I am looking to prototype an illumination system. My objective is to use a small halogen filament bulb and end up with a beam of light. What would be the best lens or lens combination to give me this projected spot of light?

Generally speaking, the use of a Holographic Diffuser, Condenser Lens, and Plano-Convex (PCX) Lens would satisfy the application. The holographic diffuser (elliptical in this case) would take the filament and image it as an approximate circular blur at a particular distance from the diffuser (based on diffusing angle chosen). By placing a filament 1 focal length from the condenser lens, the resulting output will be approximately collimated. Short focal length lenses tend to work best for this type of application. The PCX lens would then be used to refocus the light a given distance (EFL of the PCX Lens) from the lens. The distance of the PCX lens from the condenser lens is arbitrary. You can pick and choose relative to the application.

For example, you could use a Condenser lens (EFL = 13mm) and PCX Lens (EFL = 100mm). Place the condenser lens 13mm from the filament, and place the diffuser between the filament and condenser lens. The spacing between the condenser and PCX lenses can be 87mm (arbitrary). Overall, the resulting filament to final focal position is 200mm. Using geometry and trigonometry, you can determine exactly where the PCX lens needs to be placed in order to produce a 20mm spot at a distance of 200mm from the filament.

The system you are trying to model is basically a Koehler Illumination setup. We have a great EO Tech Tool – Koehler Illumination - that helps you choose the best lens parameters from a few simple variables.