Fly’s eye arrays are typically used in pairs along with a condenser lens to provide uniform irradiance at the illumination plane.  The first fly’s eye array is often called the objective array and the second array along the optical axis is called the field array.  For now we will consider only the objective array. The function of the objective array is to act like an objective lens on a camera and form an image of an object, or light source in our case, at the focal plane of the objective lens, as shown below.  In our case we will form an image of the collimated light source at the focal plane of the objective array.

If an objective array is used with collimated light and we place a condenser lens at the focal plane of the objective array as shown above, we will obtain a uniform irradiance at the illumination plane as shown in Figure 5.  Unfortunately we are not lucky enough to have point sources of light so it is very difficult to obtain collimated light from a lamp assembly with a parabolic reflector.  The light from lamp assemblies with a parabolic reflector has some divergence or angle because the fire ball of the lamp is an volume light source and not a point.  We can see the results of using only an objective array and condenser lens with a diverging source  and a source with two field angles in the two screenshots below.

The axial rays are imaged to overlap at the illumination plane and provide uniform illumination.  The diverging rays shown in the left figure above as green rays are imaged to a different location and therefore do not overlap with the collimated beam rays at the illumination plane.  This imaging at a different axial location causes a nonuniformity at the illumination plane because the full beams from the axial rays are overlapping and only half of the illumination from the diverging rays is illuminates the same plane as the on-axis (blue) rays. 

For the figure on the right above, the two field angles get imaged to different object heights at the condenser lens and therefore get imaged to a different object height by the condenser lens at the illumination plane.  If the images from all fields are not overlapped at the illumination plane we will have a nonuniform illumination plane. 

In both of these cases we can improve the uniformity at the illumination by adding a second fly’s eye array called a field array.  This field array is a second fly’s eye array and is located at the image plane of the objective array.  The function of the field array is to provide overlapping images at the illumination plane for different fields from the source.  To be uniform at the same plane we need the full width of the illumination plane illuminated by both the axial and diverging rays to be the same.  We can see what the addition of the field arrays do for our two situations in the figures below.  In both the diverging rays and the field rays the field array of fly’s eye lenses acts like a field lens and works with the condenser lens to keep the illumination so that it will still overlap at the illumination plane.