A simpler, more reliable alternative to the integrating shoebox

QReciprocity42 · May 14, 2026, 3:02am

This post is partly motivated by a discussion with @Dc38, who proposed diffusion as an alternative to integration for the purpose of measuring total luminous output. TL;DR: the diffusion device below is easier to build than the integrating shoebox, and has a 6x better worst-case precision guarantee (+/- 3% compared to +/- 19%) over different beam profiles.

The idea is very simple: if you have a device that converts any beam profile into a fixed, universal beam profile (like the uniform or Lambertian), then the peak intensity of the resulting beam profile is proportional to the total flux by a universal constant, which can in turn be deduced with a single calibration measurement.

My version of this device is very simple: a piece of printed paper framed in cardboard.

This device attempts to convert any beam profile into a universal beam profile close to the Lambertian, with a constant multiplicative loss in total output (which is absorbed during calibration). Let’s see how well it fares!

**Experiment setup for paper screen**

A sensor (OnePlus 12 phone running the latest ceilingbounce release) is placed 1m behind the paper screen. The test light is a custom Fresnel lens thrower (modified S2+) running the 17mm 3V 5A buck driver at 10%, to ensure stability of output.

The reason for choosing this nonstandard light is that its rays are extremely close to being parallel, unlike that of a reflector light, which usual span 60 degrees. This consistency in emission angle allows quantification of how much the measured output is affected by emission angle of the light, which equals the incidence angle to the screen. Recall that a beam profile is simply a probability distribution on the space of emission angles.

I will take 3 measurements: with the beam pointing directly onto the screen, with the beam offset by 30 degrees, and with the beam offset by 60 degrees. For each angle, readings will be taken for 30 seconds, and an interval containing the persistent maximum and minimum raw readings is recorded.

Here are photos of the beam offsets by 0, 30, and 60 degrees:

**Results for paper screen**

Below are raw readings from ceilingbounce.

0 degrees offset: 13.75 +/- 0.10
30 degrees offset: 12.95 +/- 0.10
60 degrees offset: 11.34 +/- 0.10

For all lights that have a beam radius less than 30 degrees (which consists of pretty much every thrower, LED or LEP, and pretty much every reflector light), we see that the beam profile can introduce a variation of at most 13.75/12.95 -1 = 6.2%; if you calibrate to their average, you get a variation of +/- 3.0%.

For floodier lights with most emission bounded within a 120-degree cone (which includes pretty much everything that’s not a mule), you get a variation of at most 13.75/11.34-1 = 21.3%; if you calibrate to their average, you get a variation of 9.6%. In practice, the measurements will be more precise than this worst-case performance guarantee suggests, since most of the beam has an offset angle close to 0.

Now how does this compare to an integrating shoebox?

**Integrating box measurements**

I put together a quick paper-lined box, and refocused the test light so that it converges prematurely to an image that is small enough to go through the hole in the box. An advantage of the paper screen is the ability to measure lights with large heads.

Since the box is not radially symmetric, specifying an offset angle is not enough–one needs to also specify the direction of offset. I have taken 5 measurements: 1 with no offset plus 4 with offsets of 30 degrees in the left, right, up, and down directions. Here are the results:

No offset: 835 +/- 1
30 degrees left: 1020 +/- 2
30 degrees right: 857 +/- 3
30 degrees up: 1237 +/- 3
30 degrees down: 1036 +/-3

For lights with emission confined in a 30-degree radius cone, the worst-case variation due to beam profile is as high as 1237/835-1 = 48.1%; calibrating to the average turns that into +/- 19.4%. That’s over 6 times the 3.0% error of the paper screen!

Dc38 · May 14, 2026, 1:05pm

Awesome! If possible, would you be willing to test with a second diffusion layer with varying spaces between the diffusion layers? i.e., “contact layer” remains the lambertian emitter, a second layer smooths the beam *further?

jon_slider · May 14, 2026, 3:08pm

thanks for your efforts

It is also possible to calibrate the device, by increasing the number of diffusion layers (paper or other material) , so the lux meter reads directly in equivalent value to lumens. With no need for conversion factor calculations..

That is how the Texas Ace tube is calibrated..

he added diffusion layers to convert the lux meter reading to equivalent lumens, so his device can be read in lumens directly.

QReciprocity42 · May 14, 2026, 5:24pm

Increasing the number of diffusion layers is a good suggestion, and I will test it at some point. My main concerns are

A single diffusion layer is already very lossy, and adding more layers causes the measurable lux to drop exponentially, making the measurement of relatively low-output sources difficult. In my setup, I already cannot measure the 1% mode because the lux registered is low enough to be hardly separable from noise.

However, this is not a limitation in actually using this device to measure lumens on low settings, because a single measurement at max output establishes the proportional constant between lumens and lux for this light; from this point on, it suffices to simply measure lux and convert to lumens via that constant.

For most lights, a single diffusion layer already narrows the measurement precision to +/-3%, which I’m almost certain is below the error introduced by other, uncontrollable sources, such as the accuracy of the lux meter itself, or just variation among different samples of the same light. For this reason, I see little point in chasing even better precision–recall that this method already has a 6x better precision guarantee than the integrating shoebox!

Regarding calibration: I worry that increasing diffusion to calibrate might be more trouble than it’s worth, since the diffusion material needs to be very finely tunable/adjustable, which printer paper isn’t. Multiplication by a constant is really not a hassle, and even that can be automated away by entering the “lux per lumen” factor in the ceilingbounce app.

I also like that my current setup has no moving parts whatsoever, which ensures consistency of measurements every time it is picked up or taken out of storage.

Oli · May 15, 2026, 3:27am

The problem as I see it is that you can’t measure how much you are losing. How much light is reflected and how much light is absorbed by the paper. There are likely many variables that go into making paper. Different types of wood supplies that could change on a weekly or monthly basis. Fillers and brighteners. Manufacturers are not really concerned with how much light can pass through. They do measure the opacity and try to keep that consistent but that is just measuring how much light is reflected. How much of it is absorbed is not measured and how much is transmitted is not measured
They do want to make sure that print can’t be seen from the other side. Cooler colors will reflect off of that white surface more than warmer colors.

QReciprocity42 · May 15, 2026, 1:50pm

This turns out to not be a problem for a simple reason: calibration with a known light absorbs away all that uncertainty.

Suppose you have a light that you know is 1000 lumens. You get a piece of paper from batch A, and get a raw reading of, say, 20. Then you get a calibration factor of 1000/20=50 lumens per lux. If you get a piece of paper from batch B with only half the transmissivity, you’ll get a raw reading of 10, and get a calibration factor of 100 lumens per lux. In either scenario, you get to move on and measure other lights with no problems.

There are many things we can’t measure about the whole process–how much you’re losing, as stated, but also things like how much the diffuser reduces the intensity, and how many lux the phone detects per lumen emitted. But it all doesn’t matter because the purpose of performing calibration is to produce a single constant that takes care of all that uncertainty.

Another way to think about it is: doesn’t matter what the transmissivity is, the relation between lumens emitted (E) and lux measured (M) is: E = cM for some constant c that absorbs the uncertainty you stated in the measurement process. If you have a calibration light with known E and get a measurement M, you now have enough data to deduce what c is.

Does this make sense?

I’m not concerned with brighteners because they only affect light with UV output, which excludes all white LEDs. There may be some unevenness in transmission/reflection rate across the visible spectrum, but the variation should be small enough, and if this is a problem, then it also affects integrating shoeboxes.

Oli · May 15, 2026, 3:39pm

After bouncing some of this off of an AI and better understanding how paper manufacturers mostly want to concentrate on opacity. They use a black background behind the paper and a white background behind the paper and a color temperature that closely matches an incandescent bulb and another standard that some use that is closer to sunlight.
I don’t think the transmissive rate will be equal amongst all color temperature lights. I also think you should experiment with different distances with the light meter to potentially rule out some variables. I would suggest 10 or 20 cm and also move it a few cm in either direction off of center to rule out patches or different areas of the paper that may not be basically the same. And I really think that the beam shape is going to introduce large variances. Although this should be fairly easy to test with any single light where you can change the TIR.

QReciprocity42 · May 15, 2026, 6:15pm

That is a reasonable concern, and I will think of some experiments to test this. My initial sense is that this should not introduce enough error to significantly throw off measurements, because the color of light transmitted through the paper is not noticeably different from the light pre-transmission. But of course, it would be good to get quantitative estimates.

Also a great suggestion. Ideally, the sensor should be as far from the paper screen as possible, so that different parts of the screen are roughly equidistant from the sensor. At 1m distance (as in the experiment), a piece of letter paper (with semimajor axis 140mm) introduces a distance variation no more than sqrt(1^2+0.14^2)-1=0.975%, and thus an intensity variation no more than (1^2+0.14^2)-1 = 1.96%. This is significant but can be reduced by moving the sensor back. Also, most lights have a beam that puts most of the light in a much smaller circle, much smaller than 280mm diameter.

The stated purpose of this experiment is to quantify how much different beam profiles can mess with the readings. The conclusion was that all beam shapes contained in a 60-degree cone can introduce variation no more than +/-3%, and that beams shapes contained in a 120-degree cone can introduce variation no more than +/-10%.

Whether these variances are large is up to debate, but the +/-3% variance of the paper screen is certainly better than the +/-19% variance of the shoebox, for the 60-degree containment cone.

Boaz · May 16, 2026, 6:13am

Posting in the thread…
thinking I’m in a private message .