Off-axis limiting photometric distance of Lambertians and narrow beams

The near-field - far-field disparity of light sources leads to discussion within the lighting community. In the far-field, light sources are approximated by a point source with a luminous intensity distribution from which other photometric quantities can be computed. In the near-field, light sources must be considered as extended sources and no closed-loop analytical solutions can be found for the illuminance in off-axis directions. The illuminance allows to compute the apparent intensity used for the assessment of the limiting photometric distance (LPD), that is, the threshold between near- and far-field regions. Numerically, the illuminance can be determined through a discretisation of the luminous surface. This approach is verified for on- and off-axis directions through direct comparison with ray tracing simulations based on near-field goniophotometric measurements. A good match is observed for Lambertian and narrow beam light sources. Using the numerical approach, the LPD in all directions is assessed. For light sources of which the luminous intensity strictly decreases while moving away from the optical axis, the LPD decreases as well until a minimal value is reached. After this, the LPD increases again, in certain scenarios up to values resulting in the invalidity of the inverse square law.