# Ensuring the manufacturability of a freeform surface

The article is devoted to the main manufacturability parameters of freeform surfaces manufactured using a single-point diamond turning machine. It is explained how the manufacturability parameters are connected to instrument parameters. It is shown how to check and control those manufacturability parameters in OpticStudio. Also, it is explained how to deal with the behavior of a freeform surface outside its active area. As an example, a plastic freeform lens (an Alvarez lens element) is used.

Authored By DynaOptics

## Surface parameters to control

The parameters of the lens to be controlled depend on the manufacturing method and equipment. One of the most popular and widely available ways to manufacture plastic optics is using a 3-axis diamond turning machine (Fig. 1) for either direct cutting or, which is more often, cutting an insert to mold the lens.

Fig. 1 - 3-axis diamond cutting machine (left) diamond cutting tools (right)

## Slope angles

Let’s look at the limitations coming from the instrument (Fig. 2). Cutter’s flank angle limits maximum possible slope angle along any radial cross-section. Because such a radial cross-section coincides with the tangential plane the corresponding slope is called the “tangential slope” in Optic Studio. Comparably a rotationally symmetrical surface tangential slope of a free-form surface has different distributions along different radial cross sections.

Another parameter is a “sagittal slope” angle. When we cut a free-form surface on a 3-axis diamond turning machine, the cutter is moving back and forth along Z-axis at every turn of the workpiece in order to produce the non-rotationally symmetrical shape of the lens. In this, the clearance angle of the cutter limits how fast the surface can change along each circle on the lens, this is called sagittal slope. To be more precise, the cutter produces a spiral track on the surface but the step of the spiral is so small that in most cases it is OK to consider the cutter track as a series of circles. For rotationally symmetrical lenses sagittal slope is just zero.

Fig. 2 – tangential and sagittal slopes,
yellow line shows along which direction slope is measured

Sometimes it is reasonable from the manufacturing standpoint to place the workpiece not along the rotation axis of the stage, but off-axis so the cutters track on the workpiece looks like almost straight lines. In this case, we should control so-called “X-slope” and “Y-slope” (Fig. 3).

Fig. 3 – X and Y slopes,
yellow line shows along which direction slope is measured

Let’s see how to control those parameters in Zemax OpticStudio. As an example, we took a freeform lens with one plane of symmetry, so-called Alvarez lens. If we go into OpticStudio to Analyze -> Surface -> Surface Slope we can check slope distribution across the surface. This plot displays the tangential, sagittal, x and y slopes of a surface as a 2D color or contour map, or as a 3D surface plot (please refer to the OpticStudio User Manual for more details).

Fig. 4 – Tangential and Sagittal slopes of the 1st surface of the example Alvarez lens

Fig. 5 -X and Y slopes of the 1st surface of the example Alvarez lens

Let’s suppose we have the following drawing of the cutter from a manufacturer (Fig. 6). What is important to learn for our optical design on this drawing is the cutter clearance angle and flank angle.

Fig. 6 – cutting instrument drawing

Clearance angle = 7 ± 0.5 degrees.

The flank angle can be calculated as:

Flank angle = (90 – Included_angle/2) = 65 ± 0.5 degrees.

During the optical design, we should limit the slopes of our surfaces correspondingly, so it will be possible to manufacture the lens with the provided instrument.

In case of axial cutting, the maximum tangential slope angle of the surface must be less than the cutter flank angle, e.g. less than 64.5 degrees (we took into account ± 0.5 degrees tolerance here). The maximum sagittal slope angle must be less than the cutter clearance angle, e.g. less than 6.5 degrees.

In case of off-axis cutting, the maximum X-slope angle of the surface must be less than the cutter flank angle, e.g. less than 64.5 degrees, and the maximum Y-slope angle must be less than the cutter clearance angle, e.g. less than 6.5 degrees.

OpticStudio has the DSLP operand which can be included in the Merit Function Editor in order to control slope angles of the surface during optimization. DSLP operand has various parameters and different outputs; you can find more information in the OpticStudio User Manual. In the figure below (Fig. 7) it is shown how to get maximum values of tangential, sagittal, X and Y slopes.

Fig. 7 – using DSLP operand to get maximum values of the surface slopes

The slope is calculated as a gradient at a point or in other words it is calculated by finding the ratio of the sag change to the coordinate change between (any) two distinct points. If we want to get a corresponding slope angle in degrees we should calculate an arctangent of the slope value.

For the 1st surface of our Alvarez lens we have:

Maximum tangential slope = 12.539 deg < Cutter flank angle

Maximum sagittal slope = 6.056 deg < Cutter clearance angle

Maximum X slope = 11.229 deg < Cutter flank angle

Maximum Y slope = 8.468 deg > Cutter clearance angle

As we can see this surface can be cut by the provided instrument in axial mode but cannot be cut in off-axis mode because Y slope exceeds the cutter clearance angle.

Please pay attention to sampling. For the sake of optimization speed, you may want to use smaller sampling in the merit function for the DSLP operand but in this case it may provide a smaller value than the actual slope. It is ok to optimize with a smaller sampling to increase optimization speed, but we would recommend from time to time, especially at the final optimization stage, to check slopes with a high enough sampling.

Also in OpticStudio we can get values of tangential, sagittal, X and Y slopes at an exact coordinate on the surface using SSLP operand (Fig. 8) to control slopes at an exact surface point.

Fig. 8 - using SSLP operand to get slopes at an exact surface point

If we take a close look at the tip of the cutter we can see that it is actually not a single point, its facet has a circular shape which is characterized by the tip radius. And in the other direction (see cross section A-A on the Fig. 9) cutter tip is characterized by the cutting edge radius.

Fig. 9–close look at cutter’s tip

Similar to slope angles local radiuses of the surface must be larger than the corresponding radiuses of the cutter, otherwise, the surface will be overcut and spoiled.

In case of axial cutting, the maximum tangential local radius of the surface must be larger than the cutter tip radius. The maximum sagittal local radius must be larger than the cutter edge radius.

In case of off-axis cutting maximum X local radius of the surface must be larger than the cutter tip radius. Maximum Y local radius must be larger than the cutting edge radius.

If we go in Optic Studio to Analyze -> Surface -> Curvature we can check curvature distribution across the surface. This plot displays the tangential, sagittal, x and y curvature of a surface as a 2D color or contour map, or as a 3D surface plot (please see the OpticStudio User Manual for more details). The dependency between local radius and curvature is simply:

Usually, the cutter tip radius and cutter edge radius are significantly smaller than local radiuses of an optical surface, so in most cases, we don’t have to control it during optimization but in some particular cases of complex shape surfaces it may be needed. It is a good practice to check local radiuses of free-form surfaces at the final stage of optimization and make sure there are no problems with a chosen cutting instrument.

Fig. 10 - Tangential and Sagittal curvature of the 1st surface of the Alvarez lens

Below is the summary table of surface parameters to be controlled depending on the cutting method on a 3-axis diamond turning machine:

 Axial cutting Off-axis cutting Tangential slope angle < cutter flank angle - Sagittal slope angle < cutter clearance angle - X slope angle - < cutter flank angle Y slope angle - < cutter clearance angle Tangential curvature < (1/cutter_tip_radius) - Sagittal curvature <(1/cutter_edge_radius) - X curvature - < (1/cutter_tip_radius) Y curvature - <(1/cutter_edge_radius)

## The behavior of a free-form surface outside the active area

From an optical design standpoint, it is better not to limit the behavior of a free-form surface outside the active area because additional limitations decrease optimization speed, produce additional local minima, and increase the risk of non-convergence to the optimal system performance.

If we look at our Alvarez lens beyond the active area (Fig. 12) we can see that the shape is too aggressive and definitely is not manufacturable.

Fig. 12– Alvarez lens shape beyond its active area

In order to place the lens to a barrel we should add a flange to our Alvarez lens. Since the shape of our Alvarez lens beyond the active area is too aggressive we should replace it with some smooth transition area which would connect the active area and the flange. At DynaOptics we use our own software called uVo which can automatically create such a transition area. We just specify the flange Z position and the radial gap between our surface and flange and uVo will automatically create a smooth transition area (Fig. 13).

Fig. 13 – transition area automatically created by uVo software

Also uVo is showing Sagittal and Tangential slope angles of the resulting surface, so we can check on the spot if this surface can be manufactured with the provided instrument (Fig. 14).

Fig. 14–uVo software screen to create a smooth transition area

## Conclusion

Understanding the manufacturing method and corresponding instrument limitations helps to build effective communication between the optical design team and the manufacturing team. Taking manufacturing method limitations into account during the optical design stage helps to save time and money. DynaOptics uVo software can significantly decrease the engineering time needed to design a smooth transition area between the working surface and flange, especially for complex freeform surfaces.

## References

1. https://www.contour-diamonds.com/
2. https://en.wikipedia.org/wiki/Slope
3. https://support.zemax.com/hc/en-us/articles/1500005491161-Optical-Zoom-based-on-Alvarez-Freeform-Elements
4. Fabrication of Hexagonal Microlens Arrays on Single-Crystal Silicon Using the Tool-ServoDriven Segment Turning Methodby Mao Mukaida and Jiwang Yan https://www.mdpi.com/2072-666X/8/11/323
5. Technology of Manufacture of the Negative Matrices for Linear Fresnel Lenses, S. Grubyy, V.V. Lapshin, E.M. Zakharevich https://www.researchgate.net/publication/301939103_Technology_of_Manufacture_of_the_Negative_Matrices_for_Linear_Fresnel_Lenses/download