Étendue
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Étendue describes the extent of a bundle of light in both position and propagation direction. In geometrical optics it combines an area with an angular range, weighted by the refractive index and the orientation of the surface.[1] In near-eye displays, limited étendue constrains the combination of field of view and eyebox: a design must deliver the image over an angular range while accommodating different eye positions.[2]
Geometrical definition
For an infinitesimal ray bundle crossing a surface, the three-dimensional optical definition is:[1]
- dG = n2 cos(θ) dA dΩ
| Symbol | Meaning |
|---|---|
| n | Refractive index of the medium containing the rays. |
| dA | Element of surface area. |
| θ | Angle between the ray direction and the surface normal. |
| dΩ | Element of solid angle occupied by the bundle. |
The étendue of a finite bundle follows by integrating over its occupied areas and directions. The cosine accounts for projected area, so an unqualified area-times-angle product omits part of the general definition.[1]
For geometrical propagation through ideal refraction and specular reflection, étendue is conserved. With no absorption, scattering or reflection losses, luminous flux is also conserved. Luminance divided by n2, sometimes called basic luminance, remains invariant across a refracting interface. Ordinary luminance itself is conserved when the medium's refractive index is unchanged. These assumptions matter when applying the conservation law to a display's optical path.[1]
Field of view and eye position
For a conventional holographic display, the finite number of spatial light modulator (SLM) pixels constrains the supported combination of field of view and eyebox. Enlarging one within that fixed arrangement requires reducing the other.[2]
A waveguide display can use exit pupil expansion to distribute copies of an input pupil over a larger output region. Light is guided through a thin substrate and coupled out at multiple positions. This improves the range of eye positions from which an image can be viewed, but the distribution and efficiency of the output remain design constraints. Jang and colleagues' 2024 waveguide-holography experiment, for example, explicitly optimized a balance between output uniformity and overall throughput efficiency.[3]
Pupil replication also has implications for image formation. In conventional waveguides, copies following different optical paths can produce ghost images when finite-distance imagery is injected. The 2024 experiment used a calibrated propagation model to compensate for the waveguide's effects and render three-dimensional holographic content. Its discussion still identified SLM artifacts, restricted field of view and sensitivity during calibration as limitations.[3]
Holographic displays and space-bandwidth product
A holographic display reconstructs imagery by controlling a light wavefront. Its dynamic SLM has a finite active area and pixel pitch. The pitch limits the diffraction angle, while the active area limits the spatial extent of the modulation. In a conventional configuration, these jointly constrain display étendue. Relay optics can exchange image size and angle, but do not supply additional independently addressable SLM pixels.[4]
Display research therefore relates étendue to the SLM's space-bandwidth product. However, an expanded viewing region and independently controllable image information across that region are separate considerations. Merely repeating a wavefront does not produce a different perspective at every eye position. Chao and colleagues distinguished pupil replication from illumination and modulation methods designed to reproduce a light field with view-dependent parallax.[5]
Expansion methods and research
Static diffractive elements
In 2024, Tseng and colleagues reported a neural étendue expander: a fabricated static diffractive element optimized jointly with SLM patterns using natural images. Its smaller feature pitch broadens the diffracted angular range. The team demonstrated a 64-fold expansion, corresponding to eightfold expansion along each angular axis in its experiment. The optimization incorporates a model of retinal spatial-frequency filtering; this result concerns the tested holographic reconstruction and perceptual model, rather than an increase in the number of controllable SLM pixels.[4]
Multiple-source illumination
Chao and colleagues' 2024 system combined multiple illumination directions with a phase SLM and a second SLM controlling amplitude in the Fourier plane. Jointly optimized, time-multiplexed patterns suppressed unwanted copies while reproducing different views across an expanded eyebox. The authors evaluated a benchtop prototype and simulations. Miniaturization and real-time light-field hologram synthesis remained future work; the reported optical method was not a finished consumer headset.[5]
Synthetic aperture waveguide holography
Choi and colleagues' 2025 Nature Photonics paper combined a steered laser, a holographic waveguide, an SLM and a holographic eyepiece. A MEMS mirror scanned the illumination to form a synthetic aperture from multiple smaller apertures. The rendering model accounted for partial coherence and used light-field targets to represent changes in perspective and occlusion across the viewing region.[6]
The research prototype supported a 38° diagonal field of view, a 9 × 8 mm eyebox and an eye-relief range of 23-33 mm. Its approximately 3 mm panel-to-eyepiece optical stack did not represent the thickness of every component in the complete apparatus. The authors compared a three-dimensional eyebox roughly two orders of magnitude larger than the intrinsic SLM-limited volume without steered illumination. That comparison concerns the scanned system and its specified baseline, rather than a universal increase for all holographic displays.[6]
Interpreting an expansion claim
An expansion factor needs its baseline and mechanism: angular scattering, pupil replication and aperture scanning describe different changes. Image quality, brightness uniformity and view-dependent information remain separate requirements.[4][3][5][6]
See also
References
- ↑ 1.0 1.1 1.2 1.3 R. A. M. van Gestel, M. J. H. Anthonissen, J. H. M. ten Thije Boonkkamp and W. L. IJzerman (2021-09-12). "An Energy Conservative hp-method for Liouville's Equation of Geometrical Optics". Journal of Scientific Computing, volume 89, article 27. https://doi.org/10.1007/s10915-021-01612-x. Retrieved 2026-10-21.
- ↑ 2.0 2.1 Grace Kuo, Laura Waller, Ren Ng and Andrew Maimone (2020). "High resolution étendue expansion for holographic displays". ACM Transactions on Graphics, volume 39, issue 4, article 66. https://doi.org/10.1145/3386569.3392414. Retrieved 2026-10-21.
- ↑ 3.0 3.1 3.2 Changwon Jang, Kiseung Bang, Minseok Chae, Byoungho Lee and Douglas Lanman (2024-01-02). "Waveguide holography for 3D augmented reality glasses". Nature Communications, volume 15, article 66. https://doi.org/10.1038/s41467-023-44032-1. Retrieved 2026-10-21.
- ↑ 4.0 4.1 4.2 Ethan Tseng et al. (2024-04-22). "Neural étendue expander for ultra-wide-angle high-fidelity holographic display". Nature Communications, volume 15, article 2907. https://doi.org/10.1038/s41467-024-46915-3. Retrieved 2026-10-21.
- ↑ 5.0 5.1 5.2 Brian Chao, Manu Gopakumar, Suyeon Choi, Jonghyun Kim, Liang Shi and Gordon Wetzstein (2024). "Large Étendue 3D Holographic Display with Content-adaptive Dynamic Fourier Modulation". SIGGRAPH Asia 2024 Conference Papers. https://doi.org/10.1145/3680528.3687600. Retrieved 2026-10-21.
- ↑ 6.0 6.1 6.2 Suyeon Choi, Changwon Jang, Douglas Lanman and Gordon Wetzstein (2025-07-28). "Synthetic aperture waveguide holography for compact mixed-reality displays with large étendue". Nature Photonics, volume 19, pages 854-863. https://doi.org/10.1038/s41566-025-01718-w. Retrieved 2026-10-21.