Diffraction grating
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A diffraction grating is an optical component made of many reflecting or transmitting elements spaced at a distance comparable to the wavelength of light, such as a row of slits in an opaque screen or a set of grooves on a substrate. Its basic physical feature is a spatial modulation of refractive index that changes the amplitude, the phase, or both of an incident light wave, so that the light leaves the grating as a set of discrete beams called diffraction orders.[1] Because the angle of every order except the zero order depends on wavelength, gratings separate light into its colors, and for most of their history their main use has been spectroscopy.[1][2]
In augmented reality (AR) hardware, gratings are the in-couplers, pupil expanders and out-couplers of diffractive waveguide displays. They bend light from a small projector into a thin transparent plate, spread it across the eye box, and send it toward the eye, while the wearer still sees the world through the plate. The Microsoft HoloLens and the Magic Leap One both used surface relief grating waveguides.[2][3] The same dependence on wavelength that makes gratings useful in spectrometers is behind the color non-uniformity and "rainbow" artifacts of grating-based AR glasses.[3]
This article covers the grating as an optical element: its physics, types, fabrication and its role in AR displays. The general wave phenomenon is described in Diffraction, the wider family of diffractive components in Diffractive optics, and complete waveguide combiner systems in Waveguide display.
How it works
The grating equation
Light diffracted by each groove or slit of a grating combines with the light from every other groove. At most angles the contributions cancel, but at a discrete set of angles the path difference between neighboring grooves is a whole number of wavelengths and the contributions add up. In the form given by the Diffraction Grating Handbook published by MKS Instruments, these directions obey the grating equation
- mλ = d (sin α + sin β)
where λ is the wavelength, d is the groove spacing (also called the pitch), α is the angle of incidence, β is the angle of diffraction, and m is an integer called the diffraction order. The handbook also writes the equation with the groove density G = 1/d, usually given in grooves per millimeter.[1] For normal incidence on a transmission grating, introductory textbooks use the simpler form d sin θ = mλ.[4]
The zero order (m = 0) is plain specular reflection or straight-through transmission, and in it all wavelengths travel in the same direction. Only orders for which |mλ/d| < 2 can propagate, so a grating sends a given wavelength into a finite number of orders; when the pitch is not much larger than the wavelength, only a few orders exist.[1] In a white-light spectrum the central maximum stays white while the higher orders spread the light into a rainbow of colors. OpenStax notes that gratings can have more than 1,000 lines per millimeter, and that regular microstructures acting as reflection gratings produce the iridescence of butterfly wings, opals and hummingbird feathers.[4]
The basic equation assumes that the incident and diffracted rays lie in a plane perpendicular to the grooves. When light arrives at an angle to that plane, the diffracted orders lie on a cone instead, a case called conical diffraction. A common special arrangement is the Littrow configuration, in which light is diffracted back along its incoming direction and the equation becomes mλ = 2d sin α.[1]
Resolving power
The resolving power of a grating, R = λ/Δλ, measures its ability to separate two nearby spectral lines. The textbook expression is R = mN, where N is the number of grooves illuminated. The handbook shows that this can be rewritten in terms of the ruled width W of the grating and the two angles, so that the maximum attainable resolving power is 2W/λ whatever the order or groove count.[1]
Efficiency and blazing
How much light goes into each order depends on the power and polarization of the incident light, the angles of incidence and diffraction, the refractive indices of the grating materials, the groove spacing and the groove profile. A complete treatment requires solving Maxwell's equations for the corrugated surface.[1] The most widely used rule of thumb for reflection gratings is the blaze condition, mλ = 2d sin θB, where the blaze angle θB is the angle between the groove facets and the grating plane. At the blaze condition the incident and diffracted rays obey the law of reflection with respect to each facet, which is why the facets are often said to act as tiny mirrors; the handbook cautions that this is not strictly true, because the facets are often about as large as the wavelength itself.[1]
In 1969 Herwig Kogelnik published, in the Bell System Technical Journal, a coupled wave analysis of Bragg diffraction by "thick" hologram gratings, in which the grating is a modulation of refractive index or absorption through the depth of a recording layer. The theory gives algebraic formulas for diffraction efficiency and for angular and wavelength sensitivity, and remains valid when the incident wave is strongly depleted. For gratings that only modulate absorption, it predicts a maximum efficiency of 3.7 percent in transmission and 7.2 percent in reflection. For a lossless dielectric (phase) transmission grating without slant, used at the Bragg angle, the efficiency is the square of a sine whose argument grows with the index modulation and the layer thickness; Kogelnik noted that measurements by Shankoff and Lin on dichromated gelatin holograms had reached efficiencies approaching 100 percent, in agreement with the theory.[5]
Gratings whose features are comparable to or smaller than the wavelength are usually modeled numerically. In 1981 M. G. Moharam and T. K. Gaylord published rigorous coupled-wave analysis (RCWA), a state-variable matrix method for diffraction by planar gratings, including slanted ones, bounded by two different media.[6]
Types of grating
Gratings are classified in several overlapping ways. A reflection grating has its pattern on a reflective surface and a transmission grating on a transparent one. A master grating has a pattern made "from scratch", by mechanical ruling or holographic recording, while a replica grating is cast or molded from another grating.[1]
| Type | Structure | Notes and XR relevance |
|---|---|---|
| Ruled grating | Grooves burnished one at a time with a diamond tool into a thin evaporated metal coating | The first commercial gratings; replicas are used in lasers, spectroscopic instruments and fiber-optic telecommunications equipment[1] |
| Holographic (interference) grating | Interference fringes of two laser beams recorded in photoresist, giving sinusoidal grooves | Made since the late 1960s; can be coated and replicated like ruled masters[1] |
| Blazed grating | Sawtooth grooves with facets tilted at the blaze angle | Concentrates light into one order; Magic Leap One in-couplers are metal-coated blazed gratings[1][2] |
| Binary grating | Rectangular, vertical-walled ridges | Magic Leap One out-couplers are shallow binary gratings with depth modulation[2] |
| Slanted grating | Ridges tilted relative to the substrate normal | Used for both in- and out-couplers in the Microsoft HoloLens (first generation)[2] |
| Multilevel or quasi-analog surface relief grating | Stepped or continuous profiles approximating a designed phase function | Kress and Pace name Dispelix (multilevel) and WaveOptics (quasi-analog surface relief computer-generated holograms)[2] |
| Volume (Bragg) grating | Refractive-index modulation through the depth of a photopolymer, gelatin or liquid-crystal layer | Strong angular and wavelength selectivity, described by Kogelnik's coupled wave theory; used as holographic waveguide couplers, for example by DigiLens[5][2] |
| Polarization grating | Periodic pattern of liquid-crystal orientation | Diffraction depends on circular polarization; reflective polarization volume gratings have been demonstrated as waveguide couplers[3][7] |
| Two-dimensional (crossed) grating | Periodic in two directions, for example a hexagonal lattice | Can expand and out-couple a waveguide image in one region[3] |
| Metasurface grating | Subwavelength nanostructures designed by optimization | Research waveguide couplers; see Metasurface[8] |
Liquid-crystal polarization gratings behave differently from conventional ones. In 2008 Chulwoo Oh and Michael J. Escuti reported an achromatic polarization grating made from polymerizable liquid crystals (reactive mesogens). It acted as a broadband thin-film polarizing beam splitter, with diffraction efficiency of approximately 100 percent, an extinction ratio of at least 1000:1, and an operating bandwidth Δλ/λ0 of about 56 percent, roughly the visible range and more than four times that of conventional polarization gratings. The design stacks two chiral polarization gratings with opposite twist.[9]
History
According to the Diffraction Grating Handbook, the first diffraction grating was made by the American astronomer David Rittenhouse in 1785, who reported a half-inch wide grating with fifty-three apertures but did not develop it further. In 1821 Joseph von Fraunhofer, probably unaware of Rittenhouse's work, began making gratings for spectroscopy. His gratings were good enough to measure the absorption lines of the solar spectrum, now called Fraunhofer lines, and he derived the equations for the dispersive behavior of gratings.[1]
The Prussian instrument maker F. A. Nobert was supplying better gratings by 1850, and around 1870 the New York lawyer and amateur astronomer L. M. Rutherfurd ruled reflection gratings in speculum metal that surpassed the most powerful prisms. H. A. Rowland, a physics professor at Johns Hopkins University, built precise ruling engines; his 1882 work established the grating as the primary optical element of spectroscopy, and he invented the concave grating. Bausch & Lomb decided in 1947 to make precision gratings commercially and produced its first high-quality ruled master in 1950, on a rebuilt engine that originated in Albert A. Michelson's laboratory at the University of Chicago. Interferometric and servo control of ruling engines later pushed accuracy beyond the mechanical limits of the machines.[1]
Holographic gratings have a separate lineage. Aimé Cotton produced experimental holographic gratings in 1901. A few decades later Michelson considered the interferometric generation of gratings obvious, but recognized that the intense monochromatic light source and the sufficiently fine-grained photosensitive material it required did not yet exist. Ion lasers and photoresists, both available by the mid-1960s, supplied these, and the handbook credits work by D. Rudolph and G. Schmahl (1967) and by A. Labeyrie and J. Flamand (1969) for the method in use since the late 1960s.[1] Kogelnik's coupled wave theory of thick hologram gratings was published in 1969.[5]
Kress and Pace write that surface relief gratings became a commodity technology once mastering and mass replication were established in the early 1990s.[2] Their use as light-guide couplers for near-eye displays was studied in the 2000s, for example in Tapani Levola's 2006 paper on diffractive optics for virtual reality displays and in a 2007 paper by Levola and Pasi Laakkonen on slanted gratings for in- and out-coupling, before grating waveguides reached commercial mixed reality headsets with the HoloLens and Magic Leap One.[10][11][2]
Fabrication
Ruled masters require extreme mechanical precision. The handbook states that high spectral resolution needs groove spacing held to better than 1 nm, and that its ruling cells keep temperature stable to within ±0.01 °C for weeks at a time.[1] Holographic masters are recorded by intersecting two coherent laser beams; the fringe spacing is d = λ/(2 sin θ), where θ is half the angle between the beams, so the finest possible spacing is λ/2 and visible-light recording can produce thousands of fringes per millimeter. After development, a positive photoresist leaves sinusoidal ridges.[1]
Gratings for AR waveguides are much finer than most spectroscopic gratings. Kress and Pace give typical periods below 500 nm for grating couplers working by total internal reflection in the visible spectrum, with features of only a few tens of nanometers for multilevel structures. Masters are written by electron-beam lithography, i-line or deep-ultraviolet lithography, or interference lithography, and the structures are replicated in volume by nanoimprint lithography (NIL). Slanted-grating NIL with slants up to 50 degrees, they write, "has been mastered by many foundries around the world", and typical NIL yields for slanted gratings are about 90 percent for materials with refractive index below 1.8 but lower for indices of 1.9 and 2.0.[2] Xiong and colleagues cite a report that surface relief gratings 300 nm high with slant angles of up to 50 degrees can be replicated with high yield and reproducibility.[3] Levola and Laakkonen showed in 2007 that slanted gratings could be made in large quantities on plastic light guides from a high-index material by UV replication, and that the design could favor either the reflected or the transmitted out-coupling order.[11]
Very high index substrates need other methods. For the silicon carbide waveguides of its Orion AR glasses prototype, Meta states that the grating had to be slant-etched into the material, with the grating lines "sloped diagonally" rather than vertical. Research manager Nihar Mohanty said that "the whole industry used to rely on nano imprint, which doesn't work for substrates with such a high refractive index".[12]
Applications in AR displays
Waveguide couplers
In a diffractive waveguide, the in-coupler is usually a grating, and the out-coupler is usually a grating with the same period, though it can also be an off-axis lens with slight curvature that places the image at a finite depth.[3] Xiong and colleagues name three main coupler families: surface relief gratings, photopolymer gratings, and liquid-crystal polarization volume gratings. The in-coupler should be efficient, while the out-coupler is usually weak, so that light leaves the plate evenly; designers vary the local grating height and duty cycle to grade the out-coupling efficiency across the plate. For surface relief out-couplers, a reflective grating with a large slant angle is preferred because it suppresses the transmission orders.[3]
Because the in-coupler and out-coupler apply equal and opposite grating vectors, the dispersion of one cancels that of the other, so the output image generally shows no color dispersion even with a broadband LED source. In two-dimensional pupil expansion schemes the grating vectors form a closed loop, with the same result. The color problem is instead one of field of view and uniformity: red light travels inside the plate at a steeper angle than blue light from the same in-coupler, so designs usually stack two or three waveguide layers with different grating pitches.[3] Kress and Pace note that spectral spread is most critical with broadband LED illumination, used in HoloLens 1, Vuzix, Magic Leap, DigiLens and Nokia devices, and name the laser MEMS display engine of the HoloLens 2 as a notable difference.[2] Xiong and colleagues give a typical optical efficiency for diffractive waveguide combiners of about 50 to 200 nit per lumen, and state that current diffractive waveguide combiners generally achieve a field of view of about 50 degrees.[3]
Pupil expansion
A single in-coupled beam is too narrow to fill a usable eye box, so gratings are also used to replicate it across the plate, a process called exit pupil expansion. One arrangement uses two consecutive one-dimensional expansions: a turning grating duplicates the light in one direction and turns it, and the out-coupler then expands it in the other direction. The alternative is a single two-dimensional grating, such as a hexagonal lattice that provides grating vectors in six directions, which out-couples while it expands and gives more design freedom at the cost of heavier optimization. Xiong and colleagues add that unslanted geometries of this kind usually cause large light leakage and possibly low efficiency, and that adding slant helps but makes fabrication harder.[3]
Magic Leap's US patent 10,451,799, granted in October 2019 from a January 2017 priority filing, describes an eyepiece for virtual, augmented or mixed reality systems built around an input coupler region, orthogonal pupil expander (OPE) regions and an exit pupil expander (EPE) region, each made of diffractive features such as diffraction gratings; the patent also covers crossed-grating and lattice input couplers.[13] WaveOptics described its waveguides as using two diffraction regions, an input grating and an output structure made of a two-dimensional array of nanostructures, where rival designs typically used three gratings.[14]
Grating designs in commercial headsets
Kress and Pace describe the grating architectures of two early mixed reality headsets in detail.[2]
| Device | In-coupler | Out-coupler | Other details |
|---|---|---|---|
| Microsoft HoloLens (first generation, 2015) | Uncoated slanted gratings, each acting on one spectral band and passing the rest to the next waveguide | Slanted gratings, modulated in depth for a uniform eye box | Input and output slants close to 45 degrees, redirection gratings at about half that; periods tuned per color waveguide; symmetric in- and out-couplers compensate spectral spread[2] |
| Magic Leap One (2018) | Strong blazed gratings coated with reflective metal such as aluminum, one per color input pupil | Shallow top-down binary gratings, depth modulated | Input pupils spatially separated by color, each carrying the whole field of view[2] |
| Meta Orion (prototype) | Gratings slant-etched into the silicon carbide waveguide; Meta's description does not detail the in- and out-couplers separately | Meta states silicon carbide has a refractive index of 2.7 and gives Orion a field of view of about 70 degrees with a single plate per lens[12] | |
Volume gratings are an alternative to surface relief. Kress and Pace write that DigiLens' holographic polymer dispersed liquid crystal (H-PDLC) material has the largest index swing of current hologram materials, giving strong coupling in a layer typically four microns thick or less and allowing operation in transmission mode with the hologram sandwiched between two plates.[2] In a 2017 paper, Yun-Han Lee, Kun Yin and Shin-Tson Wu demonstrated a reflective liquid-crystal polarization volume grating as a waveguide coupler, with a measured coupling efficiency of 90 percent at 650 nm and a 50 degree deflection angle.[7] Xiong and colleagues note that reflective polarization volume gratings are preferred as in-couplers because transmissive ones are much more difficult to fabricate owing to liquid-crystal alignment issues.[3]
Metasurface gratings
In a 2024 Nature paper, researchers from Gordon Wetzstein's group at Stanford University, the University of Hong Kong and NVIDIA built a holographic AR display with inverse-designed full-color metasurface gratings as in- and out-couplers. The gratings were etched directly into high-index SF6 glass with a period of 384 nm and a height of 220 nm, and the authors state that an index of 1.8 or higher is needed for red, green and blue light to pass through a single coupler. Dispersion was compensated by the waveguide geometry rather than the gratings: with couplers of equal and opposite momentum on a 5 mm thick waveguide, red, green and blue light undergo one, three and five internal reflections before reaching the out-coupler. The authors report a see-through efficiency of about 78.4 percent and note that because red light is diffracted at a very large angle, further improving the efficiency of the red channel is more difficult.[8]
Leakage and rainbow artifacts
A grating in front of the eye diffracts light from both sides. Xiong and colleagues describe two unwanted effects that out-coupler design must suppress: light leakage, where guided light is diffracted outward toward the world, and the rainbow, where ambient light is diffracted into the user's eye.[3] Kress and Pace compare the world-side leakage (sometimes called eye glow) of different coupler types:[2]
| Coupler type | Leakage figure given by Kress and Pace |
|---|---|
| Surface relief gratings, binary profiles | About 50% of the eye-side leakage |
| Surface relief gratings, slanted profiles | About 40% of the eye-side leakage |
| Holographic grating couplers | About 8-10% of the eye-side leakage |
| Reflective waveguides (no grating) | 5% without anti-reflection coating, under 1% with it |
Meta describes rainbows as colorful streaks created when ambient light reflects off the waveguide. Barry Silverstein, Meta's director of research science, said that silicon carbide "gets rid of those rainbows".[12] Waveguides that use embedded partial mirrors instead of gratings, such as those of Lumus, are covered under Reflective waveguide.[2]
See also
References
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 Christopher Palmer (2020-03). "Diffraction Grating Handbook, eighth edition". Richardson Gratings. MKS Instruments, Inc.. https://www.edmundoptics.jp/ViewDocument/MKS%20Diffraction%20Grating%20Handbook%20(8th%20edition).pdf. Retrieved 2026-10-11.
- ↑ 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 Bernard C. Kress, Maria Pace (2022). "Holographic optics in planar optical systems for next generation small form factor mixed reality headsets". Light: Advanced Manufacturing, vol. 3, no. 4. doi:10.37188/lam.2022.042. https://www.light-am.com/article/doi/10.37188/lam.2022.042. Retrieved 2026-10-11.
- ↑ 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 Jianghao Xiong, En-Lin Hsiang, Ziqian He, Tao Zhan, Shin-Tson Wu (2021-10-25). "Augmented reality and virtual reality displays: emerging technologies and future perspectives". Light: Science & Applications, vol. 10, article 216. doi:10.1038/s41377-021-00658-8. https://doi.org/10.1038/s41377-021-00658-8. Retrieved 2026-10-11.
- ↑ 4.0 4.1 Samuel J. Ling, Jeff Sanny, William Moebs. "4.4 Diffraction Gratings". University Physics Volume 3. OpenStax. https://openstax.org/books/university-physics-volume-3/pages/4-4-diffraction-gratings. Retrieved 2026-10-11.
- ↑ 5.0 5.1 5.2 Herwig Kogelnik (1969-11). "Coupled Wave Theory for Thick Hologram Gratings". Bell System Technical Journal, vol. 48, no. 9, pp. 2909-2947. doi:10.1002/j.1538-7305.1969.tb01198.x. https://archive.org/details/bstj48-9-2909. Retrieved 2026-10-11.
- ↑ M. G. Moharam, T. K. Gaylord (1981-07). "Rigorous coupled-wave analysis of planar-grating diffraction". Journal of the Optical Society of America, vol. 71, no. 7, pp. 811-818. doi:10.1364/JOSA.71.000811. https://doi.org/10.1364/JOSA.71.000811. Retrieved 2026-10-11.
- ↑ 7.0 7.1 Yun-Han Lee, Kun Yin, Shin-Tson Wu (2017-10-20). "Reflective polarization volume gratings for high efficiency waveguide-coupling augmented reality displays". Optics Express, vol. 25, no. 22, p. 27008. doi:10.1364/OE.25.027008. https://doi.org/10.1364/OE.25.027008. Retrieved 2026-10-11.
- ↑ 8.0 8.1 Manu Gopakumar, Gun-Yeal Lee, Suyeon Choi, Brian Chao, Yifan Peng, Jonghyun Kim, Gordon Wetzstein (2024-05-08). "Full-colour 3D holographic augmented-reality displays with metasurface waveguides". Nature, vol. 629, pp. 791-797. doi:10.1038/s41586-024-07386-0. https://pmc.ncbi.nlm.nih.gov/articles/PMC11111399/. Retrieved 2026-10-11.
- ↑ Chulwoo Oh, Michael J. Escuti (2008). "Achromatic diffraction from polarization gratings with high efficiency". Optics Letters, vol. 33, no. 20, p. 2287. doi:10.1364/OL.33.002287. https://doi.org/10.1364/OL.33.002287. Retrieved 2026-10-11.
- ↑ Tapani Levola (2006-05). "Diffractive optics for virtual reality displays". Journal of the Society for Information Display, vol. 14, no. 5, pp. 467-475. doi:10.1889/1.2206112. https://doi.org/10.1889/1.2206112. Retrieved 2026-10-11.
- ↑ 11.0 11.1 Tapani Levola, Pasi Laakkonen (2007). "Replicated slanted gratings with a high refractive index material for in and outcoupling of light". Optics Express, vol. 15, no. 5, pp. 2067-2074. doi:10.1364/OE.15.002067. https://doi.org/10.1364/OE.15.002067. Retrieved 2026-10-11.
- ↑ 12.0 12.1 12.2 "Crystal Clear: Our Silicon Carbide Waveguides & the Path to Orion's Large FoV". Meta Blog. Meta Platforms. 2025-03-06. https://www.meta.com/blog/orion-silicon-carbide-waveguides-ar-glasses-large-field-of-view/. Retrieved 2026-10-11.
- ↑ Michael Anthony Klug, Robert Dale TeKolste, William Hudson Welch, Eric Browy, Victor Kai Liu, Samarth Bhargava (2019-10-22). "US10451799B2: Eyepiece for virtual, augmented, or mixed reality systems". Google Patents. Magic Leap, Inc.. https://patents.google.com/patent/US10451799B2/en. Retrieved 2026-10-11.
- ↑ "WaveOptics eyes production ramp with $13M top-up". optics.org. 2019-09-11. https://optics.org/news/10/9/16. Retrieved 2026-10-11.