The case for angular extent X (Invited Paper)

How does a quantity emerge in science and engineering? First, it must be measurable – radiance is measurable; beauty is not. Second, it must appear in an important law of nature – energy does; the “quantity” m²T/v³ does not. When enough people agree about the importance of a quantity, and a name, that name will eventually stick. But names and symbols must be well chosen. They should (i) be concise, (ii) convey immediate meaning and (iii) be unique. Not all well-known quantities (intensity) and symbols (E) meet all three criteria, but when they do, they are immensely helpful in communicating effectively, and in clarifying our thinking. In illumination optics, the quantity "refractive index squared times projected solid angle" is very important: it forms one half of the definition of etendue and is therefore as ubiquitous as etendue itself. It is also conserved before and after refraction at an infinitesimal surface element. Whenever we think of etendue, we implicitly think about this quantity, which therefore qualifies for a name and a symbol. We propose: Angular extent. Unique, short, and conveying immediate meaning: "extent" is the literal English translation of "etendue", and "angular" means direction. As a symbol, we suggest X: Concise, associates "extent" and is unique in our field. Differential etendue becomes dU = dA dX, and in many important simple cases, etendue itself is simply U = A · X. Moreover, for constant luminance L, irradiance E is simply E = L · X. Thus, in imaging optics, relative image illumination is proportional to X, including all pupil aberrations.