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Étendue is a measure of the area and directions occupied by a bundle of rays. In an ideal passive lossless optical system it cannot be compressed away: if a ray bundle is made smaller in area, it must generally occupy a wider angular range. That is why a lens or collector cannot gather every ray from a broad source into an arbitrarily small target.
For a flat aperture of area A fully occupied over a cone of half-angle θ in a uniform medium of refractive index n, the geometrical-optics expression is G = π n² A sin²θ. A fictional 1 cm² entrance accepting a 30° half-angle cone in air has G ≈ 7.85 × 10⁻⁵ m²·sr. A 0.25 cm² exit requires an entire 90° half-angle acceptance in air to pass that same ideal phase-space bundle without loss.
Understand the light bundle
Define étendue and radiance · Separate collecting area, direction and measured power.
Derive and calculate
Work the filled cone · Integrate the projected solid angle and test a collector.
Set a physical limit
Apply the acceptance bound · Find where concentration requires clipping or another change.
Investigate a real instrument
Check the measurement path · Nonuniform beams, alignment, losses and numerical aperture.
Practise and connect
Solve a new aperture case · Carry the reasoning into optical design and sources.
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CHAPTER 1 OF 12 · Photonics
1. Ask which rays a collector can actually accept
An optical collector may be a lens, concentrator, objective, sensor entrance or fibre coupler. Its effectiveness depends not only on clear aperture but also on the range of incoming directions. A wide lens does not promise perfect delivery to a small detector, and a small detector does not necessarily measure all the power passing the lens.
Ray optics describes each elemental bundle by where it crosses a plane and in which direction it travels. A system can make one dimension smaller while enlarging another. The joint phase-space extent is what étendue tracks under appropriate geometrical-optics assumptions.
The key design question is therefore not simply, ‘How much can the lens magnify or focus?’ It is, ‘Which input positions and directions must be transmitted into which permitted output positions and directions?’ That is the distinction from the related Gaussian-beam aperture-clipping guide, which calculates a detector’s captured fraction for a specified intensity profile.
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CHAPTER 2 OF 12 · Photonics
2. Keep power, irradiance and radiance distinct
Optical power P is energy per unit time, measured in watts. Irradiance is power incident per unit illuminated area, commonly W/m². Radiance additionally resolves direction and projected area, using W/(m²·sr). These are not interchangeable labels for brightness.
A detector of a given area can record more power if the angular distribution into it increases, even when basic radiance is not increased. Similarly, a focused spot may have high irradiance without creating radiance beyond what passive optics permits.
The refractive index matters. With the étendue convention used here, G = ∫ n² cosθ dA dΩ. Across lossless refractive components, accounting for n properly preserves the optical invariant. Dropping n² while comparing air and glass can produce an apparent violation that is only a bookkeeping mistake.
Scattering, absorption, reflections, clipping and diffraction change what a real instrument transmits. Geometrical conservation gives a necessary ideal limit; it does not guarantee that a practical setup reaches the limit.
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CHAPTER 3 OF 12 · Photonics
3. Derive the circular acceptance-cone expression
Consider a flat patch of physical area A in a medium of uniform n. For rays at polar angle α to the patch normal, projected area contributes cosα. The differential solid angle is sinα dα dφ. For a rotationally symmetric fully occupied cone, integrate α from 0 to θ and azimuth φ from 0 to 2π.
G = n²A ∫₀²π dφ ∫₀θ cosα sinα dα = π n² A sin²θ. The result has area times solid angle units. The π comes from the projected angular integral, not from the geometrical area of a circular opening.
This relation describes the maximum geometrical étendue of the filled area-and-angle domain. It is not a statement that every arbitrary source uniformly fills that domain in radiance. A narrow laser beam travelling through a large empty aperture occupies less phase space than the aperture could accept.
For small angles measured in radians, sinθ ≈ θ, so G ≈ πn²Aθ². For high numerical aperture, retain sinθ rather than silently replacing it with θ. A full hemisphere reaches sin²θ = 1; no larger angular acceptance exists on that side of a flat plane.
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Take an invented ideal light collector in air with entrance area Aᵢ = 1.00 cm² = 1.00 × 10⁻⁴ m² and incoming filled half-angle θᵢ = 30°. Its incoming étendue is π × 1.00×10⁻⁴ × sin²30° = π×2.50×10⁻⁵ ≈ 7.854×10⁻⁵ m²·sr.
Suppose the target face is only Aₒ = 0.25 cm² = 2.50×10⁻⁵ m². Preserving the entire bundle in air would require sin²θₒ = G/(π Aₒ) = 1. Therefore θₒ must reach 90°. This limiting output cone has no further angular headroom.
If the target is reduced again to 0.10 cm², the required sin²θₒ becomes 2.5. Since sin²θ cannot exceed 1, the premise of lossless delivery of the entire filled input bundle fails. No clever passive lens shape removes that mathematical conflict.
| Location | Area | Filled half-angle | Étendue or requirement |
|---|---|---|---|
| Input in air | 1.00 cm² | 30° | G = 7.854×10⁻⁵ m²·sr |
| Ideal output A | 0.25 cm² | 90° required | At the hemispherical bound |
| Proposed smaller output | 0.10 cm² | Would need sin²θ = 2.5 | Not possible for same entire ray bundle |
| Same aperture, θ=20° | 1.00 cm² | 20° | G ≈ 3.675×10⁻⁵ m²·sr |
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Numerical aperture in a uniform incident medium is NA = n sinθ for the accepted ray cone under this geometric definition. Thus G = π A NA² for the filled, axisymmetric cone. This relation makes the trade-off useful when reading fibre, objective or sensor coupling specifications.
Be careful: NA is not the same as a detector’s area or its optical power sensitivity. An optical component may accept a large cone yet have losses, an incompatible field distribution or alignment restrictions. A nominal NA and a physical core diameter describe potential acceptance, not guaranteed coupling efficiency.
When the refractive index differs between input and output, use n²A sin²θ, not merely A sin²θ. Refraction can make rays occupy different angular ranges without violating conservation. The invariant counts the change in medium as part of the geometry.
A real fibre also supports modes whose number and coupling depend on wavelength, core size and refractive-index profile. Étendue offers a powerful ray-optics intuition but cannot replace a wave-optical mode-overlap calculation when coherence and diffraction dominate.
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CHAPTER 6 OF 12 · Photonics
6. Ask what happens if the receiving area is halved
Keeping the same incoming filled cone and medium, reducing receiver area by a factor of two requires doubling sin²θ at the receiver to preserve étendue. This is possible only while the resulting value remains no more than one. There is a hard geometric ceiling.
For our example, the 0.25 cm² output already reaches the hemispherical ceiling. Halving it to 0.125 cm² would require sin²θ = 2. A passive lossless device cannot squeeze the entire specified bundle into that output while retaining the same single-sided, same-medium description.
The device may instead reject some input directions, enlarge the illuminated target, change the index environment, or divide the flow among multiple outputs. Clipping and absorption reduce transmitted power rather than defeating the invariant. A correct engineering report should name which compromise was chosen.
A bright spot on a camera does not show that all incoming optical power has been preserved. Measure or infer the total transmitted power and compare angular distributions if the design claim concerns full collection.
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CHAPTER 7 OF 12 · Photonics
7. Recognise the difference between étendue and throughput
Étendue is geometrical acceptance in position-and-direction space. Throughput is a system-dependent term sometimes used for accepted flux or geometrical throughput; the exact convention matters. Transmitted optical power also depends on the source’s radiance distribution, coatings, attenuation, scattering and reflectance.
For a source with uniform radiance L in one medium and a filled accepted domain, the collected power can be written P = (L/n²)G with the convention used here. For a nonuniform source, integrate the radiance over the accepted positions and directions instead of multiplying one nominal number.
An instrument with high theoretical étendue can still measure very little if the scene contributes light only into a narrow subset or if its transmission is low. Conversely, a small-area, high-NA component may accept a useful part of a directional source.
When comparing products, document whether a vendor’s quoted ‘collection efficiency’ includes Fresnel losses, detector quantum efficiency, wavelength dependence, surface vignetting or purely geometric acceptance.
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CHAPTER 8 OF 12 · Photonics
8. Find the hidden assumptions in a laboratory reading
Alignment errors can displace the angular distribution relative to the input acceptance. A coating’s transmission may vary with angle and wavelength. A finite detector can clip a beam whose centroid or waist changes with time. All can produce a power change even when the light source itself is stable.
Inspect the system in three separable stages: characterise the source’s position-angle distribution; establish the physical apertures and NA along the path; then measure optical transmission and detector response. Each measurement rules out a different class of explanation.
For strongly coherent or diffraction-limited optical fields, rays are no longer a complete description. Mode count, diffraction and interference introduce requirements beyond the simple filled-cone estimate. Yet the phase-space argument remains an excellent first check on claims of arbitrarily efficient concentration.
Make an explicit choice between an energy-budget question (‘Where did the watts go?’) and a geometric-acceptance question (‘Which rays were ever eligible?’). Treating them as the same question often leads to unjustified corrections.
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CHAPTER 9 OF 12 · Photonics
9. Create a measurement and design evidence trail
An informative collector test records entrance aperture dimensions, usable exit area, numerical apertures or acceptance angles, source spectrum, alignment, detector linearity and optical throughput. It also states whether the assumption of uniform angular filling is a teaching approximation or an observed condition.
If two instruments collect unequal power, check their receiving solid angles before inferring that one detector is less sensitive. If a beam is narrowed by upstream optics, note whether the angular divergence widens. If a lens is replaced, check the effective stop rather than only its advertised diameter.
The table separates what a basic instrument measurement demonstrates from what it does not. This makes a notebook more useful than an attractive but unexplained ‘efficiency’ percentage.
| Observation | Supports | Does not establish alone |
|---|---|---|
| Measured entrance and exit apertures | Geometric area boundaries | Full source angular occupation |
| Specified NA or acceptance cone | Potential angular range | Lossless delivery of all light |
| Detector power reading | Captured optical power under calibration | Source radiance or total emitted power |
| Angle-resolved scan | Directional distribution at a plane | Performance across every wavelength |
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CHAPTER 10 OF 12 · Photonics
10. Questions that prevent false concentration claims
Several common statements about optical focusing are correct only after specifying the source, medium and angular domain.
Does conservation of étendue prohibit focusing light?
No. Focusing can reduce area while increasing angular spread. The restriction concerns the full occupied position-and-direction domain, not an isolated spot diameter.
Is an aperture’s area all I need to calculate collection?
No. Angular acceptance, source radiance, wavelength, vignetting and transmission also matter. The same area can collect different amounts of power.
Does a higher numerical aperture always mean a better detector?
No. It can support a larger acceptance cone, but sensitivity, noise, field requirements, calibration and losses may be more important for the actual task.
Can coherent optics get around this argument?
Wave-optical descriptions use modes and field overlap. A passive device does not manufacture radiance or information arbitrarily; the simple ray-cone calculation is not itself a complete theorem for every coherent field.
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CHAPTER 11 OF 12 · Photonics
11. Transfer the rule to a different entrance cone
Try a fictional entrance of 2.00 cm² in air with a 20° half-angle filled cone. Because sin²20° ≈ 0.1170, G ≈ π × 2.00×10⁻⁴ × 0.1170 ≈ 7.35×10⁻⁵ m²·sr.
A 0.50 cm² exit is four times smaller in area. Lossless acceptance would require sin²θₒ = 4 sin²20° ≈ 0.4679, giving θₒ ≈ 43.2°. This is geometrically possible because the required value remains below one.
Change the exit to 0.10 cm² instead. The area ratio becomes twenty, so the required sin²θₒ ≈ 2.34; the same ideal ray bundle cannot be transmitted into that output. The right answer is to identify the lost assumption, not to report an impossible angle.
A final challenge: list one method for reducing the accepted phase-space content without mislabelling it a lossless concentrator. Examples include accepting a narrower source cone or collecting only part of an extended source. Specify exactly which source rays no longer pass.
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CHAPTER 12 OF 12 · Photonics
12. Use the invariant to ask the next optical question
The gold standard is a design statement that records both the relevant area and the accepted direction range. Without that pairing, an apparently impressive focusing claim may be physically incomplete. Étendue is the early calculation that prevents building an impossible requirement into a downstream specification.
Return to the Photonics root to place this geometrical limit beside photon energy, waveguiding, detection and optical metrology. For a distinct finite-detector question, continue to aperture clipping and captured Gaussian power; that guide integrates the light present at a particular sensor.
Together these two cases show why source emission, ray acceptance, captured power and detector conversion must remain separate stages. A correct calculation is not just a number; it tells the reader which stage has been tested and what observation would be needed next.
Sources and further reading
The references below support the physical principles and the scope of the methods. All numerical cases in this article are constructed educational examples; they are not field measurements, product specifications or a substitute for specialist analysis.
| Reference | What to verify |
|---|---|
| NIST: Photonic Radiometry | Radiometric measurement and calibration principles |
| MIT OpenCourseWare: Fundamentals of Photonics | Optical fields, sources and geometrical-wave foundations |
| RP Photonics: Radiance | Directional radiometric definitions and projected-area concept |
| Scientific Reports: Étendue and radiance conservation (2026) | Phase-space form of optical-invariant and concentration bounds |
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References checked 21 October 2026. Fictional worked examples state their simplifying assumptions so their calculations can be checked or challenged.