What the Inverse Square Law Does to a Lighting Layout

Why this matters

Everyone can recite that light falls off with the square of distance. Almost nobody applies the second half of the same calculation, and the second half is the one that decides whether the space between two fixtures goes dark. Drop it and every spacing check you run comes out flattering, in the same direction, by roughly a sixth. That is the difference between a layout that passes on paper and a customer standing in the aisle telling you it is fine under the lights and dim in between.

The law, and the two things it holds constant

E = I / d^2, with intensity I in candela and distance d in feet giving illuminance E in footcandles, or d in meters giving lux.

It holds two things constant, and both get violated in the field:

1. I is the intensity in the one direction you are looking along, not the fixture's peak and not its nadir value. Move the point and the angle changes, so I changes, and you must go back to the photometric file for a new candela value.
2. The source is a point, dimensionless. The working boundary for that is five times the maximum luminous dimension: past that distance the error from treating a real source as a point stays small, a couple of percent for typical distributions, and inside it the law overstates the falloff.

That second rule is where most misapplication lives. A 4 ft linear luminaire has a maximum luminous dimension of 4.0 ft, so the point method needs 20 ft of separation before it is honest, and a 10 ft mounting height is not 20 ft.

Three source geometries, three exponents

Name which one you have before you calculate anything.

• Point source, at distances beyond five times its luminous dimension: falls as 1 / d^2.
• Line source, a continuous illuminated row much longer than the distance to it: falls closer to 1 / d in the near field, and only reverts to the inverse square once you are far enough away that the row subtends a small angle.
• Large area source, a luminous ceiling or a wall wash close up: barely falls off at all with distance, because moving away loses intensity and gains apparent area at nearly the same rate.

Apply the point exponent to a continuous row at close range and you will predict a dip that the installed job does not have. That is the one direction this error runs conservative, which is why it survives.

The cosine term is the one people drop

Illuminance on a plane depends on the angle the light strikes it. Light arriving at 31 degrees off vertical spreads its flux over a larger patch of floor than the same beam arriving straight down, and the horizontal illuminance falls by the cosine of that angle.

Combine that with the inverse square and, holding the mounting height above the work plane constant while the lateral offset varies, the two terms collapse into one convenient form:

E (horizontal) = I x cos^{3(theta) / h}2

where h is the mounting height above the work plane and theta is the angle from straight down to the point. The cube is not a typo: one power of cosine is the incidence angle on the plane, and two more come from converting the slant distance back to the fixed height.

            luminaire
                *
               /|
    slant     / |
    distance /  | mounting height h
       d    /   | above work plane
           /    |
    theta /     |
---------*------+--------------- work plane
     offset point directly
     point below luminaire
     |<-- x --->|

The calculation sheet, filled in

Two compact luminaires in a row. Maximum luminous dimension 1.5 ft. Mounting height above the work plane h = 10.0 ft. Spacing S = 12.0 ft, so the spacing-to-mounting-height ratio is 1.2. Candela values read off the photometric file for this distribution; treat them as illustrative stand-ins for the file you actually have.

Point A, directly beneath luminaire 1.

• From L1: theta = 0 deg, cos^3 = 1.000, E = 1,200 x 1.000 / 100 = 12.00 fc.
• From L2: offset 12.0 ft, theta = arctan(12.0 / 10.0) = 50.2 deg, cos = 0.640, cos^3 = 0.262, E = 620 x 0.262 / 100 = 1.63 fc.
• Total at A = 13.63 fc, horizontal, initial, direct component only.

Point B, midway between them.

• Offset from each luminaire = 6.0 ft, theta = arctan(6.0 / 10.0) = 31.0 deg, cos = 0.857, cos^3 = 0.630.
• From each: E = 1,050 x 0.630 / 100 = 6.61 fc. Two contributions = 13.22 fc, horizontal, initial, direct component only.

Max to min on the direct component = 13.63 / 13.22 = 1.03. At a spacing-to-height ratio of 1.2, the dip between fixtures is three percent, which is invisible.

Now the cost of dropping the incidence cosine. At point B the slant distance is sqrt(10.0^{2 + 6.0}2) = 11.66 ft, and a bare inverse-square calculation gives 1,050 / 136.0 = 7.72 fc per luminaire against the correct 6.61 fc. That is 16.6 percent high per contribution, which is exactly 1 / cos(31 degrees), and it is high every time, never low.

The same sheet at wider spacing

Change one input. Spacing S = 20.0 ft, ratio 2.0, everything else identical.

• Point B: offset 10.0 ft, theta = 45.0 deg, cos^3 = 0.354, E = 780 x 0.354 / 100 = 2.76 fc from each, 5.52 fc total.
• Point A: 12.00 fc from L1, plus L2 at offset 20.0 ft, theta = 63.4 deg, cos^3 = 0.089, E = 240 x 0.089 / 100 = 0.21 fc. Total 12.21 fc.
• Max to min = 12.21 / 5.52 = 2.21.

The spacing went up by a factor of 1.67 and the uniformity ratio went from 1.03 to 2.21. Almost none of that came from the square term, which only moved the slant distance from 11.66 ft to 14.14 ft. It came from the cosine cube: 0.630 at 31 degrees against 0.354 at 45 degrees, and from the intensity itself falling from 1,050 cd to 780 cd as the angle opened. The falloff between fixtures is an angle problem wearing a distance problem's name.

The same geometry on a vertical plane, which is a different formula

Everything above is horizontal illuminance. A great deal of real work happens on a vertical surface: a pick face, a shelf label, a panel schedule, a person's face in a corridor. Swap the plane and the trigonometry changes, holding the same mounting height above the point:

E (vertical, facing the luminaire) = I x sin(theta) x cos^{2(theta) / h}2

One power of cosine has been traded for a sine, because the angle a ray makes with a vertical surface's normal is the complement of the angle it makes with a horizontal one.

Run it at point B in the sheet above, on a face turned toward luminaire 1. Theta is 31.0 degrees, sin = 0.515, cos^2 = 0.735, so the product is 0.378. E = 1,050 x 0.378 / 100 = 3.97 fc vertical, at the height of the face. Only L1 contributes, because L2 sits behind the surface and lights its back.

That is 30 percent of the 13.22 fc horizontal reading at the same point. A meter laid on the floor in that aisle is reporting a number roughly three times the one the picker's task is actually working with, which is why the plane has to be stated on every value.

The vertical form also peaks somewhere unexpected. The product sin(theta) x cos^2(theta) is largest at about 35 degrees off nadir, not straight down, so a luminaire placed directly above a vertical surface is close to the worst place for it and one offset to the side does better. That single fact explains most aisle layouts that look wrong on a reflected ceiling plan and work in the building.

What this sheet cannot see

Every figure above is the direct component only. It counts photons that travel from the luminaire to the point without touching anything. In a room with light surfaces the interreflected component adds a fairly even wash on top, and it lifts the minimum far more than it lifts the maximum, so measured uniformity comes out better than a point calculation predicts. In a dark high-bay with exposed structure there is almost nothing to bounce off and the direct calculation is close to the whole story.

So the point sheet is a floor on uniformity, not an estimate of it. That is exactly what makes it useful for a spacing check and useless as a substitute for a full calculation. What flips the recommendation: if the room has 80 percent ceiling and 50 percent wall reflectance, a 2.21 direct ratio may measure closer to 1.6 once bounced light arrives, and the layout may be acceptable. If the surfaces are dark, 2.21 is what you get and the aisle is visibly striped.

Checking your own figures

• Point-source boundary respected. Required 5 x 1.5 ft = 7.5 ft. Shortest distance used anywhere in the sheet is 10.0 ft, at point A under L1. Passes.
• Geometry named before the exponent was picked. Compact luminaires, so 1 / d^2. Had these been a continuous row, the near-field exponent is 1 / d and the sheet would not apply.
• Cosine applied at every off-axis point. Angles used: 0.0, 31.0, 45.0, 50.2 and 63.4 degrees, each with its own candela value pulled at that angle rather than reusing the nadir 1,200 cd.
• The cosine correction printed, not implied. 7.72 fc bare against 6.61 fc corrected at point B, a 16.6 percent overstatement of the correct value if dropped, which is 1 / cos(31 degrees), running high.
• Plane switched with the formula, not just with the label. The vertical figure used sin x cos^{2, not cos}3, and dropped L2 entirely because it lights the back of that surface. 3.97 fc vertical against 13.22 fc horizontal at the same point, a ratio of 0.30.
• Convention on every result. All horizontal values horizontal, initial, direct component only; the vertical value marked vertical at the height of the face. None of them may be compared to a maintained target without the re-basing the maintained-illuminance card owns.
• Held-constant variable stated with the exponent. The cos^{3 / h}2 form holds mounting height above the work plane fixed at 10.0 ft and varies lateral offset. Change the mounting height and every cosine in the sheet changes with it.
• No constant appeared in an example that was not defined above it. The intensity table is the only new data, and it is labeled illustrative and sourced to the photometric file.

References

• IES recommended practice covering point calculations, spacing criteria and the point-source approximation, in the edition your specification or employer standard names, which binds through that document rather than on its own
• Manufacturer photometric report and candela table for the specific luminaire, which owns every intensity value in a point calculation
• See related: Why Lumens and Lux Are Not Interchangeable; What a Photometric File Actually Describes; What Light Is Measured In, and Why the Units Confuse People