Model description:
The model and resulting maps enable rapid, meaningful comparison of key flashlight performance metrics—output, range, beam shape, and hotspot size—either individually or in concert.
The Goldilocks Map has a simple interpretation: equal horizontal or vertical distances correspond to equal practical differences in spottiness potential and maximum output of the lights, measured on the natural scales of Acuity Index and Output Index. This means the map shows not only which flashlight is 'better' than another, but by how much (or how little). Together, these two parameters determine a light’s Useful Reach, shown by the red contour lines.
The Overview Map uses the same values as the Goldilocks, but compresses both scales (but not values) logarithmically to extend the range and fit diverse lights on one plot. Interpretation of numerical values is identical to Goldilocks, but the relative distances between them are distorted and may not reflect by how much the lights really differ.
The hotspot size formulation (used in the Hotspot Map) is agnostic to how the emitted light is formed at the flashlight aperture. It is intended to apply across a range of optical designs, including reflectors, TIRs, and other optics, as well as multi-emitter systems.
The model assumes that, sufficiently far from the flashlight, the full angular intensity profile—including the hotspot, beam (corona), and spill—is adequately approximated by a Gaussian distribution truncated at ±90° relative to the optical axis.
If the hotspot angle is small and the amount of light in truncated tails negligible, the hotspot spread angle [FWpM or 'full width at a fraction p of the maximum intensity'] can be approximated by this cute, closed-form expression: 2×√(ln(1/p))/[Acuity Index].
The Acuity Index, which decouples a flashlight ability to concentrate light from its maximum output, is defined as: √(π×[max cd]/lm). The value of 1 corresponds to an ideal Lambertian mule.
The parameter p has a clear physical interpretation: it is the fraction of the maximum hotspot intensity that defines the edge of the hotspot, as we see it.
For the special case of single smooth reflectors, when additional parameters such as LED diameter, rim diameter, and reflector height are known, the choice of p can be guided by ray-tracing considerations.
The hotspot size formulation exhibits a special property for isotropic 2-D Gaussian beams: the fraction p defining the hotspot boundary is numerically equal to the fraction of light excluded from the hotspot cone. Thus, an FWHM definition (p = 50%) implies that half of the total flux is contained within the hotspot cone, while defining the boundary at p = 20% (FW(20%)M) results in 80% of the flux being enclosed. This equivalence applies only to Gaussian profiles and not to the Super-Gaussian case discussed below.
To evaluate the model predictions for the hotspots, I measured hotspot diameters of a few flashlights on a wall 8.2 m away, as they appeared to me, than standardised them to 10 m distance. Based on this small sample (n=5) the model predictions should be within some ±20 cm of the actual hotspot diameter at least half of the times. There may also be some systematic bias (the model computing hotspots that are too large overall) of similar magnitude, which can be mostly fixed by adjusting the hotspot cutoff intensity fraction (p), if required. A larger calibration sample is needed to diagnose it more confidently.
Some other relations used in the Map (Gaussian variant):
[max cd] = 4ln(1/p)/(π×[FWpM]²)×lm
[Useful Reach] = √[max cd]
FWHM = 2√(2ln2)×σ ≈ 2.35×σ, where σ is the standard deviation of the Gaussian angular light intensity distribution.
FWpM = 2√(2ln(1/p))×σ
For the hotspot-on-the-wall calibration measurements: measured [hotspot angle] = 2atan(H/(2L)) and H(at L=10 m) = 10/L×H, where H is the measured hotspot diameter and L is the distance to the wall.
Etendue conservation limit - Lambertian to Gaussian: [FWpM] ≥ 2√ln(1/p)×d/D ≈ 1.7×d/D for FWHM, where d is the LED diameter and D is the optics front aperture diameter. Applies to TIRs and reflectors. For Super-Gaussian the limit becomes: [FWpM] ≥ 2(ln(1/p))^(1/s)×d/D.
All angles are expressed in radians. Multiply by [180°/π] to convert to degrees.
Super-Gaussian extension
While flexible, a Gaussian intensity distribution may provide a suboptimal fit for some real flashlights. In principle, it can be replaced by any other unimodal distribution with finite variance—including one obtained directly from goniophotometric measurements—but doing so requires prior calibration to establish that distribution. The resulting need for numerical solution is a secondary complication, but does not preclude implementation.
Instead, the model adopts a Super-Gaussian intensity distribution, which generalizes the Gaussian by adding an additional shape parameter. This parameter simultaneously sets the flatness of the hotspot core and the steepness of the angular fall-off, effectively controlling the sharpness of the hotspot–spill transition.
At present, the shape parameter is fixed at 2 for all lights on the Map, reducing the Super-Gaussian to the standard Gaussian case described above. This parameter can be calibrated to better match measured beam profiles and potentially improve model predictions.
Below is an interactive simulation illustrating how beam shape and hotspot size depend on the input parameters. It can also be used to compute hotspot sizes—both in angular units (degrees) and as diameters at 10 m—for individual lights without modifying the global Map parameters.