Optical etendue calculator

This calculator multiplies a source's projected area in square metres by the solid angle of the beam in steradians, returning etendue in square-metre steradians. The quantity is central to illumination, imaging, concentrator, and radiometry calculations because an ideal lossless optical system cannot reduce it. Enter finite, non-negative SI values to obtain a transparent result with the formula shown alongside the normalized inputs.

What optical etendue measures

Etendue combines two properties that are easy to consider separately but must be tracked together: the projected area occupied by a source or beam and the range of directions over which its rays travel. A wide emitter confined to a narrow cone can have the same etendue as a small emitter spreading light through a broad cone. For the uniform, aligned case handled here, the relationship is G = A × Ω, where A is projected source area and Ω is solid angle. The resulting SI unit is square-metre steradian, written m²·sr. This calculator is useful when the area and angular extent are already known and you need a direct geometric result. It does not infer projected area from a tilted surface, derive solid angle from lens dimensions, or model spatial and angular variation across a nonuniform source. Those cases require integration over position, direction, and the appropriate cosine projection factor before their effective area and solid angle can be represented by this simplified product.

How to enter area and solid angle correctly

Provide source_area_m2 as the projected emitting or beam area in square metres, not merely a convenient mechanical dimension. For a circular spot, convert diameter to radius and calculate πr²; for a rectangle viewed normally, multiply its side lengths. If the source plane is tilted relative to the optical axis, the projected area may differ from its physical surface area. Provide solid_angle_sr in steradians. A steradian measures angular spread over a sphere and is not the same as an angle stated in degrees or radians. If you have the half-angle θ of a circular cone instead, first calculate Ω = 2π(1 − cos θ), with θ expressed in radians. The calculator accepts zero to represent a limiting zero-area or zero-angle case and rejects negative quantities, non-finite numbers, and solid angles greater than 4π, because 4π steradians covers the complete sphere. Keep every conversion explicit so the returned m²·sr value remains dimensionally meaningful.

Interpreting conservation and practical results

In ideal geometrical optics, etendue is conserved through a lossless optical system and cannot be compressed by ordinary passive components. A lens may reduce beam area only by increasing angular spread, or narrow the angular spread only by increasing area. This is why a large, diffuse source generally cannot be focused into an arbitrarily small spot with a narrow output cone. Compare the calculated source etendue with the acceptance etendue of downstream apertures, fibres, detectors, or concentrators. If the receiving system accepts less etendue, some rays must be rejected, so geometric throughput cannot reach one hundred percent even before absorption, reflection, scattering, and alignment losses are considered. The simple product is a geometric benchmark rather than an optical-power prediction: it says nothing directly about watts, radiance, spectrum, polarization, or component transmission. Use the output to identify phase-space mismatch, then apply efficiency and loss factors separately when estimating delivered power. Via the API, each deterministic calculation costs $0.002 and returns the exact inputs, formula, value, and unit for auditable workflows.