Laser Brightness Calculator

Background.

An LED and a laser can each emit 5 watts of light, yet only the laser can be focused to a spot that chars wood. The difference is not how much light there is but how tightly it is packed, both in space and in direction. Brightness, in the physicist's sense, measures exactly that: power divided by the area it leaves through and by the solid angle it spreads into, in watts per square metre per steradian. Radiometry calls the same quantity radiance.

Lasers win on both counts. A typical beam leaves through a spot a millimetre or so across and spreads by only a milliradian or two, so the product of area and solid angle is tiny. The example here, 5 W from 1 mm² into 10⁻⁵ sr, comes to 5 × 10¹¹ W/(m²·sr). The Sun delivers about 1,361 W/m² at the Earth from a disc about 0.53° across, a radiance of roughly 2.0 × 10⁷ W/(m²·sr). That modest laser is some 25,000 times brighter than the surface of the Sun.

Brightness matters because optics cannot raise it. A lens can squeeze a beam into a smaller spot, but the beam then spreads into a proportionally wider cone, and the product of area and solid angle, called étendue, stays the same or grows. So the brightness of the source sets the smallest, most intense spot any optical system can make from it, which is why laser makers quote beam quality alongside power.

The figure calculated here is a uniform average: total power over the whole beam area and the whole solid angle. Real beams are brighter at the centre than at the edge, pulsed lasers peak far above their average, and bandwidth, coherence and the competing conventions for measuring a Gaussian beam's width are all left out.

What is laser brightness calculator?

Laser brightness is the optical power emitted per unit source area per unit solid angle, measured in W/(m²·sr); it is the quantity radiometry calls radiance. The area is the cross-section the power leaves through, such as the beam waist. The solid angle is the cone the beam spreads into, in steradians, where a complete sphere is 4π, about 12.57 sr. A source can be powerful yet dim in this sense, if its light comes from a large area or spreads in every direction.

How to use this calculator.

1. 01Enter the optical power in watts. For a pulsed laser, the average power gives an average brightness; the peak during a pulse is higher by the ratio of pulse period to pulse length.
2. 02Enter the beam area in square metres. A circular beam of radius w has area π × w², so a radius of 0.564 mm gives 1 mm², which is 10⁻⁶ m². Note which radius convention you used, such as the 1/e² radius, because it changes the answer.
3. 03Enter the solid angle the beam fills, in steradians. For a narrow cone with half-angle θ in radians, Ω ≈ π × θ², so a half-angle of 1.784 mrad gives 10⁻⁵ sr.
4. 04Read the average brightness with the scope note beside it, and check a datasheet's conventions before comparing your figure with one quoted there.

The formula.

L = P ÷ (A × Ω). P is the optical power in watts, A the area of the emitting cross-section in square metres and Ω the solid angle of the emitted cone in steradians, so L comes out in W/(m²·sr). The product A × Ω is the étendue, the size of the beam in position and direction combined, and brightness is simply power per unit étendue. Power sits alone in the numerator, so doubling it doubles the brightness, and halving either the area or the solid angle does the same. In a lossless optical system the étendue cannot shrink: a lens that focuses the beam to a hundredth of its area makes the cone a hundred times wider, and the brightness stays where it was. The formula assumes the power is spread evenly over both A and Ω; for a real beam with a bright centre it gives an average, not the peak.

A worked example.

Frequently asked questions.

References& sources.

1. [1]ISO 80000-7:2019, Quantities and units — Light and radiation (BIBLIOGRAPHIC/PAYWALL).
2. [2]RP Photonics Encyclopedia, Brightness of Laser Beams.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.