For a complex lens, it can be convenient to add dummy surfaces at various locations: at the entrance and exit pupils, the nodal points, or the principal planes. This article demonstrates how to insert dummy surfaces at the front and rear principal planes of a Cooke Triplet. A sketch of the final system is shown in Figure 1.
Figure 1: A layout plot of a Cooke Triplet that includes dummy surfaces at the front and rear principal planes.
The Lens Data Editor is shown in Figure 2. Surfaces 2 and 3 are used to go out to the front principal plane and return from it. Surfaces 11 and 12 are used to go to the rear principal plane and return from it. (Because those surfaces don’t contain any materials, they do not bend the rays and are referred to as “dummy” surfaces.)
Figure 2: The Lens Data Editor for the system, showing configuration 1.
The lens file contains three configurations. The first configuration is the lens as it will be operated. This is the configuration one would use to analyze and/or optimize the lens performance in the usual way. The second configuration is used to find the location of the rear principal plane. The third configuration is used to find the location of the front principal plane.
2 Finding the Rear Principal Plane
Configuration 2 is used to locate the rear principal plane. This will determine the thickness of line 11, which travels out to the rear principal plane. The thickness of surface 12 is picked up from surface 11, to return from the principal plane to the image plane.
Figure 3: The Lens Data Editor for Configuration 2, used to locate the rear principal plane.
The rear principal plane, by definition, is at the location where a ray in object space, traced from an object at infinity, intersects the same ray in image space. So, in configuration 2, we set the object distance to infinity using the Multi-Configuration Editor, as shown in Figure 4.
Figure 4: In the Multi-Configuration Editor, the object distance is set to infinity.
This gives a Layout plot as shown in Figure 5. With the object at infinity, the image plane is out of focus. That’s okay, because we’re using this configuration just to locate the principal plane, and not for any other purpose.
Figure 5: The layout plot for Configuration 2.
In the Merit Function, we can use Configuration 2 to optimize for the thickness of surface 11. The thickness of line 11 is set to a variable. Then, in the Merit Function, we measure the Y height of a paraxial ray in object space using the PARY operand in line 34. And we measure the Y height of a paraxial ray in image space (line 35). In line 36, we ask that the difference between the two measurements is zero.
Figure 6: The Merit Function lines that are used to locate the rear principal planes.
After optimizing this, the location of the rear principal plane is correctly captured in the thickness of surface 11, and is found to be -97.3 mm before the image plane.
3 Finding the front principal plane
Configuration 3 is used to locate the front principal plane. The front principal plane, by definition, is at the location where a ray in collimated image space intersects the same ray in object space.
The first step is to place the object at a location that produces a collimated beam in image space. To do that, we set the thickness of line 1 to be a variable, and the thickness of line 0 to zero. The Lens Data Editor for Configuration 3 and the Multi-Configuration Editor are shown below.
Figure 7: The Lens Data Editor for Configuration 3, used to locate the front principal plane.
Figure 8: The Multi-Configuration Editor, with a variable on the thickness of surface 1.
In the Merit Function Editor, we trace a few paraxial rays to check the collimation of the beam in image space. This is done using the paraxial ray angle operands, PARB, in lines 43-47.
Figure 9: The Merit Function Editor is used to locate the object so that a collimated beam is produced in image space.
After optimizing, Configuration 3 now contains a system with the object at the front focal plane and a collimated beam in image space.
Figure 10: In configuration 3, the object is located in the front focal plane of the lens and there is a collimated beam in image space.
The second step in locating the Front Principal Plane is to optimize to find the thickness of surface 2, which is set to a variable (see Figure 7). In the Merit Function, working in Configuration 3, we measure the height of a paraxial ray in object space and another in image space. We then use the DIFF operand to require that the difference in the two measurements goes to 0.
Figure 11: The Merit Function uses lines 52-54 to locate the front principal plane.
After optimizing to find the thickness of surface 2, then, surface 3 will be correctly located at the front principal plane of the lens. For this system, the front principal plane is about 12.6 mm to the right of the first lens surface. (See Figure 2).
4 Checking the results
We can confirm the locations of the principal planes by looking at Analyze \ Reports \ Prescription Data and selecting “Cardinal Points” in the settings.
In the Lens Data Editor, the front principal plane location is +102.6 mm from the object. And the rear principal plane is -97.3 mm from the image plane. The Prescription Data lists the same values, confirming our results.
Figure 12: Selecting Cardinal Points in the settings of the Prescription Data window.
Figure 13: Confirming the locations of the Principal Planes at the primary wavelength.
To add dummy surfaces at the principal planes of a thick lens, do the following:
- Insert pairs of dummy surfaces after the object and before the image plane.
- Create a configuration with the object at infinity to find the rear principal plane.
- Create a configuration with the object at the front focal plane to find the front principal plane.
- Set up the Merit Function using the separate cofigurations to optimize for the locations of the dummy surfaces.