{"id":3648,"date":"2026-10-23T09:37:00","date_gmt":"2026-10-23T01:37:00","guid":{"rendered":"http:\/\/www.agrctmachinefmco.com\/blog\/?p=3648"},"modified":"2026-10-23T09:37:00","modified_gmt":"2026-10-23T01:37:00","slug":"how-to-calculate-the-dispersion-of-a-plane-ruled-grating-49c1-c6e12b","status":"publish","type":"post","link":"http:\/\/www.agrctmachinefmco.com\/blog\/2026\/10\/23\/how-to-calculate-the-dispersion-of-a-plane-ruled-grating-49c1-c6e12b\/","title":{"rendered":"How to calculate the dispersion of a plane ruled grating?"},"content":{"rendered":"

If you\u2019ve ever stared at a prism splitting sunlight into a rainbow, you\u2019ve seen wavelength-dependent light dispersion in action\u2014but for high-precision applications like spectrophotometry, laser tuning, or astronomy, prisms simply can\u2019t match the consistency and versatility of a plane ruled grating. As a supplier of plane ruled gratings, I talk to researchers, engineers, and lab technicians every week who struggle to wrap their heads around the math behind dispersion: it\u2019s not just a number we print on a spec sheet, it\u2019s the foundation of how your instrument will perform. Too many teams grab a grating based only on groove density, then run into problems when their expected spectral resolution doesn\u2019t land. Today, I\u2019m breaking down exactly how to calculate a plane ruled grating\u2019s dispersion, what variables matter most, and how we help customers avoid common missteps when ordering. Plane Ruled Grating<\/a><\/p>\n

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First, let\u2019s ground this in what a plane ruled grating actually is: a flat, glass (or metal-coated) surface etched with thousands of parallel, equally spaced grooves. When polychromatic light hits these grooves, it reflects (for a reflection grating, the most common type we supply) at angles determined by the grating equation. That equation is the starting point for every dispersion calculation, so I\u2019ll write it out clearly, no skipped steps:
\nn\u03bb = d(sin\u03b8_i + sin\u03b8_m)
\nHere, n = the order of diffraction (1st order is the most commonly used, unless you\u2019re working with extreme UV or very narrow bandwidths), \u03bb = the wavelength of incoming light, d = the groove spacing (the inverse of groove density\u2014so if you have 1,200 grooves per millimeter, d = 1\/1200 mm = 833.33 nanometers), \u03b8_i = the angle of incidence (the angle between the incoming light and the grating\u2019s normal line), and \u03b8_m = the angle of diffraction (the angle between the diffracted light and the normal). That equation works because each groove acts as a point source of light, and the reflected waves interfere with each other to reinforce only wavelengths that align with that path difference.<\/p>\n

Dispersion, put simply, is how spread out the diffracted wavelengths are per unit angle. If two wavelengths (say, 500 nm and 501 nm) hit your grating, dispersion tells you how many degrees apart their diffracted beams will be. The formula for angular dispersion (the dispersion we use most for aligning instruments) is the derivative of the grating equation with respect to wavelength. Let\u2019s do that math together: differentiate both sides with respect to \u03bb, and you get:
\nd(sin\u03b8_i + sin\u03b8_m) \/ d\u03bb = n
\nWhich rearranges to angular dispersion (let\u2019s call it D_\u03b8, in radians per nanometer) = d\u03b8_m \/ d\u03bb = n \/ (d cos\u03b8_m)
\nWait\u2014there\u2019s a catch here: \u03b8_m isn\u2019t a constant. It changes with \u03bb, so you can\u2019t plug in a single static value for cos\u03b8_m unless you\u2019re working at a very narrow wavelength range, or near the Littrow condition (\u03b8_i = \u03b8_m, which is where most gratings operate for maximum efficiency). For example, if you\u2019re using a 600 groove\/mm grating (d = 1,666.67 nm) in 1st order (n=1) at a wavelength of 600 nm, and your angle of incidence is 20 degrees, you first solve for \u03b8_m using the original grating equation: 1600 = 1666.67<\/em>(sin20 + sin\u03b8_m). Sin20 is ~0.342, so rearranged: sin\u03b8_m = (600 \/ 1666.67) \u2013 0.342 = 0.36 \u2013 0.342 = 0.018, so \u03b8_m \u2248 1.03 degrees. Then cos\u03b8_m is ~0.9998, so angular dispersion would be 1 \/ (1666.67 * 0.9998) \u2248 6e-4 radians per nm, or about 0.034 degrees per nm. That means a 1 nm wavelength difference is spread over 0.034 degrees of diffraction angle.<\/p>\n

But here\u2019s where many customers go wrong: they confuse angular dispersion with linear dispersion, which is what matters for the slit or detector in your instrument. Linear dispersion (D_l, usually in mm per nm) is the dispersion you measure along the focal plane of your spectrograph, right where your detector sits. To get from angular dispersion to linear dispersion, you multiply by the focal length of your collimating or focusing mirror (f, in mm). The formula is D_l = f * D_\u03b8 = n*f \/ (d cos\u03b8_m). If the same 600 groove\/mm grating above is paired with a 300 mm focal length mirror, linear dispersion is 0.0006 rad\/nm * 300 mm = 0.18 mm\/nm. That means a 1 nm wavelength interval will span 0.18 mm on your detector. For a customer who needs to fit 1 nm intervals across a 25 mm detector, that would work perfectly\u2014but if they used a 1200 groove\/mm grating (smaller d) with the same setup, d would be 833.33 nm, so angular dispersion doubles to ~0.0012 rad\/nm, and linear dispersion becomes 0.36 mm\/nm, requiring a longer detector. That\u2019s the kind of calculation we walk customers through all the time when they\u2019re choosing a grating for a new system.<\/p>\n

Now, let\u2019s talk about the variables you can control when ordering a plane ruled grating, because that\u2019s what makes our job as suppliers so collaborative. Groove density (the number of lines per mm) is the most obvious, but it\u2019s not the only one. The order of diffraction (n) is another: higher orders (2nd, 3rd) give higher dispersion, but you have to watch for overlapping orders. For example, 400 nm light in 2nd order overlaps with 800 nm light in 1st order\u2014so if you\u2019re working across a wide range of wavelengths, you might need a order-sorting filter, or stick to lower orders to avoid confusion. \u03b8_m (the diffraction angle) is tied to your instrument\u2019s optical layout: if you\u2019re building a Czerny-Turner spectrograph, \u03b8_m will shift across your wavelength range, so you\u2019ll need to calculate dispersion at both the shortest and longest wavelengths in your target band to get a full picture, not just a single number.<\/p>\n

Wait, let\u2019s test a real example from a recent customer to make this concrete. A team at a biotech lab was building a Raman spectrometer, which needs to measure wavelengths from ~532 nm (the laser line) to ~800 nm (Stokes shift). They initially ordered a 1200 groove\/mm grating, expecting a linear dispersion of ~0.6 mm\/nm for their 500 mm focal length detector. But when we walked through the calculation, we pointed out that at 800 nm, the angle of diffraction would be much steeper, so cos\u03b8_m would be smaller, making actual dispersion higher. Let\u2019s run that: 1200 groove\/mm = d = 833.33 nm, n=1, \u03b8_i is fixed at 15 degrees for their setup. For \u03bb=800 nm: 800 = 833.33*(sin15 + sin\u03b8_m). Sin15 \u22480.2588, so sin\u03b8_m = (800\/833.33) \u2013 0.2588 = 0.96 \u2013 0.2588 = 0.7012, so \u03b8_m \u224844.5 degrees, cos\u03b8_m \u22480.713. Then angular dispersion = 1\/(833.33 * 0.713) \u2248 0.00168 rad\/nm, linear dispersion = 500 * 0.00168 = 0.84 mm\/nm. That\u2019s 40% higher than their initial estimate, which would have made their 512-pixel detector (each pixel 15 \u00b5m wide) only cover ~5120.015 =7.68 mm, so they\u2019d only get ~9 nm of spectral range. We suggested a 600 groove\/mm grating instead: d=1666.67 nm, at 800 nm \u03b8_m would be sin\u03b8_m = (800\/1666.67) \u20130.2588 =0.48 \u20130.2588=0.2212, \u03b8_m\u224812.8 degrees, cos\u03b8_m\u22480.975, linear dispersion= (1<\/em>500)\/(1666.67*0.975)\u22480.307 mm\/nm, which would let them cover ~25 nm of range with the same detector\u2014perfect for their Raman application. That\u2019s the kind of tailored calculation we do for every customer, because a spec sheet number is only useful if it matches your actual optical setup.<\/p>\n

Now, what about common mistakes that lead to wrong dispersion calculations? The biggest one is ignoring that \u03b8_m changes with wavelength, especially for wide spectral ranges. If you\u2019re working with a narrow band (say, 10 nm around 633 nm for a helium-neon laser), approximating cos\u03b8_m as a constant works fine, but for a 300 nm band, you need to calculate dispersion at both ends to capture the full spread. Another mistake is mixing groove density with groove spacing: remember, d is the distance between grooves, so d = 1 \/ (groove density in lines per mm) * 1e6 to convert to nanometers. So 600 lines\/mm is 1\/600 mm = 1,666,666 nm, wait\u2014no, wait, 1 mm is 1e6 nm, so 1\/600 mm = 1e6 \/600 \u22481,666.67 nm, that\u2019s right, I\u2019ve seen customers slip up and divide by 1e9 instead of 1e6, leading to a d value a thousand times too small, and a dispersion number way off. We always confirm d with customers, and even provide a custom calculation for their specific \u03bb range to avoid that.<\/p>\n

Another point for plane ruled gratings, specifically: ruled gratings have slightly different efficiency curves than holographic gratings, which can affect how usable your dispersion is. Ruled gratings work best at Littrow condition (\u03b8_i = \u03b8_m) for maximum efficiency, which is also where dispersion is highest for a given wavelength. If you\u2019re operating away from Littrow, your dispersion will be lower, and efficiency will drop too\u2014so we\u2019ll adjust the groove density or angle of incidence to align with your desired dispersion and efficiency. For example, a customer building a tunable laser needed high dispersion in the visible range, so we customized a 1800 groove\/mm grating optimized for Littrow at 550 nm, giving angular dispersion of ~0.0018 rad\/nm, which was exactly what they needed to achieve 0.1 nm tuning resolution.<\/p>\n

Let\u2019s also cover how dispersion relates to spectral resolution, since that\u2019s what most customers care about. Spectral resolution (\u0394\u03bb) is the smallest wavelength difference your system can distinguish, and it\u2019s tied to dispersion, but also to the number of grooves illuminated on your grating (N) and the slit width (w). The formula for resolution is R = \u03bb \/ \u0394\u03bb = nN, so \u0394\u03bb = \u03bb \/ (n<\/em>N). If you have a 1200 groove\/mm grating and you illuminate a 10 mm length of it, that\u2019s N=12,000 grooves, so resolution at 600 nm is 600 \/ 12,000 = 0.05 nm. To get that, your dispersion needs to spread that 0.05 nm difference over at least one pixel on your detector, so linear dispersion * \u0394\u03bb > pixel size. For a 15 \u00b5m pixel, that means linear dispersion needs to be > 0.015 mm \/ 0.05 nm = 0.3 mm\/nm. That\u2019s another check we do: if your dispersion is too low, your resolution will be limited by the detector, not the grating; if it\u2019s too high, you\u2019re wasting detector space on a narrower range than you need.<\/p>\n

As a plane ruled grating supplier, our role isn\u2019t just to ship a product\u2014it\u2019s to help you get the right performance. Too many labs order a grating based on generic spec numbers, then run into alignment issues or resolution limits because they didn\u2019t calculate dispersion for their exact setup. We keep a library of standard groove densities (300, 600, 1200, 1800, 2400, 3600 grooves\/mm) and can also make custom groove spacings for specialized applications like extreme UV or terahertz, where standard densities don\u2019t work. We also provide detailed calculation sheets for every custom order, so you can verify our work, and we can adjust the coating (aluminum, gold, dielectric) to boost efficiency at your target wavelength, which in turn affects how well you can measure dispersed wavelengths.<\/p>\n

<\/p>\n

If you\u2019re planning a new spectrograph, laser, or spectroscopic sensor, and you need help calculating dispersion, choosing the right plane ruled grating, or verifying your current setup\u2019s performance, we\u2019re here to assist. Our team has worked with researchers across academia, biotech, aerospace, and industrial manufacturing, and we tailor every solution to your specific requirements. For more information on our plane ruled gratings and to discuss your project, feel free to reach out to connect for procurement and technical guidance.<\/p>\n

Plane Ruled Grating<\/a> References<\/p>\n

    \n
  1. Palik, E. D. (1985). Handbook of Optical Constants of Solids. Academic Press.<\/li>\n
  2. Hutley, M. C. (1982). Diffraction Gratings. Academic Press.<\/li>\n
  3. Loewen, E. G., & Popov, E. (1997). Diffraction Gratings and Applications. CRC Press.<\/li>\n
  4. Kogelnik, H. (1969). Coupled wave theory for thick hologram gratings. The Bell System Technical Journal.<\/li>\n<\/ol>\n
    \n

    Jilin Juyao Technology Co., Ltd.<\/a>
    As one of the leading plane ruled grating manufacturers and suppliers in China, we offer a wide range of products with superior quality. Please feel free to wholesale customized plane ruled grating from our factory. Welcome to view our website for more information.
    Address: Room 101, No. 2 Huiwen Road, Nanguan District, Changchun City, Jilin Province, China
    E-mail: jyoptix@outlook.com
    WebSite:     https:\/\/www.jyoptix.com\/<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

    If you\u2019ve ever stared at a prism splitting sunlight into a rainbow, you\u2019ve seen wavelength-dependent light … How to calculate the dispersion of a plane ruled grating?<\/span>Read more<\/a><\/p>\n","protected":false},"author":886,"featured_media":3648,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[3611],"class_list":["post-3648","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-industry","tag-plane-ruled-grating-4da3-c71e45"],"_links":{"self":[{"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/posts\/3648","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/users\/886"}],"replies":[{"embeddable":true,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/comments?post=3648"}],"version-history":[{"count":0,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/posts\/3648\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/posts\/3648"}],"wp:attachment":[{"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/media?parent=3648"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/categories?post=3648"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.agrctmachinefmco.com\/blog\/wp-json\/wp\/v2\/tags?post=3648"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}