With freeform surfaces it is possible to shape light. Look at these two examples employing a freeform reflector and a freeform lens, respectively:

On the left is shown a freeform reflector that transforms a Lambertian light distribution from a laser-illuminated phosphor layer to a circular and uniform distribution of light 6 meters away from the reflector. In this case only a single freeform surface shapes the light.

On the right is shown a freeform, injection molded polymer lens that transforms a Lambertian light distribution from an LED to a rectangular, uniform illumination that is supposed to match a road stretch between two light poles. In this case two refractive surfaces shape the light.

How are these freeform surfaces designed? Let’s take a look at the reflector, as an example:

We build up the reflector using segments. Each reflector segment is responsible for illuminating a corresponding segment on the road, as illustrated above.

We assume that the LED has a Lambertian light distribution, i.e.:

$$I(\theta ) = {{\cos (\theta )} \over \pi },$$

-where *I*(θ) is the radiant intensity (W/sr) and θ is the angle with respect to the optical axis of the LED.

If we integrate *I*(θ) over the entire hemisphere we get:

$$\mathop \smallint \limits_{Hemisphere} I(\theta )d\Omega = \mathop \smallint \limits_0^{90} {{\cos (\theta )} \over \pi }2\pi \sin (\theta )d\theta = 1W,$$

-where Ω is the solid angle. Hence, the total power is normalized to 1 W.

Consider the first reflector segment in the figure above. It spans the first 5 Degs. of the hemisphere. Thus, energetically, this segment distributes:

$$\mathop \smallint \limits_{85}^{90} 2\cos (\theta )\sin (\theta )d\theta = 0,0076W.$$

This implies, that segment 1 needs to redirect its central light ray at the outer 0.76% of the area on the road. If the illuminated area is a uniform circle with, say, a radius of 1m, this outer rim of light would span from radius √(1-0.0076) = 0.996m to 1m.

So, now that we know the target position on the road, it is a simple geometrical matter of tilting the first reflector segment according to this requirement. The second reflector segment is tilted using the same procedure and is then attached to the first segment to form a continuous shape. This procedure is continued until the whole hemisphere is covered.

The technique can be used to any desired light distribution. Below is shown the result of three designed reflectors that generate light distributions in the shape of sin^{2}(r), sin^{2}(2r), and sin^{2}(3r), where r is the normalized distance from the center:

Freeform reflectors for generating light patterns without rotational symmetry are somewhat harder to design, even though the principle is very similar to the one outlined above. Freeform lenses (including non-rotationally symmetric TIR lenses) are also designed using a similar technique.