-1\"><span class=\"field-title\"><span>Transmitted light intensity<\/span><span class=\"result-term\">I<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><div class=\"field-result is-column\" id=\"output-2\"><span class=\"field-title\"><span>Transmission fraction and details<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Copy link to this calculation<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"66977\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Malus's Law Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ I = I_0 \\cos^2\\theta \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(I_0\\) \u2014 intensity of the linearly polarized light incident on the analyzer (W\/m\u00b2)<\/li>\n<li>\\(I\\) \u2014 transmitted intensity after the analyzer (W\/m\u00b2)<\/li>\n<li>\\(\\theta\\) \u2014 angle between the light's polarization direction and the analyzer's transmission axis<\/li>\n<\/ul>\n<p>The law states that a polarizer transmits only the component of the electric field parallel to its transmission axis, so the transmitted intensity scales with the square of the cosine of the angle between the two directions.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Malus's Law \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> linearly polarized light with an intensity of 100 W\/m\u00b2 passes through an analyzer whose transmission axis is rotated 30\u00b0 relative to the light's polarization direction.<\/p>\n<p>\\[ I = I_0 \\cos^2\\theta = 100 \\times \\cos^2(30^\\circ) \\]<\/p>\n<p>\\[ \\cos(30^\\circ) \\approx 0.8660, \\quad \\cos^2(30^\\circ) = 0.75 \\]<\/p>\n<p>\\[ I = 100 \\times 0.75 = 75 \\; \\text{W\/m}^2 \\]<\/p>\n<p><strong>Result:<\/strong> the analyzer transmits 75 W\/m\u00b2, which is 75 % of the incident light; the remaining 25 W\/m\u00b2 is absorbed or blocked.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter the <strong>Incident intensity (I\u2080)<\/strong> \u2014 the intensity of the linearly polarized light before the analyzer \u2014 and choose its unit: W\/m\u00b2, mW\/cm\u00b2, or W\/cm\u00b2.<\/li>\n<li>Enter the <strong>Angle between axes (\u03b8)<\/strong> \u2014 the angle between the light's polarization direction and the analyzer's transmission axis \u2014 in degrees or radians. You can enter negative or values larger than 360\u00b0; only the cosine matters.<\/li>\n<li>Read the <strong>Transmitted light intensity<\/strong>, shown in all three intensity units, and the <strong>Transmission fraction and details<\/strong>, including cos \u03b8, cos\u00b2\u03b8, and the blocked intensity.<\/li>\n<\/ol>\n<h2>What is Malus's law?<\/h2>\n<p>Malus's law, published by \u00c9tienne-Louis Malus in 1809, describes how much linearly polarized light passes through a polarizing filter (often called an <em>analyzer<\/em> when it is used to examine polarized light). Light is an electromagnetic wave, and its electric field oscillates in a direction perpendicular to the direction of travel. In linearly polarized light, that oscillation happens along one fixed plane.<\/p>\n<p>A polarizer only transmits the component of the electric field that lies along its transmission axis. If the light's polarization direction makes an angle \u03b8 with that axis, the field component that survives is \\(E_0 \\cos\\theta\\). Because intensity is proportional to the square of the field amplitude, the transmitted intensity becomes \\(I_0 \\cos^2\\theta\\) \u2014 Malus's law.<\/p>\n<p>The angle dependence has some important special cases. At \u03b8 = 0\u00b0 the axes are aligned and all the light passes through. At \u03b8 = 90\u00b0 the polarizers are <em>crossed<\/em> and, ideally, no light passes at all \u2014 this is the principle behind adjustable sunglasses and LCD shutters. At \u03b8 = 45\u00b0 exactly half the intensity is transmitted. Because of the squared cosine, the transmission is also periodic with a period of 180\u00b0, so 30\u00b0 and 150\u00b0 give the same result.<\/p>\n<p>Malus's law is central to polarization optics and appears in physics courses alongside other wave phenomena such as <a href=\"https:\/\/wpcalc.com\/en\/physics\/double-slit-interference-calculator\/\">double-slit interference<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/physics\/diffraction-grating-calculator\/\">diffraction gratings<\/a>. If you work with intensity on a logarithmic scale, the related <a href=\"https:\/\/wpcalc.com\/en\/physics\/sound-intensity-level-calculator\/\">sound intensity level calculator<\/a> converts between ratios and decibels.<\/p>\n<h2>Assumptions, limitations, and practical tips<\/h2>\n<ul>\n<li><strong>Perfect polarizer:<\/strong> the law assumes an ideal analyzer that transmits 100 % of light polarized along its axis and blocks 100 % perpendicular to it. Real polarizers absorb 10\u201320 % of the aligned component, so measured values are somewhat lower than the ideal result.<\/li>\n<li><strong>Fully polarized input:<\/strong> the incident light must be linearly polarized. If unpolarized light of intensity \\(I_0\\) enters a polarizer, exactly half is transmitted on average (\\(I = I_0\/2\\)), regardless of the angle \u2014 you can then apply Malus's law to that first polarizer's output.<\/li>\n<li><strong>Energy conservation:<\/strong> the blocked component is not lost; it is absorbed as heat (as in Polaroid film) or deflected (as in polarizing beam splitters). The calculator reports this blocked intensity as \\(I_0 - I\\).<\/li>\n<li><strong>Angle periodicity:<\/strong> since \\(\\cos^2\\theta\\) has a period of 180\u00b0, any angle can be reduced modulo 180\u00b0 without changing the result. Angles above 90\u00b0 give the same transmission as their supplement (e.g. 120\u00b0 behaves like 60\u00b0).<\/li>\n<li><strong>Partial polarization:<\/strong> for partially polarized light, treat the input as the sum of a polarized part (which follows Malus's law) and an unpolarized part (which always contributes half of its intensity).<\/li>\n<\/ul>\n<h2>Reference table: transmission vs. angle<\/h2>\n<table>\n<thead>\n<tr>\n<th>Angle \u03b8 (degrees)<\/th>\n<th>cos\u00b2\u03b8<\/th>\n<th>Transmitted fraction<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>0\u00b0<\/td>\n<td>1.000<\/td>\n<td>100 %<\/td>\n<\/tr>\n<tr>\n<td>15\u00b0<\/td>\n<td>0.933<\/td>\n<td>93.3 %<\/td>\n<\/tr>\n<tr>\n<td>30\u00b0<\/td>\n<td>0.750<\/td>\n<td>75 %<\/td>\n<\/tr>\n<tr>\n<td>45\u00b0<\/td>\n<td>0.500<\/td>\n<td>50 %<\/td>\n<\/tr>\n<tr>\n<td>60\u00b0<\/td>\n<td>0.250<\/td>\n<td>25 %<\/td>\n<\/tr>\n<tr>\n<td>75\u00b0<\/td>\n<td>0.067<\/td>\n<td>6.7 %<\/td>\n<\/tr>\n<tr>\n<td>90\u00b0<\/td>\n<td>0.000<\/td>\n<td>0 %<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Use this table to sanity-check your calculator inputs: the transmitted fraction should drop smoothly from 100 % to 0 % as \u03b8 goes from 0\u00b0 to 90\u00b0.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate how much linearly polarized light passes through a polarizer using Malus's law. Enter the incident intensity and the angle between the polarization direction and the analyzer axis.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[62],"tags":[87],"class_list":["post-66977","post","type-post","status-publish","format-standard","hentry","category-physics","tag-waves-optics"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66977","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=66977"}],"version-history":[{"count":1,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66977\/revisions"}],"predecessor-version":[{"id":66978,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/66977\/revisions\/66978"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=66977"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=66977"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=66977"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}