Hey ​@Michael Cheng , thanks for this great writeup.

There is one more very important consideration for angle definitions in OpticStudio, namely when using Angle as the Field Type definition.  As you mentioned, there is a conversion between the direction cosines L, M, N and the Field Data Editor’s X Angle & Y Angle, namely:

This is a slightly simplistic conversion and only works when either L or M is 0 (the field point falls along one of the axes).  I’m going to call this case the “ground truth”.  If we measure the angle of incidence the ground truth makes with the paraxial Entrance Pupil (Object plane is not tilted), we get the following equation:

AOI=acos⁡(N)=α

So, if we write (0, 50) for the X/Y Angle, then the angle of incidence on the Entrance Pupil will be 50deg.  

However, if we keep the normalized field the same and we rotate the field to be along one of the diagonals such that the new coordinate is (35.35, 35.35), then the angle of incidence becomes 45.1deg (note that by selecting Rectangular Normalization in the FDE, we can decouple the Hx & Hy normalization and use Hx=Hy=1 to represent the diagonal)

If we plot both of these fields in the FDE, we can see that the normalized field coordinates are the same but the AOI is different:

So, it’s only the ground truth that accurately reflects the Field Data Editor value as the desired Angle of Incidence, even when the normalized field coordinate is preserved.  I believe that most users expect to have the same angle of incidence (same “cone” of light) regardless of the rotation of the field, so an extra conversion is needed.

This issue arises due to compound angles and decomposing our alpha angles from the FDE to direction cosines.  Simply taking the arctan of the direction cosines will only provide the angle as projected on the X & Y axis.  To get the true starting field angles which will provide a constant angle of incidence, you need to account for the field rotation angle:

Where θ is the clockwise rotation of the field from the +y axis, AOI is the ground truth angle.  The diagram below illustrates the X/Y projected angle as well as the rotated angle which provides a constant AOI:

This will produce a “distorted” normalized field coordinate circle but it ensures that every blue point below will map to a constant angle of incidence on the Entrance Pupil.  This becomes extremely important for a lot of custom analysis and Monte Carlo post processing where you’re trying to accurately capture off-axis field points.

Hi ​@MichaelH,

Thank you for much more details. It’s exactly one important point I was trying to explain, while probably I didn’t explain well. I think your explanation will help to more people.

Just for providing different viewpoint, which I think it’s same as you explained, I rather would think it’s mainly two different coordinate systems: (αx, αy) vs. (theta, phi), where cos(theta) = α as you described. Intuitively, we may think αx^2+αy^2=theta^2, however, it’s not true because they are different coordinate system.

I don’t know the initial reason why we choose (αx, αy) for Field Angle, but I can see it has a benefit. Here is an example (well just repeating what I shared above):

1. Assuming we have an rectangular object plane with side of 23.8 * 23.8
2. Now we have defined two Field Angle for the two points (-23.8, 0) and (0, -23.8) on the object plane, which are the maximum value on X and Y axes. Let’s say the corresponding Field Angle is (50, 0) and (0, 50) as shown below.
3. Now if I want to define a new field for the object plane point (-23.8, -23.8), I can simply set up (50, 50) Field Angle and it exactly give the ray that comes from (-23.8, -23.8).

This coordinate system gives you an easy way to set up when you consider in a rectangular object plane.

This will be true even when the αx=/=αy.

Hey Michael,

Thanks for the clarification.  What I’m talking about is slightly different.  When OpticStudio initially maps a normalized field coordinate (regardless of what the X & Y coordinates are), you will get a different (x, y) value on the image plane if you have a different clocking angle about the Y axis in the Field Data Editor.  This is due to a difference in projecting angular space onto a 2D object plane and the current mapping for alpha angles to direction cosines.  

In the first picture you have above, there are 3 fields:

1. (50, 0)
2. (0, 50)
3. (50, 50)

The first 2 fields have the same normalized radius of 50, but the last field has a normalized radius of 70.17.  If you change the last field to a coordinate of (35.35, 35.35), you will have all 3 fields laying on the same normalized circle:

For a rotationally symmetric system, all three of these fields should produce the same radial coordinate on Surface 1 (or the same Angle of Incidence on the Entrance Pupil).  If you look at the radial coordinate on Surface 1 from the Spot Diagram, you can see that Field 1 and Field 2 (which both lie on one of the axes such that either L or M is 0), the radial coordinate is 11.918mm.  However, if you look at the radial coordinate for Field 3 (which has the same normalized field angle of 50deg), you get a radial coordinate of 10.026mm.

The concept with radial symmetry is you should be able to choose any clocking angle for the input field and get the same output.  However, the way OpticStudio converts from alpha angle to direction cosine means that you have a periodic symmetry that only matches with reflections about either the axes or the diagonals.

When converting from angular space to linear space (lens units), there will be an inherent mis-mapping of values.  So, if you keep a perfectly circular set of field coordinates, you will get a distorted set of image coordinates (this is the red and blue circles I had at the end of my original post).  In order to get a circular set of image coordinates, you will need to use the equation I mentioned:

This doesn’t affect any individual built-in analysis but it becomes important when you start looking at system level designs where you might have different camera measurement angles for a camera module or if you’re trying to create a custom analysis.  I also think this equation provides more insight to my article from a few years ago about Image Simulation and using Angle vs Object Height:

https://www.linkedin.com/pulse/anamorphic-object-pixels-michael-humphreys

Hi ​@MichaelH,

Thank you for more information. Sorry I think I probably really didn’t explain well, but what you explained is exactly what I wanted to say. Above I mentioned many people would think αx^2+αy^2=theta^2, where the theta is the AOI you mentioned, while this is not true. In other words, in your syntax, αx^2+αy^2 will never be equal to AOI^2.

I think it’s already good enough to show αx^2+αy^2 != AOI^2, but the example you showed with image height explains it better.

The only point I added is I think the way we define the coordinate (αx, αy) actually has a benefit, although it can be confusing when people find αx^2+αy^2 != AOI^2.

I don’t have more to add on top of what you explained. It’s very clear to and I believe this is very useful for users who check this thread. Thank you for patiently adding picture and examples. :)