Aside from given preconditions, the core concept of relative illumination is the ratio of exit pupil area between off-axis fields and on-axis fields within the direction cosine space. The diagram below is a widely referenced illustration for explaining relative illumination: the X-axis denotes the direction cosine L, and the Y-axis denotes the direction cosine m. Under this definition, relative illumination equals the area ratio of an ellipse to a circle.

We will then apply this definition to calculate relative illumination for a sample optical system, and cross-verify the result against the RELI (Relative Illumination) metric in ZEMAX. The 3D layout and MTF (Modulation Transfer Function) evaluation plots of the target optical system are provided below.

The corresponding relative illumination curve of this optical system, along with its tabulated data, are also attached.

Next, we perform calculations using ZEMAX operands based on the aforementioned relative illumination formula:

Such discrepancies stem from multiple contributing factors: apodization, vignetting, physical aperture constraints, optical aberrations at both the image plane and pupil plane, varying F-number across the field, chromatic aberration, image surface geometry, chief ray incidence angles, as well as anti-reflection coating performance on every optical surface.