Etendue (also called optical throughput or AΩ product) is a conserved quantity in optical systems. It equals the product of source area and the solid angle of emission, scaled by the refractive index squared: E = n²AΩ. No passive optical system can reduce etendue; it can only stay the same or increase.

How does numerical aperture relate to etendue?

For a circularly symmetric beam, the projected solid angle defined by numerical aperture NA = n·sin(θ) is Ω = π·sin²(θ) steradians. The etendue then becomes E = n²·π·A·sin²(θ) = π·A·NA². This calculator converts NA to Ω automatically.

Why does refractive index matter?

Etendue is defined as n²AΩ, not just AΩ. When light passes from a medium with index n into air (n=1), AΩ must increase to conserve n²AΩ. This is why oil-immersion microscope objectives achieve higher light collection than air objectives of the same physical aperture.

What units does etendue have?

Etendue has units of area × solid angle, typically mm²·sr or cm²·sr in photonics, or m²·sr in radiometry. The n² factor is dimensionless, so the base unit remains area × steradian.

What happens when source etendue exceeds system etendue?

The system becomes the bottleneck: it can only accept light up to its own etendue. The coupling efficiency is E_system / E_source. Excess light is lost at the aperture stop or field stop.

Etendue Calculator

Calculate optical etendue from source area and solid angle or numerical aperture — the fundamental conserved quantity in optical systems.

Inputs

Input Mode

Source Area

Solid Angle (sr, geometric)

Geometric solid angle: full hemisphere = 2π ≈ 6.2832 sr

Refractive Index (n)

Air ≈ 1.0 · Glass ≈ 1.5 · Water ≈ 1.33 · Oil ≈ 1.515

Results

Enter values and click Calculate

Common Presets

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Summary

Etendue is the fundamental conserved quantity of an optical system — the product of source area, solid angle, and refractive index squared (E = n²AΩ). No passive optical system can increase brightness beyond what the source etendue allows. Use this calculator to find etendue from either a solid angle or a numerical aperture, evaluate coupling efficiency between a source and an optical system, and understand how immersion media affect light collection.

How it works

Select input mode: Solid Angle or Numerical Aperture.
Enter the source area and unit (mm², cm², or m²).
Enter the solid angle in steradians, or the numerical aperture and refractive index.
The calculator computes etendue using E = n² × A × Ω and displays it in multiple units.
Use the results to compare with your detector or fiber etendue to assess coupling efficiency.

Use cases

Check whether a fiber bundle can capture all light from an LED source.
Determine the minimum detector size needed to collect all flux from a collimator.
Compare source and system etendue to find the coupling efficiency bottleneck.
Design illumination optics for projectors, microscopes, or laser systems.
Verify throughput conservation when switching between immersion and air optics.
Calculate the maximum brightness transfer possible through an optical train.

Frequently Asked Questions

Last updated: 2026-06-19 · Reviewed by Nham Vu