Radiation plots describe LED intensity over angle characteristics. There are many different representation formats used by different suppliers. The units in these plots are typically W/sr or lm/sr (candela). The two most useful angles for analysis are the 50% intensity (FWHM) and the 10% intensity angle.
Many radiation plots are normalized. Some are normalized to one, as the example below, and some are normalized to cd/klm. Normalizing to one makes it easy to read the full width half max (FWHM) angle and 10% intensity angle directly from the plot. Normalizing to cd/klm is convenient for calculating the peak candela for different lumen inputs.
Visualization examples using optical software
We will first compare the beam patterns for three types of light sources with our optical software set at 100 lumens for each source:
- A point light source, typical of a light bulb
- A Lambertian light source, typical of an LED
- A narrow beam light source, typical of an LED combined with focusing optics
and make some observations.
Due to software limitations, not all of the plot legends are legible. We will attempt to describe what is going on and if you need better resolution graphics, send an email to techsupport@luminus.com. All three of these light sources have circular symmetry. There are many light sources that use optics to achieve non-symmetrical beam patterns that are not considered here.
The figure below shows rendering examples for these light sources.
The point light source has a uniform angular distribution. The intensity is the same in all angular directions. The Lambertian light source has a cosine distribution. This is typical of a surface light source with random light emission directions. The 23-degree light source has an optical element modifying the beam pattern. This light source has a significant amount of side light, but the majority of the light is in the desired 23-degree beam pattern.
The figures below show three different formats for quantifying radiation patterns. All of these have intensity units divided by a solid angle. Since we are using 100 lumens for the source optical power in this model, these plots are in absolute candelas (lumens/steradian). If radiometric units were used, the units would be watts/steradian.
In the figure above, the 3D representation of the beam patterns were shown. In the first row of plots below another form of 3D representation, Raster diagrams, are used. The luminous intensity (cd) is plotted against rotational angles on both axes. In the second two rows of plots, the luminous intensity (cd) over angle for rotational slices are shown. These are called Polar and Cartesian diagrams, and in this example there are two slices 90 degrees apart.
Raster plots are useful for asymmetric beam pattern visualization such as below. For symmetric beams, the polar and cartesian plots are generally used. The beam angles that correspond to 50% and 10% intensity are tabulated below along with the minimum, maximum, and average candela values calculated by the model using a 100-lumen input.
Raster plot for an asymmetric light source.
Now we will use planar detectors in optical software to visualize the beam patterns (in lux) for these three cases. In the figures below, the three detector distances are 100, 200, and 400 mm and the length of each side of the square have the same values. The projected Lux patterns clearly form a cone with similar shapes at each distance.
The table below shows the values for this set of models. The incident lumens are constant because the model detector size is scaled with the distance from the source. Note that the average and minimum lux values in the table below depend on the size of the detector. For analysis of real objects, these are useful metrics. In this example, we used detectors scaled to best show the cross-sectional shapes of the projected beam.
Calculating lux from the radiation plots
The lux for some LEDs can be estimated using the definitions of candela, lux, steradians, combined with the inverse square law. This works best for wide beam angles with circular symmetry and a rounded top. This method does not work for beam patterns with flat tops or collimated beams.
Example calculation, given:
- 100 lumen light source
- 120° FWHM LED
- Distance = 400 mm = 0.4 m
- Calculate the solid angle of the light source using the FWHM angle theta. Use
Ω = 2*3.141*(1-cos(120/2) = 3.141 steradians.
- Calculate the candelas using the definition: 1 cd = 1 lm/sr
candelas = 100 lumen / 3.141 steradian = 31.8 lm/sr
note that candela is not a function of distance.
- Use the definition of the steradian to convert this value to lux.
At one meter, one steradian has an area of one square meter (on a spherical surface). Therefore, at one meter, the candela value is numerically equal to the lux value so at one meter
the lux value = 31.8 lm/m2.
- Use the inverse square law to calculate the lux at other distances.
At 400 mm (0.4 m),
lux = 31.8 * (12 / 0.42) = 198.9 lux
The figures below show verification results of this theoretical method. The red points are the optical software values and the black dashed lines are the inverse square calculation. The error reported is the % difference between the theoretical calculation and the optical software results. The Max (lx) in the table are the values calculated using the theoretical method. We see that this method works best for wider beam angles. The error term is highly dependent on the settings in the optical software and should be simply considered an estimate, especially in the near field.
Note that online inverse square calculators do not all give the same results, and the underlying method is not always discussed.
Calculating areas from the radiation plots
As an example, we will show the beam area at one meter. Using the polar and cartesian plots for radiation patterns, it is easy to read off the 50% (FWHM) and 10% angles. Using the sine of the half angle it is easy to calculate the radii associated with these intensity levels.
The figure below is a schematic of a Lambertian source (120° FWHM beam angle). The 10% angle is 169° which is called the Field Angle. This is the angle where the illumination circle is considered to effectively end. The annulus between the beam and field light is called the spill light. Using these values, we can calculate the diameters for a 1 m distance as 20.8 m for the field diameter and 3.5 m for the beam diameter. The maximum intensity is at the center of the beam and decreases as the distance from the center of the beam increases. The profile of this decrease is shown in the inset and is similar to the shape of polar plots. If you need more accurate calculations on real object surfaces, you will need to use optical software.
Useful References:
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Luminus Website https://www.luminus.com/
Luminus Product Information (datasheets): https://www.luminus.com/products
Luminus Design Support (ray files, calculators, ecosystem items: [power supplies, lenses, heatsinks]): https://www.luminus.com/resources
Luminus Product Information sorted by Applications: https://www.luminus.com/applications
Where to buy Samples of Luminus LEDs: https://www.luminus.com/contact/wheretobuy.
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