Photometry
Measuring light the way the eye experiences it.
Radiant & Luminous Flux
Watts measure energy; lumens measure how bright that energy looks.
Two lamps both radiate 10 watts of light, one yellow and one red. The yellow one looks far brighter. Photometry is the measurement of light as the human eye judges it.
Total radiant flux
The eye’s sensitivity
How bright light looks depends on its wavelength as well as its power. Under normal lighting, an average eye is most sensitive near 555 nm (yellow-green), and the sensitivity falls off gradually toward violet and red. There is no sharp edge to the visible range: radiation is visible wherever this luminosity isn’t zero.
Luminous flux and the lumen
Luminous flux measures a source’s capacity to produce the sensation of brightness. Its unit is the lumen (lm):
A 5.0 W source emits monochromatic light of 600 nm, where the relative luminosity is 0.60. Find its luminous flux.
A source emits 10 W at 630 nm (relative luminosity 0.27) and 6.0 W at 570 nm (0.95). Find its radiant flux and luminous flux.
- 1Total radiant flux: energy per second at all wavelengths (W).
- 2The eye is most sensitive near 555 nm; relative luminosity falls off on both sides.
- 3Luminous flux in lumens: 1 W at 555 nm = 685 lm; multiply by the relative luminosity for other wavelengths.
Check yourself
Luminous Efficiency & Intensity
How efficiently a lamp makes light, and how much it sends in each direction.
A lamp’s packaging gives its lumens; a torch is rated by how much light it throws in one direction. These are two different quantities, and the second needs the idea of a solid angle.
Luminous efficiency
For a lamp, what matters is how well the electrical power becomes brightness:
A filament lamp wastes much of its power heating the filament, and much of its radiation is infrared, which the eye can’t see. Its overall efficiency is rarely above about 50 lm/W.
Solid angle
Luminous intensity
The luminous intensity of a source in a given direction is the luminous flux per unit solid angle sent in that direction:
An ideal point source radiates equally in all directions, so . Real, extended sources have different intensities in different directions. Luminous intensity is also called illuminating power.
A small disc of radius 1.5 cm, 3.0 m from a uniformly radiating point source and facing it, receives 6.0 × 10⁻³ lm. Find the source’s intensity and total flux.
A 60 W lamp has an overall luminous efficiency of 15 lm/W and emits evenly into the forward half-space only. How much flux falls on a 4.0 cm² card 0.40 m away, facing it?
- 1Luminous efficiency = luminous flux / radiant flux; overall efficiency = luminous flux / power input.
- 2Solid angle ω = A/R² (steradian); whole sphere 4π sr.
- 3Luminous intensity I = dF/dω, in candela; a uniform point source has I = F/4π.
Check yourself
Videos, simulations & further reading (3)
Illuminance & the Lighting Laws
Half as far means four times as bright; tilt the page and it dims.
How brightly lit your page looks depends on how much light lands on each square centimetre, which depends on how far the lamp is and at what slant its light arrives.
Illuminance
The inverse-square and cosine laws
A small area ΔA is at distance r from a point source of intensity I, and its normal makes an angle θ with the line to the source. Seen from the source, the area subtends a solid angle
so it receives flux ΔF = IΔω, and its illuminance is
A small bulb radiating 1260 lm evenly in all directions hangs 0.60 m above a table. Find the illuminance (a) directly below it, (b) at a point on the table 1.0 m from the bulb.
Lambert’s cosine law
A small flat source (a glowing patch, or a sheet of paper lit from behind) radiates only into the half-space in front of it, and not equally in all those directions. Its intensity is greatest along the normal and falls off away from it. For many surfaces:
A perfectly diffusing small source has intensity 200 cd along its normal. How much flux does it send into a cone of 0.015 sr around a direction 45° from the normal?
- 1Illuminance E = dF/dA, in lux (lm/m²).
- 2From a point source: E = I cos θ/r², the inverse-square law with a cosine factor.
- 3Lambert's cosine law for a flat source: I = I₀ cos θ (perfectly diffusing surfaces).
Check yourself
Videos, simulations & further reading (3)
Photometers
Make two screens equally bright and the distances tell you which lamp is stronger.
The eye is a poor judge of how bright something is, but an excellent judge of whether two things are equally bright. Photometers use that talent to compare light sources.
The principle
Two screens sit side by side; one is lit only by source S₁, the other only by S₂, both at the same angle. Move the sources until the screens look equally bright. Then their illuminances are equal:
Bunsen’s grease-spot photometer
An optical bench carries three stands on a rail with a metre scale. The middle stand holds white paper with a grease spot; the other two carry the sources S₁ and S₂. Two plane mirrors behind the paper let you see both sides of the spot at once.
The grease spot lets some light through. When one side is lit more strongly, the spot looks darker than the paper on that side and brighter on the other. Keep one source fixed and slide the other until the spot looks the same on both sides: the illuminances match. Measure d₁ and d₂ and use the formula above.
A 40 cd lamp and an unknown lamp balance a photometer when they are 0.80 m and 1.20 m from the screen. Find the unknown intensity.
- 1Equal brightness of two screens means equal illuminance: I₁/d₁² = I₂/d₂².
- 2Bunsen photometer: a grease spot on paper between the two sources, viewed in two mirrors; balance when the spot vanishes against the paper.