Optics 017 · Light a desk evenly

Illumination & Radiometry Studio

How do you light a flat surface evenly when light fades with distance and angle? Spread several lamps along it and hang them at the right height. Light from one lamp fades with the square of the distance and with the slant at which it lands, so the spot right below is brightest. Raising the lamp evens the surface but dims all of it. Several lamps fill each other’s dim edges; a broad panel mostly softens the shadows and evens the light only a little.

Hang one to four lamps over a desk, from a bare LED to a broad panel, and watch the illuminance, its evenness and a bottle’s shadow change.

Average over the desk (target 500 lx) $\text{—}$

Uniformity, dimmest over average (target 0.60) $\text{—}$

Illuminance at the probe $\text{—}$

Light left at the darkest point of the bottle’s shadow $\text{—}$

Luminance of each lamp’s face $\text{—}$

Dimmest and brightest point of the desk $\text{—}$

Probe’s distance and angle from the nearest lamp $\text{—}$

At the probe if each lamp were a point of the same flux $\text{—}$

Share of the lamps’ light that lands on the desk $\text{—}$

Desk cells computed for this record $\text{—}$

Each lamp is a square face that emits equally in every direction from every point, a Lambertian source, facing straight down; its luminance is its flux divided by π times its area. The illuminance it gives each point of the desk is computed exactly from the configuration factor of a parallel rectangle, Hottel’s 1931 result as catalogued by Howell, with no small-source approximation. Only direct light is counted: the desk, walls and ceiling reflect nothing back, as if the room were black, so a real room would read higher and more even. Lumens and lux are photometric units, weighted by the eye’s daylight sensitivity, as defined in NIST Special Publication 250-37 (Ohno 1997). The desk is 1.2 m by 0.7 m and its map has 1 cm cells; the average, minimum and uniformity are taken over the empty desk on a 2 cm grid. The bottle is an opaque cylinder 7 cm across and 22 cm tall; what part of a lamp it hides is found by splitting the lamp’s face into 24 by 24 squares, 48 by 48 for the probe and the cut through the shadow. The task targets, an average of 500 lx with a uniformity of at least 0.60, are the EN 12464-1 entry for writing, typing, reading and data processing as reproduced in public summaries; this Lab applies them to the whole desk without the standard’s maintenance factor, surroundings or glare limits, so it is a design exercise, not a compliance calculation.

Physics tutorial

Light a desk evenly with lamps whose light fades with distance

BackgroundIlluminance is the light landing on each square metre of a surface, measured in lux: one lumen per square metre. A reading desk is usually lit to about 500 lx. Light spreads out as it travels, so the same lamp lights a near surface more brightly than a far one.

Why it mattersA lighting designer wants the whole desk bright enough and nearly equally bright everywhere, without hard shadows. The European standard EN 12464-1 asks for an average of 500 lx for writing and reading, and a uniformity, the dimmest point divided by the average, of at least 0.60. With a single lamp these two goals pull against each other.

Start with the essentials

Focus question: One 800 lm LED hangs 0.6 m above a 1.2 m desk. Why is the desk five times brighter under it than at its corners, and what does it take to meet both targets?
One-sentence intuition: A small lamp facing down sends out the intensity $I_0\cos\theta$ at an angle $\theta$ from straight down. On its way to a point of the desk at distance $d$ , that light spreads over an area growing as $d^2$ , and it lands at the slant $\theta$ , so $E=I_0\cos^4\theta/h^2$ . At the desk’s far corners, 0.92 m from a lamp 0.6 m up, $\cos^4\theta$ is only 0.18. Raising the lamp narrows the range of angles and evens the desk, but dims all of it as $1/h^2$ . Several lamps spread along the desk fill each other’s dim edges, so the desk can be bright and even at once.

Core mathematical model

A small lamp facing down

\begin{gathered}I(\theta)=I_0\cos\theta,\quad I_0=\frac{\Phi}{\pi}\\E=\frac{I(\theta)\cos\theta}{d^2}=\frac{I_0}{h^2}\cos^4\theta\end{gathered}

A flat face that looks equally bright from every direction, a Lambertian source, sends its greatest intensity $I_0$ straight down and less at a slant; its whole flux $\Phi$ is $\pi I_0$ . The illuminance $E$ falls as $1/d^2$ and with the slant at which the light lands. NIST Special Publication 250-37 (Ohno 1997) defines these photometric quantities.

A lamp with a real face

\begin{gathered}L=\frac{\Phi}{\pi s^2},\qquad E=\pi L\,F\\E_0=4L\,\frac{a}{\sqrt{1+a^2}}\arctan\frac{a}{\sqrt{1+a^2}}\\a=\frac{s}{2h}\end{gathered}

A square face of side $s$ has luminance $L$ , its brightness as the eye sees it. The illuminance it gives a point is $\pi L$ times the configuration factor $F$ , the share of the point’s view the face fills, weighted by slant; Hottel (1931) gave it in closed form for a parallel rectangle. Straight below the centre this is $E_0$ . For a small face $E_0\approx(I_0/h^2)(1-s^2/3h^2)$ , the inverse square law; under a face much wider than its height, $E_0\to\pi L$ .

The task targets

U_0=\frac{E_{\min}}{\bar E},\qquad\bar E\ge500\,\mathrm{lx},\quad U_0\ge0.60

EN 12464-1 lists these for writing, typing, reading and data processing, as maintained values over the task area. The Lab applies them to the whole desk and counts direct light only, with nothing reflected from the room, so a real room would read higher.

The edges of a soft shadow

\begin{gathered}x_{\pm}=\frac{(\rho+r)\,h\mp\tfrac{s}{2}\,h_{\mathrm{b}}}{h-h_{\mathrm{b}}}\\w=x_{-}-x_{+}=\frac{s\,h_{\mathrm{b}}}{h-h_{\mathrm{b}}}\end{gathered}

Along the desk from the point below the lamp’s centre, through the bottle’s axis at distance $\rho$ : light from the face’s far side grazes the top rim of a bottle of radius $r$ and height $h_{\mathrm{b}}$ and reaches the desk at $x_{-}$ , light from its near side at $x_{+}$ . In this plane the whole face is hidden short of $x_{+}$ and none of it beyond $x_{-}$ . The soft edge between them is $w$ wide: 1.2 cm for a 2 cm LED at 0.6 m, 35 cm for a 0.6 m panel.

Common difficulties

Distance is not the whole story

Typical misconceptionIlluminance falls only with the square of the distance.

Better mental modelIt also falls with the slant of the light. For a small lamp facing down, the slant costs two more factors of $\cos\theta$ , one as the lamp sends less light sideways and one as the light lands at a slant, so the corners get 0.18 of the light straight below rather than the 0.43 that distance alone predicts.

A bigger lamp is not a brighter one

Typical misconceptionA larger lamp puts more light on the desk.

Better mental modelAt the same flux it gives out the same light, spread over more directions. In the Lab the 0.6 m panel puts 36 % of its flux on the desk against the bare LED’s 40 %, and its face is 900 times less bright.

The inverse square law has a range

Typical misconceptionThe inverse square law holds at any distance.

Better mental modelIt holds for a point, and for a real source only once the distance is several times the source’s size. NIST’s photometry guide warns that it does not hold well close to a large lamp (Ohno 1997); 0.3 m below a 0.6 m panel it overestimates the light by a factor of 2.3.

Run the experiment

01
See the bright spot
Keep the default preset, Bare LED: a bright spot and a hard shadow.
What to observe: The desk is brightest straight below the LED, 707 lx, and falls to 135 lx at the corners. The average is 385 lx and the uniformity 0.35, short of both targets. Of the lamp’s 800 lm, 40 % lands on the desk.
02
Move the probe
Slide the probe from 0 to 0.4 m along the desk.
What to observe: Straight below the lamp the probe reads 707 lx; at 0.4 m it reads 339 lx, 0.72 m from the lamp and 34° from straight down. A point source of the same flux gives the same readings to within 0.1 %, because the LED is only 2 cm wide.
03
Raise the lamp
Drag the lamp up to 1.2 m, or load Raised lamp: even but dim.
What to observe: The uniformity rises to 0.70, above 0.60, but the average falls to 145 lx and only 15 % of the light lands on the desk. In the plot of evenness against height the curve is solid, at or above 500 lx, only below 0.45 m, where the uniformity is under 0.25: one 800 lm LED cannot meet both targets.
04
Widen it into a panel
Load Wide panel: the shadow softens.
What to observe: The same 800 lm now leaves a face 0.6 m wide, 900 times less bright to look at: $707\,\mathrm{cd/m^2}$ instead of $637\,000\,\mathrm{cd/m^2}$ . The bottle’s shadow gets a soft edge 35 cm wide, and even its darkest point keeps 12 % of its light, because a 7 cm bottle cannot hide the whole panel. The uniformity improves only from 0.35 to 0.43.
05
Get close to the panel
Load Low panel: the inverse square law fails.
What to observe: With the panel 0.3 m up, the probe at 0.2 m reads 952 lx, where a point of the same flux would give 1360 lx. Straight below, the panel gives 1230 lx against 2830 lx for a point, 2.3 times less. In the distance plot the two curves agree to 1 % only beyond about six panel widths, 3.4 m.
06
Spread the lamps
Load Three panels: both targets met.
What to observe: Three 0.4 m panels 0.85 m up and 0.4 m apart give an average of 638 lx and a uniformity of 0.66, so both targets are met; any height from 0.70 to 1.00 m works. Three bare LEDs in the same places do as well, 661 lx and 0.65, but cast hard shadows: the spacing and the height even the desk, the panels soften its shadows.

Illumination & Radiometry Studio

Light a desk evenly with lamps whose light fades with distance

Core mathematical model

A small lamp facing down

A lamp with a real face

The task targets

The edges of a soft shadow

Common difficulties

Distance is not the whole story

A bigger lamp is not a brighter one

The inverse square law has a range

Run the experiment

See the bright spot

Move the probe

Raise the lamp

Widen it into a panel

Get close to the panel

Spread the lamps

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