Excluding given prerequisite conditions, the core concept is that relative illuminance refers to the area ratio of the exit pupil for the off-axis field of view to that of the on-axis field of view within the direction cosine space. The figure below is a classic diagram widely used to explain relative illuminance. The X-axis represents the direction cosine L, and the Y-axis represents the direction cosine m. On this basis, relative illuminance equals the area ratio of an ellipse to a circle.

Then, we will apply this principle to calculate the relative illuminance of a practical case, and compare the calculated result with the RELI (Relative Illuminance) data in ZEMAX. The 3D layout diagram and MTF evaluation curve of this optical system are presented below.

The relative illuminance map of this optical system is shown below, with the corresponding data text box also listed for reference.

Next, we use operands to perform calculations based on the relative illuminance formula above.

In the RELI operand, the first number 2 corresponds to wavelength index 2, and the second 2 corresponds to field index 2.

Field index 2 represents the maximum field of view with (hy=1). It can be seen that the relative illuminance obtained by the RELI operand is 91.672%, which is consistent with the relative illuminance curve.

The calculated result derived from the relative illuminance formula is 91.627%. Although there is a slight deviation between the two values, they

are highly close to each other. In addition, the relative illuminance calculated by the fourth power of the cosine of the chief ray incident angle at this field of view

is 92.667%, which shows a larger discrepancy compared with the above two results.

These discrepancies are related to apodization distribution, vignetting, aperture structure, aberrations on the image plane and pupil plane, variations in F-number, chromatic aberration, image plane shape, and incident angle. Additionally, they are also affected by the coating condition of each optical surface.