cos^n beams

Let's assume we design optics for some spotlight. Typically, the spec says e.g. "30° FWMH" and "3500 cd". When we ask for more details about the beam shape, the answer may well be "soft and nice" (rotational symmetry is assumed). That's a somewhat incomplete specification for an engineer! However, the customer is (almost) always the king, and we better be nice to the king -- for example, by proposing a precise beam shape which is, indeed, a "soft and nice" bell shape and works well in practice.

Now, a Gaussian distribution is almost synonymous to "bell shape". But Gaussian distributions cause problems: They are hard to integrate analytically during initial estimates. They are defined by "sigma", but we need "FWHM" (full width at half max. intensity), a related but different quantity. And when the beam is wide, the Gaussian distribution is nonzero at 90° -- impossible to achieve with a planar exit aperture.

Yet another drawback of the Gaussian distribution appears when we ask a key question: How much flux do we need in the beam, with the given distribution, to achieve the 3500 cd in our example? In other words, how many cd/lm (candela per lumen) does our beam deliver? I like to call this quantity "collimation strength", because it also very well describes the action of, say, an off-the-shelf TIR lens, except that for the latter, it's source flux, not beam flux that is used. Now, for a Gaussian distribution, we need the integral, which cannot be computed analytically.

Is there something better? Yes, there is! It's the cos^n distribution (cosine to the n'th power). It is easy to integrate analytically, it's always zero at 90°, and the ubiquitous Lambertian distribution is a member of the family, too. And the collimation strength is simply (n + 1)/(2 pi). Its only drawback may be that its FWHM width is not intuitively known from the exponent n. FWHM is not difficult to compute, though.

There are three major applications in illumination optics design:

• Defining a source (e.g., LED + some optics) with a cos^n distribution. Often, I like to see what such a source would do in a larger system before I go about designing the optics for it. Or, comparing it to what my real system is doing. In LightTools, for example, cos^n is available for all planar surface sources.

• Defining an optimization merit function, when we actually design the optics.

• Defining the scattering properties of a generic scatterer.

Some time ago, I created an Excel spreadsheet to help me make cos^n easy to use. Like all my shared software, It's available as public domain open source on GitHub.