How do I estimate lux at 1 meter from a 1000-lumen LED?
Asked 10/5/2013
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I’m trying to estimate the illuminance from a single Luxeon 1000-lumen white LED at a distance of 1 meter. Assuming no reflector or additional optics, and using the datasheet’s radiation pattern (roughly Lambertian / about a 120° beam), what lux should I expect on a surface facing the LED?
Originally by Photography Stack Exchange contributor. Source · Licensed CC BY-SA 4.0
Photography Stack Exchange contributor
12y ago
2 Answers
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The polar radiation pattern of Fig. 15 (p. 16 of the datasheet) for the white LEDs looks very close to a circle. It is thus reasonable to assume that these LEDs have approximately a lambertian pattern, which is confirmed by the fact that the intensity falls to 1/2 the maximum at 60° from the optical axis (cos 60° = 1/2). Based on this, you can deduce that the on-axis luminous intensity is
I = Φ / π = (1000 lm) / (π sr) = 318 cd.
At 1 m distance, and assuming the plane you are lighting is facing the LED, the illuminance is
E = I / (1 m)² = 318 lx
But this is only straight under the LED. If you are lighting an extended plane, the illuminance will fall-off with a cos⁴ law as you move out of the center of the light spot.
Edit: I am adding some rigorous derivations to support my sayings. You can skip them if you are afraid by math or you just trust me with the integrals.
Computing the on-axis intensity
Let’s assume that the luminous intensity I has an axially-symmetric distribution, i.e. it depends only on the angle θ between the direction of measurement and the axis of the LED. Then, the total luminous flux emitted by the LED is the intensity integrated over all directions of space:
Φ = ∫ I(θ) dΩ = ∫ I(θ) 2π sin(θ) dθ,
where dΩ = 2π sin(θ) dθ is the element of solid angle. Judging from Figs. 14 and 15 of the datasheet, it appears that I(θ) quite closely follows Lambert’s cosine law:
I(θ) ≈ I(0) cos(θ) for θ < π/2, zero otherwise
(the relevant curves are the ones labeled “White”, the “Royal Blue” has a different radiation pattern). Then, the total flux is
Φ = 2π I(0) ∫ cos(θ) sin(θ) dθ
The integral is for θ in [0, π/2], and it evaluates to 1/2. See Wikipedia on Lambert's cosine law for the derivation. Thus we have
I(0) = Φ / π = 318 cd.
It is worth noting that the same result can be achieved by a very crude approximation: that I(θ) is equal to I(0) inside the 120° cone, and zero otherwise. Then
Φ = ∫ I(0) 2π sin(θ) dθ for θ in [0, π/3]
By a mere coincidence, this crude approximation happens to give the very same result as the cosine law. On the other hand, if we really need more precision, we could digitize the curve from the datasheet and compute the integral numerically. I leave this as an exercise to the reader. ;-)
Computing the illuminance
Let’s assume that we have a flat surface at a distance z = 1 m, directly facing the LED, i.e. normal to the LED’s optical axis. We then have a light spot which is brighter in the center (on-axis with the LED) and fades away progressively as one moves out of the center. Let dS be an elementary surface at the center of the spot. This surface captures the light emitted over the elementary solid angle
dΩ = dS / z²
and thus the flux
dΦ = I(0) dΩ = I(0) dS / z²
The received illuminance is then
E(0) = dΦ / dS = I(0) / z² = 318 lx
This calculation can be extended to a point lying at a distance r from the center, for which the light rays arrive at an angle θ from the LED’s axis. We get:
dΩ = dS cos(θ) / (z² + r²)
dΦ = I(θ) dΩ = I(0) dS cos²(θ) / (z² + r²)
E(r) = dΦ / dS = I(0) cos²(θ) / (z² + r²)
but since cos(θ) = z / √(z² + r²), and I(0) = E(0) z²,
E(r) = E(0) cos⁴(θ) = E(0) z⁴ / (z² + r²)²
which leads to the following iluminance pattern, as a function of the distance r to the center of the spot:
r (m) E (lx)
0 318
0.5 204
1 80
1.5 30
2 13
2.5 6
Originally by user1730. Source · Licensed CC BY-SA 4.0
user1730
12y ago
0
Generated from our catalog & community — verify before relying on it.
For a bare LED with a roughly Lambertian pattern, a good estimate is based on luminous intensity, not simply spreading lumens over the full beam circle.
A Lambertian emitter has peak on-axis intensity:
I ≈ Φ / π
So for 1000 lm:
I ≈ 1000 / π ≈ 318 cd
At 1 meter, illuminance on-axis is:
E = I / d² ≈ 318 / 1² ≈ 318 lux
So you’d expect about 318 lux directly under the LED at 1 m, assuming the target surface faces the LED.
Important: that is the center value only. Illuminance falls off away from the optical axis, and over an extended flat surface it decreases significantly toward the edges. A simple “lumens divided by beam area” estimate gives an average over the lit circle, not the peak center lux, so it can produce a lower number (such as ~106 lx) that describes something different.
In short: for one 1000-lumen Lambertian LED at 1 m, expect roughly 318 lx on-axis.
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