mudman cj
Flashlight Enthusiast
How to more accurately measure relative peak lux & confirmation of inverse square law
In another thread a member questioned the applicability of the inverse square law for focused beams of light such as those produced by flashlights. They questioned whether the inverse square law holds true for lasers (I have not verified this myself in any way), and this led them to also question it for flashlights. They then presented a few Lux measurements of an 1185 in a KT4 turbohead that indeed seemed to deviate from the law.
This led me to question whether flashlights do indeed follow the inverse square relation or not. So, I decided to buckle down and take peak lux measurements at 11 distances from 1 meter to 6 meters in 0.5 meter increments using an assortment of 10 different flashlights.
110 lux measurements and a lot of data analysis later, I think I have confirmed that the inverse square law does hold for flashlights under certain conditions. I have also developed what I think is a more accurate way of measuring the approximate peak lux of a flashlight and I would like to share this with you. When I say "more accurate" what I mean is compared to the method of taking one measurement, regardless of distance, and reporting this result (typically adjusted to 1 meter).
I think the reason why the peak lux measurements do not follow the inverse square law at close distances is due to effects from the optics. This may just be due to my using the distance from the lux meter to the lens of the flashlight instead of some virtual point from which the light source seems to be emanating. Regardless, I do not feel the need to develop a model that more accurately predicts light intensity at close distances. That is not important to the goal that I have with respect to this effort.
My primary goal here was to develop a simple testing method that affords an improved degree of accuracy compared to the single measurement method. This test method is useful for improving relative peak lux measurements, but is certainly not sufficient for making absolute measurements that may be published in a scientific journal. That would require attention to many more details, some of which were listed by LuxLuthor in a link he provided in post #38 of this thread.
For one thing, there are difficulties associated with locating the peak light intensity in a beam. Then there are a host of other variables which may affect the output. LuxLuthor has done a good job of listing many of these variables in the aforementioned link, and some of them include: bulb focusing, reflector finish differences, lens material, ambient reflections, light meter angle, spectrum differences between light sources being compared, light meter spectral response, light meter linearity (calibration), bulb to bulb differences, flashlight resistance, battery state of charge and type, heating of LED die(s), bulb/LED centering within optics, etc.
One example I would like to mention is that differences between LED spectra such as between 5mm, cool white, neutral white, warm white, high CRI affect lux readings because each of these emits light that varies in intensity with frequency while a particular light meter has a fixed response with respect to each frequency. In the case of incandescent light sources, the degree of bulb overdrive or whether the bulb is of the IRC type also has an effect on the spectrum of light and therefore even when comparing two incandescent light sources these types of differences can occur.
The point is, this is at best a relative indication of differences in peak light intensity as measured by a particular meter and variables such as those mentioned need to be accounted for or at least considered when making comparisons, so keep that in mind.
How to calculate approximate relative peak lux more accurately by averaging readings taken at multiple distances
First, you need to measure the peak lux of your flashlight at many different distances. I recommend using multiple points as I have done, starting at a minimum of 3 meters and moving away in 0.5 meter increments. This allows readings to be done indoors with relative ease. The reason for starting at no less than 3 meters will become clear once we review examples.
The first example is for a Fenix TK20.
Next, generate a plot of the peak lux readings vs. distance like this:
Create a fit to the data using a power type fit and note if any points lie off of the line. Also, the R^2 value should be near 1 for a good fit. In the case of the Fenix TK20 data above, none of the points lie far from the line and R^2 is 0.998, which is close enough for something like this. Note that the exponent is also very close to -2, just as the inverse square law predicts.
For this example, the next step is to generate the calculated peak lux values using the inverse square relation from the distances and measured peak lux values. Here they are for the TK20 example:
The average of the calculated peak lux values is the value you would report.
Details: The first column is the distances at which I took measurements, the second column is the peak lux readings I got from my meter, and the third column is a calculation using the inverse square relationship: Calculated peak lux = Measured peak lux * Distance^2. Note that I switched the sign of the exponent for this equation to hold true.
Second example: 1185 in FM3X
In the second example I will examine data for the 1185 bulb in a Fivemega FM3X head because some of the data lies off of the curve and therefore some of the peak lux readings need to be thrown out. This is a good example of how up close peak lux readings can be less accurate than those at greater distances.
First of all, I could not even get a peak lux measurement at 1 meter because it exceeded the range of my meter, so the readings begin at 1.5 meters. Here is the graph of the data:
As you can see, the exponent is far from 2, the data points do not lie on the best fit curve, and R^2 is not very close to 1. Since the first data point lies furthest from the curve, I will delete that point and replot:
In this graph R^2 improved because it is closer to 1 and the exponent improved dramatically because it is much closer to 2. The exponent is still only 1.9165/2=.95825 or 95.8% of the way there, so let's delete the first point and replot again:
Notice that although R^2 did not improve, the exponent is now closer to the theoretical value of 2, and is 1.9673/2=.98365 or 98.4% the way there. This still suggests to me that the first point may be in error, so again I will delete the first point and replot:
Now, even though R^2 actually decreased a little, the exponent is very close to 2, 2.0097/2=1.00485 or only 0.485% away from the theoretical value. This tells me that the remaining data points conform to the inverse square law quite well.
Here is the data along with the calculations for the FM3X:
The first average includes all of the data; even the points that were excluded due to non-conformance with the inverse square relation. The second average excludes measurements taken at <3 m (shaded purple).
So, what can we learn from this exercise? Mainly, peak lux readings for bright lights cannot be trusted at close distances. In this example, only readings taken from greater than or equal to 3 meters actually follow the inverse square relationship closely. If I had used lights with larger reflectors, then I have no doubt that readings taken at 3 meters would also fail to follow the inverse square relation. Larger reflectors require measurements to be taken at greater distances.
How much of a difference did it really make deleting the first few points? Well, honestly not a lot. If I average the calculated peak lux values for all of the points I end up with 112,228 lux whereas if I only include the data from 3 to 6 meters I end up with 117,039 lux. If I followed the convention of only measuring at 5 meters I would have gotten a result of 121,500. This shows how averaging multiple readings reduces error that can occur in any one measurement, especially for bright lights at close distances.
I also noticed that peak lux readings taken at close distances are more difficult to acquire because the slightest movement of the meter can make the reading jump a great deal, and in some cases the reading is unstable and therefore some degree of mental averaging must be used. At greater distances the hotspot is larger, and therefore it is easier to get a stable reading that inspires confidence in the measurement.
Compendium of exponents for various flashlights:
The following data is presented as verification that the inverse square law holds for flashlights provided that in some cases measurements taken at close distances are omitted.
There is still some deviation from the inverse square relation for the 1185 in the KT4 turbohead even if I deleted points. This leads me to believe that some of my measurements must not have been the true peak values or the state of charge of the batteries changed enough during the measurements to actually reduce the light output. Either or both of these explanations could easily contribute to such an error.
This brings up another point. If you progress steadily from short to long distances or vice versa you could introduce a systematic error in the results due to diminishing light output as the batteries discharge. Of course, this is not a concern for regulated lights. I chose to randomize my testing as follows: 5m, 2.5m, 3.5m, 1m, 6m, 2m, 4.5m, 3m, 5.5m, 1.5m.
Towards developing a new standard for calculating throw using peak lux measurements:
The next thing I would like to do is to work on developing a definition for throw that yields more realistic results than the currently accepted definition of the distance at which the light intensity equals 1 lux. This is a useful definition because of its simplicity, but I feel that it underestimates useful throwing distances. Indeed, manufacturers rate their lights at much greater distances than the ones calculated in this way, but I think people are also able to use the lights to throw greater than the calculated limits of throw. This type of work will require the cooperation of other members with the ability to measure distances to targets and to conduct peak lux measurements. Hopefully by including data from multiple people we can average out the opinions regarding useful throwing distances, especially if folks can use the same lights in some instances. Is there any interest in participating? :candle:
In another thread a member questioned the applicability of the inverse square law for focused beams of light such as those produced by flashlights. They questioned whether the inverse square law holds true for lasers (I have not verified this myself in any way), and this led them to also question it for flashlights. They then presented a few Lux measurements of an 1185 in a KT4 turbohead that indeed seemed to deviate from the law.
This led me to question whether flashlights do indeed follow the inverse square relation or not. So, I decided to buckle down and take peak lux measurements at 11 distances from 1 meter to 6 meters in 0.5 meter increments using an assortment of 10 different flashlights.
110 lux measurements and a lot of data analysis later, I think I have confirmed that the inverse square law does hold for flashlights under certain conditions. I have also developed what I think is a more accurate way of measuring the approximate peak lux of a flashlight and I would like to share this with you. When I say "more accurate" what I mean is compared to the method of taking one measurement, regardless of distance, and reporting this result (typically adjusted to 1 meter).
I think the reason why the peak lux measurements do not follow the inverse square law at close distances is due to effects from the optics. This may just be due to my using the distance from the lux meter to the lens of the flashlight instead of some virtual point from which the light source seems to be emanating. Regardless, I do not feel the need to develop a model that more accurately predicts light intensity at close distances. That is not important to the goal that I have with respect to this effort.
My primary goal here was to develop a simple testing method that affords an improved degree of accuracy compared to the single measurement method. This test method is useful for improving relative peak lux measurements, but is certainly not sufficient for making absolute measurements that may be published in a scientific journal. That would require attention to many more details, some of which were listed by LuxLuthor in a link he provided in post #38 of this thread.
For one thing, there are difficulties associated with locating the peak light intensity in a beam. Then there are a host of other variables which may affect the output. LuxLuthor has done a good job of listing many of these variables in the aforementioned link, and some of them include: bulb focusing, reflector finish differences, lens material, ambient reflections, light meter angle, spectrum differences between light sources being compared, light meter spectral response, light meter linearity (calibration), bulb to bulb differences, flashlight resistance, battery state of charge and type, heating of LED die(s), bulb/LED centering within optics, etc.
One example I would like to mention is that differences between LED spectra such as between 5mm, cool white, neutral white, warm white, high CRI affect lux readings because each of these emits light that varies in intensity with frequency while a particular light meter has a fixed response with respect to each frequency. In the case of incandescent light sources, the degree of bulb overdrive or whether the bulb is of the IRC type also has an effect on the spectrum of light and therefore even when comparing two incandescent light sources these types of differences can occur.
The point is, this is at best a relative indication of differences in peak light intensity as measured by a particular meter and variables such as those mentioned need to be accounted for or at least considered when making comparisons, so keep that in mind.
How to calculate approximate relative peak lux more accurately by averaging readings taken at multiple distances
First, you need to measure the peak lux of your flashlight at many different distances. I recommend using multiple points as I have done, starting at a minimum of 3 meters and moving away in 0.5 meter increments. This allows readings to be done indoors with relative ease. The reason for starting at no less than 3 meters will become clear once we review examples.
The first example is for a Fenix TK20.
Next, generate a plot of the peak lux readings vs. distance like this:
Create a fit to the data using a power type fit and note if any points lie off of the line. Also, the R^2 value should be near 1 for a good fit. In the case of the Fenix TK20 data above, none of the points lie far from the line and R^2 is 0.998, which is close enough for something like this. Note that the exponent is also very close to -2, just as the inverse square law predicts.
For this example, the next step is to generate the calculated peak lux values using the inverse square relation from the distances and measured peak lux values. Here they are for the TK20 example:
The average of the calculated peak lux values is the value you would report.
Details: The first column is the distances at which I took measurements, the second column is the peak lux readings I got from my meter, and the third column is a calculation using the inverse square relationship: Calculated peak lux = Measured peak lux * Distance^2. Note that I switched the sign of the exponent for this equation to hold true.
Second example: 1185 in FM3X
In the second example I will examine data for the 1185 bulb in a Fivemega FM3X head because some of the data lies off of the curve and therefore some of the peak lux readings need to be thrown out. This is a good example of how up close peak lux readings can be less accurate than those at greater distances.
First of all, I could not even get a peak lux measurement at 1 meter because it exceeded the range of my meter, so the readings begin at 1.5 meters. Here is the graph of the data:
As you can see, the exponent is far from 2, the data points do not lie on the best fit curve, and R^2 is not very close to 1. Since the first data point lies furthest from the curve, I will delete that point and replot:
In this graph R^2 improved because it is closer to 1 and the exponent improved dramatically because it is much closer to 2. The exponent is still only 1.9165/2=.95825 or 95.8% of the way there, so let's delete the first point and replot again:
Notice that although R^2 did not improve, the exponent is now closer to the theoretical value of 2, and is 1.9673/2=.98365 or 98.4% the way there. This still suggests to me that the first point may be in error, so again I will delete the first point and replot:
Now, even though R^2 actually decreased a little, the exponent is very close to 2, 2.0097/2=1.00485 or only 0.485% away from the theoretical value. This tells me that the remaining data points conform to the inverse square law quite well.
Here is the data along with the calculations for the FM3X:
The first average includes all of the data; even the points that were excluded due to non-conformance with the inverse square relation. The second average excludes measurements taken at <3 m (shaded purple).
So, what can we learn from this exercise? Mainly, peak lux readings for bright lights cannot be trusted at close distances. In this example, only readings taken from greater than or equal to 3 meters actually follow the inverse square relationship closely. If I had used lights with larger reflectors, then I have no doubt that readings taken at 3 meters would also fail to follow the inverse square relation. Larger reflectors require measurements to be taken at greater distances.
How much of a difference did it really make deleting the first few points? Well, honestly not a lot. If I average the calculated peak lux values for all of the points I end up with 112,228 lux whereas if I only include the data from 3 to 6 meters I end up with 117,039 lux. If I followed the convention of only measuring at 5 meters I would have gotten a result of 121,500. This shows how averaging multiple readings reduces error that can occur in any one measurement, especially for bright lights at close distances.
I also noticed that peak lux readings taken at close distances are more difficult to acquire because the slightest movement of the meter can make the reading jump a great deal, and in some cases the reading is unstable and therefore some degree of mental averaging must be used. At greater distances the hotspot is larger, and therefore it is easier to get a stable reading that inspires confidence in the measurement.
Compendium of exponents for various flashlights:
The following data is presented as verification that the inverse square law holds for flashlights provided that in some cases measurements taken at close distances are omitted.
There is still some deviation from the inverse square relation for the 1185 in the KT4 turbohead even if I deleted points. This leads me to believe that some of my measurements must not have been the true peak values or the state of charge of the batteries changed enough during the measurements to actually reduce the light output. Either or both of these explanations could easily contribute to such an error.
This brings up another point. If you progress steadily from short to long distances or vice versa you could introduce a systematic error in the results due to diminishing light output as the batteries discharge. Of course, this is not a concern for regulated lights. I chose to randomize my testing as follows: 5m, 2.5m, 3.5m, 1m, 6m, 2m, 4.5m, 3m, 5.5m, 1.5m.
Towards developing a new standard for calculating throw using peak lux measurements:
The next thing I would like to do is to work on developing a definition for throw that yields more realistic results than the currently accepted definition of the distance at which the light intensity equals 1 lux. This is a useful definition because of its simplicity, but I feel that it underestimates useful throwing distances. Indeed, manufacturers rate their lights at much greater distances than the ones calculated in this way, but I think people are also able to use the lights to throw greater than the calculated limits of throw. This type of work will require the cooperation of other members with the ability to measure distances to targets and to conduct peak lux measurements. Hopefully by including data from multiple people we can average out the opinions regarding useful throwing distances, especially if folks can use the same lights in some instances. Is there any interest in participating? :candle:
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