Lighting Design Basics
Lighting Design Basics
Photometric Units
A schematic representation depicting the relationship of commonly employed photometric units is shown in the figure below. Solid angle shown represents one steradian
| 1 Candela [Lm/Sr] | = | the luminous intensity of 1/60 of 1 cm2 of the projected area of a black body radiator operating at the temperature of the solidification of platinum (2045 K) |
| 1 Lumen [Lm] | = | the luminous flux contained within one steradian of solid angle from a source whose luminous intensity is one Cd. |
| 1 Lux [Lm/m²] | = | the illuminance resulting from the flux of one Lumen falling onto a one m2 surface area |
| 1 Nit [Cd/m²] | = | the luminance (brightness) from a source whose luminous intensity is one Candela per projected one m2 surface area normal to the line of observation |
Useful Conversions
Conservation Of Radiance
The design of any illumination system is based on the fundamental principle that it is physically impossible for the final radiance (brightness) at the image to exceed the initial radiance of the source. Consider the illumination system shown in the figure below which is assumed to have no optical loss.
The incremental flux, d²Φ, radiated by dA1 = dx1 dy1 into an element of solid angle dΩ1 = sin θ1 dθ1 df at the entrance pupil of the illumination system is given byd²Φ = B1(θ1, ø) dA1 sin θ1cosθ1 dθ1 dø
If the system is truly lossless, this incremental flux must pass through an area element dA2 = dx2 dy2 in image space from a solid angle dΩ2 = sin θ2 dθ2 dø, and thus the image radiance in this direction is given byB2(θ2, ø) = d2Φ ⁄ (dA2 sin θ2 cosθ2 dθ2 dø)
Because the optics of the illumination system are assumed to be lossless all rays passing through must satisfy the Abbe sine theoremN1dx1 sin θ1 = N2 dx2 sin θ2
And by changing x to y and differentiatingN1dy1 cos θ1 dθ1 = N2 dy2 cos θ2 dθ2
Combining these last three equations results in the general statement of the radiance theorem for an imageB1(θ1, ø) / N12 Ξ B2(θ1, ø) ⁄ N22
Thus no illumination system can produce an image brightness that is greater than that of the initial source.
Entendué
If an illumination system were perfectly transmitting in the sense that all of its components neither absorb nor reflect and that all rays entering the system exit without encountering any limiting stops or apertures, then the total flux collected by the system is given byΦ = ∫∫ d2Φ= ∫∫ B(r, n) dA cos θ dΩ
where B(r, n) is the source radiance at the point r in the direction of the unit vector n, where the area integral (Fourier area) extends over that portion of the surface of the source which lies within the entrance window and where the angular integral extends over the solid angle subtended by the entrance pupil of the illumination system from the location r as shown in the figure below.
For example, a Lambertian source with radiance B0 has a total flux given by Φ = B0 ∫∫ dA cos θ dΩwhich can be expressed asΦ = B0 ε ⁄ N02where the entendué of the system is defined asε = N02 ∫∫ dA cos θ dΩ
For integrated lighting systems the entendué is a purely geometrical quantity that measures the flux-gathering capabilities of a system in that the collected flux is given by the product of this quantity with the basic radiance of the source. For a lossless illumination system the entendué is invariant between the source and image planes. That is to say no system of external optics can increase the entendué of an illumination system (i.e. it is not possible to get out more light output than what you started with). Therefore evaluating the performance of an integrated lighting system design requires an understanding of the change in the entendué at various locations as the light propagates through the various secondary optics and diffusers.
