The expansion ratio

y_{3}/y_{1} = θ_{2}f_{2}/θ_{2}|−f_{1}| = f_{2}/| −f_{1}|

or the ratio of the focal lengths of the lenses. The expanded beam diameter

2y_{3} = 2θ_{2}f_{2} = 2y_{1}f_{2}/|−f_{1}|

The divergence angle of the resulting expanded beam

θ_{3} = y_{2}/f_{2} = θ_{1}|−f_{1}|/f_{2}

is reduced from the original divergence by a factor that is equal to the ratio of the focal lengths |-f_{1}|/f_{2}. So, to expand a laser beam by a factor of five we would select two lenses whose focal lengths differ by a factor of five, and the divergence angle of the expanded beam would be 1/5th the original divergence angle.

As an example, consider a Newport R-31005 HeNe Laser with beam diameter 0.63 mm and a divergence of 1.3 mrad. Note that these are beam diameter and full divergence, so in the notation of our figure, y_{1} = 0.315 mm and θ_{1} = 0.65 mrad. To expand this beam ten times while reducing the divergence by a factor of ten, we could select a plano-concave lens KPC043 with f_{1} = -25 mm and a plano-convex lens KPX109 with f_{2} = 250 mm. Since real lenses differ in some degree from thin lenses, the spacing between the pair of lenses is actually the sum of the back focal lengths BFL_{1} + BFL_{2} = -26.64 mm + 247.61 mm = 220.97 mm. The expanded beam diameter

2y_{3} = 2y_{1}f_{2}/|-f_{1}| = 2(0.315 mm)(250 mm)/|-25 mm|= 6.3 mm

The divergence angle

θ_{3} = θ_{1}|-f_{1}|/f_{2 }= (0.65 mrad)|-25 mm|/250 mm = 0.065 mrad

For minimal aberrations, it is best to use a plano-concave lens for the negative lens and a plano-convex lens for the positive lens with the plano surfaces facing each other. To further reduce aberrations, only the central portion of the lens should be illuminated, so choosing oversized lenses is often a good idea. This style of beam expander is called Galilean. Two positive lenses can also be used in a Keplerian beam expander design, but this configuration is longer than the Galilean design.

This application is one that will be approached as an imaging problem as opposed to the focusing and collimation problems of the previous applications. An example might be a situation where a fluorescing sample must be imaged with a CCD camera. The geometry of the application is shown in Figure 4. An extended source with a radius of y_{1} is located at a distance s_{1} from a lens of focal length f. The figure shows a ray incident upon the lens at a radius of R. We can take this radius R to be the maximal allowed ray, or clear aperture, of the lens.