Measuring throw: Is lux at 1 meter enough?

[I came to the light...]

The short answer is no.

On these and other flashlight boards it is commonly assumed that the throw of a flashlight can be calculated by measuring the lux at one meter and using the inverse square law to determine the distance that the beam will reach. Or, since we are just feeding different numbers into the same formula, we can be spared the algebra and use the lux measurement as an indicator of throw.

There is a fundamental problem with this logic. I recommend that you read a bit about the inverse square law, and you will likely draw the same conclusion I did. In summary, it is a strictly geometrical law concerned with a constant amount of light spread over a quadratically increasing area. The law assumes a rate of dispersion equal to that of light emitted uniformly in all directions from a point source, but the purpose of a reflector or optic is to alter that rate.

I was fortunate enough to benefit from get-lit's extensive knowledge in this area. If you really want to understand the science behind a reflector and its beam of light, or if you just want to read an explanation more eloquent and detailed than my own, his posts make good reading. The conclusion is that reflectors of any type shape light as if it were coming from a point a certain distance from its actual source, the length of which depends on the reflector, and therefore the inverse square law is accurate if one corrects for that distance.

Small reflectors require only a small correction, which yields results fairly close to an uncorrected formula. However, as reflector size increases, so does the distance between the virtual focus point and the point measured to, making the unmodified inverse square law less and less accurate.

But what is all this talk without data? I have measured two flashlights at 1 meter intervals from 1 to 5 meters, one with a small reflector and one with a larger one.
Megalennium + SureFire KT4 + WA1185 with partially depleted batteries:
1m: 35500
2m: 10100
3m: 4550
4m: 2710
5m: 1644

Barbolight T14:
1m: 12680
2m: 3220
3m: 1400
4m: 812
5m: 517

Since the Megalennium is not regulated its output declined during the testing to 34,300 lux at 1 meter. To compensate for this I based my calculations on the data corrected by the corresponding fraction of the percent different in output, assuming a constant percent decline between measurements for lack of a better method. For example, the measurement at 3 meters was multiplied by 1 + (2 * (35500 / 34300 - 1) / 5. This is the corrected Megalennium data:
1m: 35500
2m: 10171
3m: 4614
4m: 2767
5m: 1690

According to the inverse square law, the formula should be: lux = (lux at 1 meter) * distance ^ -2, or
Megalennium: lux = 35500*distance^-2
Barbolight: lux = 12680*distance^-2
The average percent difference between the value predicted by these formulas and the actual measurements, or the average percent error, is 12.61% for the Megalennium, and 1.30% for the T14. While the T14's data seems to fit the curve, presumably due to the smaller reflector, the Megalennium's data is more than a little off.

I recalculated the formulas using the form lux = a * ( distance + b ) ^ -2, where a is the theoretical lux at the vitual focus point, and b is the distance between the point measured to and the virtual focus point. The resulting formulas are:
Megalennium: lux = 47092.32 * (x + 0.1517) ^ -2
Barbolight: lux = 13084.77 * (distance + 0.0158) ^ -2
These formulas yield only 1.80% and 0.69% error respectively, which can easily be accounted for with human error and procedural inaccuracy. I believe that the Megalennium data is less accurate because of my approximate compensation for output decline over the testing period.

The most important difference between the accuracy of the two sets of formulas is that the Megalennium formula's accuracy increased sevenfold, compared to the Barbolight's two times, an effect predicted by the larger correction in the Megalennium's formula.

Using these formulas to calculate the throw of each flashlight, or the point at which the target is illuminated by 1 lux, the distances change from 188 to 217 meters for the Megalennium, and from 113 to 114 meters for the Barbolight. So while the correction may not be necessary for flashlight as small as a pocket LED flashlight, the results are significantly different in flashlights with larger reflectors. I would assume that the difference would continue to increase with reflector size and collimation factor.

Based on this theory and supporting data, I would like to propose a slight modification to the standard for reporting the throw of a flashlight: measuring the maximum lux at both one and two meters. These measurements can be compared directly to those of flashlights with similarly sized reflectors, and can also be used to calculate throw distance in order to compare to flashlights of all reflector sizes if the reader so desires. Better yet, this added accuracy comes without considerable increase in effort on the reviewer's part, and can be ignored if the reader so chooses. Reviewers, are you signed on?

 

[asdalton]

http://www.candlepowerforums.com/vb/showthread.php?t=243343

 

[I came to the light...]

http://www.candlepowerforums.com/vb/showthread.php?t=243343

Thanks for the link, I hadn't realized that mudman had already taken the discussion out of his fm3x thread. Although we are dealing with the same subject, it appears that he is taking a different approach, so I think that separate threads are justified. Give me a few minutes to go through his thread for more detail.

 

[I came to the light...]

Unfortunately, it seems that mudman_cj has set about proving the inverse square law shortly before I set about proposing an improvement. As much as I respect mudman and his work, I must say that he does not follow scientific procedure. He starts with the assumption that the unadjusted inverse square law is correct, and eliminates completely valid data points until the data set supports his original theory. He claims that they were measured too close to the emitter, yet he neglects to point out any reason why data measured close to the meter should be inaccurate, and in fact I believe that my testing shows that it is not the data that is inaccurate, but the unadjusted formula. So of course he concludes that the inverse square law is accurate; he starts with that as an assumption, and manipulates the data until it agrees with him.

My tests do not alter the data, except for the entirely separate purpose of compensating for battery use during the period of testing. I have measured several sets of data at similar distances as mudman_cj and calculated a line that fits all data points to 98.2% accuracy, and still obeys the inverse square law. The unadjusted formula cannot do this.

 

[saabluster]

he concludes that the inverse square law is accurate; he starts with that as an assumption,

I do not consider myself an expert in this area but by your own admission it is called the "inverse square law". Generally when something is called a "law" in scientific circles it means it has been throughly tested and found to be true. So it would seem to me that starting with that assumption would be the correct thing no? Maybe I missed something.:shrug:

 

[I came to the light...]

I do not consider myself an expert in this area but by your own admission it is called the "inverse square law". Generally when something is called a "law" in scientific circles it means it has been throughly tested and found to be true. So it would seem to me that starting with that assumption would be the correct thing no? Maybe I missed something.:shrug:

I certainly wouldn't put myself forward as an expert either, but I believe I can clarify this situation. I agree that it is safe to assume that the inverse square law is true for the situations it was made to apply to (which, I believe, includes flashlights with reflectors) - however, the function it describes does not change at a constant rate, and therefore it matters which point is regarded as x=0 (for a point source without a reflector, this would be the source of the light). My argument is that in a flashlight with a reflector that point is not at the bezel, or the emitter, but at a certain point determined by the reflector. I do not object to mudman_cj's assumption of the inverse square law, but of his assumption that a beam of light from a flashlight has exactly the same optical properties as a point source emitting equally in all directions (the inverse square law's ideal case).

Rereading my words, I sound more offensive than I intend to. I don't mean to attack mudman_cj, but I feel that my results are weakened by contradicting results obtained through unscientific methods. That is not to say that either experiment is or can be in our circumstances precise enough to be considered scientifically legitimate, but rather that scientific methodology should still be used as much as possible.

 

[mudman cj]

I don't think our approaches are uncomplimentary at all.

First of all, I did not assume that the inverse square law holds for flashlights and attempt to manipulate the data or exclude data in any way in order to show adherence to the inverse square law. Secondly, I did not assert that the inverse square law holds true at close distances. I found that it does not hold true under the conditions I used, such as measuring distance to the lens of the flashlight. If anything, your work confirms this finding and then adds to it.

What I did was to fit the data using a least squares regression of a power fit. I allowed the exponent to be whatever the least squares regression determined it should be. Then, I used the goodness of fit, R^2 as the sole criterion for deciding if data points should be removed. So why did I begin by removing data points obtained from peak lux measurements closest to the light? For the same reason you have decided to adjust the inverse square relation for close distances: because the optics system causes a deviation from the inverse square law when using the distance to the lens of the light in the equation. At greater distances the inverse square law just so happens to fit well using this distance metric.

You seem to have some misunderstandings regarding my work, and I would appreciate if you would provide greater consideration before continuing to make disparaging remarks about it.

 

[Justin Case]

At close range, especially for the big reflector lights at 1 meter, the assumptions for inverse square probably don't hold, i.e., the source is not a good approx of a point source. Thus, force-fitting an inverse square regression curve to data that is invalid doesn't make a lot of sense to me. What mudman did was to identify the range at which the inverse square law does start to hold and in fact his regression fit then shows almost exact agreement with a 1/R^2 falloff.

I can fit an N-1 order polynomial curve exactly to any set of N data points. But that doesn't mean that the function has any bearing on reality.

Can you give a physical explanation for the extra terms in your regression equation and why those terms also apply at long range where the source should be a good approx for a point source and thus plain old 1/R^2 should apply? I would wager that if you went to greater ranges that your regression curve that you found for 1m-5m would start to diverge from the measured data because your "correction terms" really apply only for closer ranges.

 

[qwertyydude]

Sometimes it doesn't work exactly as measured at one meter. Take a look at the beam in the fog or smoke. Some lights will actually focus down to a focal point somewhere in front of the lens thereby artificially inflating the lux at just that focal point but will quickly diverge after it. But some lights have a focal point at infinity in which the beam doesn't converge at any point and that's when the inverse square law with lux will be more applicable. I'd say it might be better to measure lux at 1 meter then again at 3 and 5 meters to get an idea of the lux drop off curve since it may not necessarily be exactly inverse square because of the focus of the light.

 

[asdalton]

I guess it was a mistake for me to post that link without an explanation. My intent was to show that people are already aware that measuring the beam-center lux at 1 meter is inadequate in many cases.

 

[2xTrinity]

Well, the important metric when determining "throw" is finding the beam intensity in Candela. In the case of an ideal point source, 1 lux (1 lumen/square meter) @ 1m distance is dimensionally equivalent to 1 candela (1 lumen/steradian of solid angle). This is where the lux @ 1m as a metric for throw came from. IMO it would be a lot less confusing if instead of converting to "equivalent lux @ 1m" we actually used the word Candela when discussing baem intensity (not illuminance on a surface).

Take a look at the beam in the fog or smoke. Some lights will actually focus down to a focal point somewhere in front of the lens thereby artificially inflating the lux at just that focal point but will quickly diverge after it.

In this case, inverse-square falloff would roughly occur, but you'd get a more accurate fit if you calculated distance from the focal point, NOT from the original location of the flashlight. So long as you measured intensity from far enough away that the light appears to be a point source.

But some lights have a focal point at infinity in which the beam doesn't converge at any point and that's when the inverse square law with lux will be more applicable. I'd say it might be better to measure lux at 1 meter then again at 3 and 5 meters to get an idea of the lux drop off curve since it may not necessarily be exactly inverse square because of the focus of the light.

In most cases it's pretty easy to tell if the light will converge to a focal point. However, even this is not toatlly that simple. A flashlight is really more like two point sources -- one of which is the "spill", or the spreading out of the light from the filament or LED die itself. This will have it's own inverse-square falloff that will be very rapid.

There will also be the image of the filament or die from the reflector depending on the focal length and other properties, which will tend to have both a different falloff rate and possible a diffferent apparent starting point compared to the actual physical location of the light.

However, at very far distance (ie, tens of meters away -- the distance of interest for most "throw" type lights) the reflector falloff (from its apaprent starting point which may be slightly in front or or behind the actaul light) will dominate, and an inverse-square decay will be roughly correct.

 

[bluepilgrim]

Note http://hyperphysics.phy-astr.gsu.edu/HBASE/vision/isql.html says 'light from a point source' -- but with optics and reflectors it's not a point source anymore. This works with electromagnetic radiation -- but transmitting radio waves is often done with directional antennas, so there we are: the actual radiation pattern and drop-off depends on the source configuration (reflectors, optics, etc.).

Note also
http://hyperphysics.phy-astr.gsu.edu/HBASE/forces/isq.html#isq
"Any point source which spreads its influence equally in all directions without a limit to its range will obey the inverse square law. This comes from strictly geometrical considerations."

The issue must include the question 'how closely does the light source resemble a point source?'.

 

[asdalton]

It's not necessary for the radiation to be uniform in all directions for the inverse square law to hold. It just needs to be spread out in a cone (or some other constant solid angle).

Even a laser beam spreads out this way eventually.

 

[LuxLuthor]

Good information. I think the key in the instances of measuring flashlights with a reflector and lens is understand that the Inverse Square Law assumes the requirement of light originating from a point source generating a uniform output. What I believe bluepilgrim is trying to address is the alteration of the point source when a reflector and/or lens adds or subtracts output.

From page 25 of Ryer's Manual:

Point Source Approximation

The inverse square law can only be used in cases where the light source approximates a point source. A general rule of thumb to use for irradiance measurements is the “five times rule”: the distance to a light source should be greater than five times the largest dimension of the source. For a clear enveloped lamp, this may be the length of the filament. For a frosted light bulb, the diameter is the largest dimension.
​

For example, if you measure the Lux at 1 meter of a bare LED protruded through a tiny hole of black cardboard, and then add a reflector around the same LED, the readings will not be the same.

Once you get into adjustable flashlight reflectors, then specific measuring guidelines must be added since various locations within a flashlight's cone output will fluctuate brightness, as well as degree of focal length adjustment, source output power, etc.

That's why I was happy to see the guidelines listed in the NEMA standards seeking to control for the many factors.

 

[xenonk]

Take a look at the beam in the fog or smoke. Some lights will actually focus down to a focal point somewhere in front of the lens thereby artificially inflating the lux at just that focal point but will quickly diverge after it.

 

[bluepilgrim]

OK -- take this with a grain of salt because I am somewhat confused about it, but it seems to me that there two things to consider:
One is the angle of the beam as it comes from the source, and the other is the natural divergence from the wave nature of light (which is seen in the case of a laser).

For the first, consider a point source with some optical focusing apparatus, such as a lens or mirror. By changing where the source is in realtion to the optic we can get a beam which emerges approximately as a parallel beam, or as a divergent beam; think of this like shooting a machine gun in one direction or sweeping an area: the bullets (photons) would end up in a tight group or over a wide area. If you used a shot gun the pellets would diverge in any case, but you could control that by using a different choke.

Looking at http://en.wikipedia.org/wiki/Beam_divergence I see something called 'far field' -- "away from any focus of the beam". It seems to me that you can focus (collimate) your light to emit a parallel beam, and that it will remain tight, but only for some distance, depending on wavelength and aperature diameter (which is affected by diffraction), so it will diverge -- but at a different rate than if the focus is varied. See http://en.wikipedia.org/wiki/Collimated_light : "A perfectly collimated beam with no divergence cannot be created due to diffraction, but light can be approximately collimated by a number of processes, for instance by means of a collimator. Collimated light is sometimes said to be focused at infinity. Thus as the distance from a point source increases, the spherical wavefronts become flatter and closer to plane waves, which are perfectly collimated."

A focused -- collimated -- beam will have a lot more throw than a point source, so the lux readings obtained will depend on the optics of the light (throw vesus flood).

It's like sending off a group of little kids -- eventually they will scatter all over, but they will stay together longer if you send them off in one direction than if you tell them to just run off.

 

[I came to the light...]

I'm sorry I have left questions hanging this long. I haven't been on the forum much, and while I have been posting, I think that the replies to this thread warrant more than a brief response.

I see three factors of throw being discussed: angle of emission, size of the emitting area, and diffraction. The first is the only issue addressed by my first post. The inverse square function relies on correct identification of the source of light (x=0), and reflectors or optics effectively move that source away from the emitter. Consider this diagram; the actual source is the dark green circle, but the rays appear to have originated at the light green circle due to the fact that the optics alter than angle of emission.

My proposal is to improve estimates of throw distance by finding where the rays have effectually originated. I say that the inverse square law aplies at all distances, but the distance must be measured from the correct point. Since the rate of change of the inverse square function decreases over distance, an inaccurate distance measurement will eventually become insigicant, thus the illusion that data fits an offset inverse square function only after a certain distance. Imagine a flashlight pointing across a field - the light is much brighter at 1m than at 2, but almost the same at 200 and 201. Here are two offset inverse square function illustrating the point - there is a large difference near 0, and a small one going right.

asdalton, you're right about the cone - because a cone is a part of a sphere. And the only reasons lasers will eventually obey the inverse square law (or always do with a very large distance offset) is that nobody makes true lasers - dispersion is artificially introduced by slightly curving the rear mirror (please excuse my simple terminology). A true laser, emitting parallel rays of light, does not obey the inverse square law, although it does diffract in the atmosphere.

qwertyydude, you have made some good points. The idea is that the emitter location correction will account for converging beams by moving the source to the point of convergence (we would run into trouble only if the point of convergence was between the two distances measured). Regarding parabolic reflectors or lasers, if the rays are truly parallel, then the only dispersion will be due to the atmosphere, or none in a vacuum. However, this cannot be achieved in a flashlight because we have no point sources of light or perfectly parabolic reflectors.

The second issue is noted by bluepilgrim: there is no point source of light; the emitter has a certain size, and the reflector or optic effectively enlarges that size. The inverse square law assumes a perfect sphere of expanding light, but in reality there is a blur of a near infinite number of spheres. mudman_cj, I believe that your experiments target this issue by finding the distances at which the inverse square law becomes more accurate (the size of the emitter becomes irrelevant), thus supporting the five times rule of thumb brought up by LuxLuthor. I was wrong to accuse you of manipulating data when in fact you were only improving the formula along a different line of thought, and I apologize. However, I do think that it is not the optical system that is causing the deviation at short distances, but the size of the emitter. By combining the distance calculation and five times rule we should be able to obtain the most accurate results.

Justin Case, my method does not fit an N-1 order polynomial to N data points - it demonstrates that the inverse square law, a 2nd order polynomial, fits N data points. The physical explanation is that I am finding the average source of the light; there is no reason to assume that the lens should be considered distance 0. Also, as distance increases, any offset will not become exaggerated, but rather insignificant - both equations will converge; see the graph above.

Diffraction in the atmosphere (the "natural divergence from the wave nature of light") will certainly increase the rate of dispersion, but the amount is so slight in a clear atmosphere that I think we can ignore it. I like the litte kids analogy 🙂

Regarding candela, the only reason I can think of for preferring lux at 1 meter is that candela implies a point source.

 

[bluepilgrim]

It's nap time, so I can't consider all the points, but I think there is a difference between inherent dispersion and atmospheric scattering -- the former being a weaker effect, I *think*, and possibly not significant over the distance of a flashlight beam; scattering can be very significant, as we see in fog or smoke. As to second order -- squaring -- I should think it would vary with what variable is being squared (i.e. 1.001 or 15 -- what the scale is) -- but that's something I'd have to look into and is beyond me at the moment. Also something I need to look up and consider when I'm more awake would be the difference in dispersion between a light beam and coherent light (laser), and maybe the color and color purity -- but I don't know how the numbers work out.

Possible the better way to do this is to devise some imperical measure instead of trying to work out all the theoreticals, such as how a car is rated by the time it takes to accelerate to some speed instead of trying to plot horsepower at the engine, the transmission, through the drive train, and all -- which is a lot like trying to rate a light in either lux or lumens (with complications with eye color sensitivity along the way).

The old paraffin block Joli photometer seems to have worked pretty well for subjective sight -- http://www.sciencebuddies.org/science-fair-projects/project_ideas/Phys_p031.shtml -- assuming we could establish a standard light source. We might use a series of numbers, say at 10 degree intervals, to rate the flood and throw. I don't see how it's possible to use just one number to describe what is actually a more complex situation.

 

[I came to the light...]

It's nap time, so I can't consider all the points, but I think there is a difference between inherent dispersion and atmospheric scattering -- the former being a weaker effect, I *think*, and possibly not significant over the distance of a flashlight beam; scattering can be very significant, as we see in fog or smoke. As to second order -- squaring -- I should think it would vary with what variable is being squared (i.e. 1.001 or 15 -- what the scale is) -- but that's something I'd have to look into and is beyond me at the moment. Also something I need to look up and consider when I'm more awake would be the difference in dispersion between a light beam and coherent light (laser), and maybe the color and color purity -- but I don't know how the numbers work out.

Possible the better way to do this is to devise some imperical measure instead of trying to work out all the theoreticals, such as how a car is rated by the time it takes to accelerate to some speed instead of trying to plot horsepower at the engine, the transmission, through the drive train, and all -- which is a lot like trying to rate a light in either lux or lumens (with complications with eye color sensitivity along the way).

The old paraffin block Joli photometer seems to have worked pretty well for subjective sight -- http://www.sciencebuddies.org/science-fair-projects/project_ideas/Phys_p031.shtml -- assuming we could establish a standard light source. We might use a series of numbers, say at 10 degree intervals, to rate the flood and throw. I don't see how it's possible to use just one number to describe what is actually a more complex situation.

I think you're mistaking atmospheric dispersion as a property of light. In a perfect vacuum an ideal laser will never disperse. On a clear night I think atmospheric dispersion is irrelevant at the ranges flashlights can operate at. If it is foggy or raining, or there is some other obstruction in the atmosphere, you are right that throw can be significantly reduced; therefore I would take it as an assumption that all measurements are taken when the air is as clear as possible.

Manually finding the range of every flashlight could be very inconvenient, and inaccurate due to instruments and ambient light. However, this should be done at least a few times to verify whatever mathematical formula results from this and other discussions.

A single throw number is meant to represent the maximum distance of the beam, no matter how small a spot at that distance is illuminated. But I agree that beam profiles are immensely useful.

 

[asdalton]

In a perfect vacuum an ideal laser will never disperse.

Not true.