The Definition of Lens Chief Ray Angle and the dreaded CRA mismatch

Lens Chief Ray Angle Mismatch and Impact on Image Quality 

The chief ray angle (CRA) of a lens and the chief ray of a sensor affect image quality factors such as color shading and vignetting.

The magnitude of impact from CRA mismatch can be approximated using the Difference of Squares. This is dependent on the sensor's pixel architecture, but is a good first order rule of thumb.

We generally recommend matching CRA within +/-10° if the sensor's CRA is <10°, +/-7° if the sensor's CRA is >10° and <20°, and within +/-4° if the sensor's CRA is >20°. This mismatch tolerance must hold across the entire field of view, so make sure to compare a full plot if the sensor's specification sheet says "non-linear" on it.

Below is an example of problematic CRA mismatch compared to proper mismatch with our CIL340 M12 Lens.

What is the Chief Ray Angle of a Lens?

The chief ray of a lens is the ray that goes through the center of the aperture stop in an optical system.

If you look into a lens from object space, the chief ray is the ray that crosses the optical axis at the entrance pupil.

If you look from image space, this is the ray at the center of the exit pupil.

Hecht's "Optics" Fifth Edition has a great explanation and description on page 185 for a general three element optical imaging system : https://www.pearson.com/en-us/subject-catalog/p/optics/P200000006793/9780137526420 

Chief rays exist for every illuminated point in object space.  Let's see how this looks for a "Real World" lens; our CIL039.

When people discuss the Chief ray angle, they typically refer to the "Maximum CRA" which corresponds to the widest field of view of a lens combination.

To accurately compare the chief ray of a lens and the chief ray of a sensor, you must consider the CRA across the usable area of the image.

Oblique Dependence of Microlenses at 25° CRA (CIL023 2.2mm F/2.2)

Oblique Dependence of Microlenses at 15° CRA (CIL039 3.9mm F/2.8)

Correcting for color shading due to CRA mismatch

CRA mismatch CAN be corrected for in post process, but ONLY in applications with well controlled static illumination such as industrial machine vision for inspection.

When the light sources change, it becomes challenging to compensate. This is due the friendly topic of metamerism. We've seen a major CRA mismatch (20° non-linear mismatch) overcome before in a regular indoor environment, so it is doable to a "good enough" extent. This requires advanced ISP tuning with a calculated pixel-level spectral energy distribution 3DMLUT approach. This in turn will slow down other performance metrics in your camera and/or require more compute, so generally not the best practice to get into this sitatuon.  

Additionally, there are only a handful of leading image quality experts with the requisite knowhow and experience to get to a "good enough" quality with a >15° nonlinear mismatch with a sensor at 33°.  I estimate <50 people in the world and it is near impossible to hire them as they are in high demand at big tech companies. So unless you are fortunate enough to be on a team with one of these experts, we highly advise against venturing down the rabbit hole of thinking you can solve >15° nonlinear CRA mismatch in software: your project will likely have a 6-12 month delay and budget overrun.

Regardless of the approach and expertise there will be more color tuning corner cases that occur with huge CRA mismatch, than when you have a well-matched lens to sensor CRA.

The Take-Away: We suggest Low Linear CRA (~<20°) Lenses/Sensors when Possible.

Otherwise Match the Lens Chief Ray Angle As Closely to the sensor as possible 

Incorrect CRA matching can result in radial red to green color shading from the center of an image to the corner.

This shading is dependent upon illumination conditions, so it makes Image Quality Tuning extremely difficult.

This is a common issue when trying to build a camera using a "Mobile" Sensor with an "Industrial" Lens or vis-versa. We've seen multiple startup projects run into this issue, resulting in extensive cost (>$100k) and schedule (>1yr) overruns.