# Technical Guide: Asphere design

## Overview

> Mathematical description

> Technological possibilities

> Guideline for asphere design

> Prelimary considerations

> Limitations of local surface curvature

> Aspherical parameters

> Edge thickness and oversize

> Tolerancing via slope error

## Mathematical description

An aspheric surface is typically described by a radius of curvature and the conic constant. Additional correction polynomials are often used for further surface adjustments. Note, that the odd coefficients are used very rarely.

## Technological possibilities

## Guideline for asphere design

**Preliminary considerations**

Even though many different glass types can be manufactured to aspheres, it is recommended to confine this selection onto a few standards whenever possible. All fused silica types are always a good choice. Recommended optical glasses have a Knoop hardness around 600 and low chemical responsiveness (e.g. Schott N-BK7 or N-SF11). Different hardness and chemical susceptibility increase production costs because of the need of alternative polishing agents or elaborate handling.

Further significant factors for economic asphere design are the lens diameter and surface tolerances. With increasing diameter, the surface to be processed increases. But in comparison to spherical lens production, the surface of aspheres cannot be treated with large-area tools. Furthermore asphere production is more complex and is tied to a close interaction between metrology and production machines. Strict tolerances lead to raised cost as in some circumstances individual production steps have to be repeated to fulfill the demanded requirements.

**Limitation of local surface curvature**

The misinformation, that inflection points in asphere surface forms cannot be manufactured, is still existent. But the real limitation behind are production tools: During asphere manufacturing, wheel based tools are used at many steps. Their spatial extend – in other words their minimal radius – limit the minimal possible local surface curvature. At Sill Optics, the smallest tool radius is 25 mm, thus all local concave surface curvatures should not be smaller than 35 mm. Otherwise manufacturing these areas is not possible. Inflection points (where the surface curvature switches from convex to concave or vice versa) on the other hand often lead to the described problem, thus are a first indicator.

**Aspherical parameters**

“Use as few aspherical parameters as possible, but as many as necessary” – This sentence reflects an important baseline in asphere design. A lot of parameters do not only complicate finding a good solution for optimizing algorithms and increase the risk for errors during documentation and production, but could also lead to very strong surface slopes towards the edges. This might even get worse, when an oversize of the diameter is necessary for production.

Additionally note, that the conic constant and the fourth order parameter are not independent of each other, which could impede with optimizing when both are set as variables at the same time. Likewise, the second order parameter is linked to the vertex radius. The second order parameter is therefore barely used.

**Edge thickness and oversize**

As mentioned before, aspheres are typically developed with 4 mm – 6 mm oversize in diameter: Production steps are facilitated as the tools are allowed to process the surface evenly beyond the final diameter. Surface from errors towards the edges are therefore avoided.

Furthermore, an edge thickness of at least 2 mm – 3 mm is recommended: First, a significant edge thickness eases handling of the lens and second, splitting of glass at the edge during manufacturing is diminished.

**Tolerancing via slope error**

For high requirements for the aspherical surface, tolerancing form deviation and irregularity might not be enough. In comparison to spheres, aspheres cannot be manufactured with large-area tools, which could result in local surface errors/ deviations. To handle these errors, an additional specification of the surface slope tolerance is recommended. It is typically given as Δz(slope tolerance/ integration length/sampling step size). Depending on the integration length, it can be used as indicator for the acceptable mid-spacial or high-spacial frequency errors.