Existing single point diamond turning machines can fabricate complicated aspheric optical surfaces. The machine tool, however, limits these surfaces to being ones of revolution. Nonaxisymmetric surfaces can be machined by introducing a high speed control system, a spindle position, two additional axes and a high bandwidth axis parallel to the spindle axis.
Researchers have carried out a limited amount of work for machining nonaxisymmetric components using conventional and diamond tools. Spivey S. Douglass recorded some of the first attempts at diamond machining of nonaxisymmetric surfaces in his Ph.D. thesis 'A Machining System for Turning Nonaxisymmetric Surfaces'. However, he faced issues with the tool actuator and control system that prevented the generation of optical quality surfaces.
In 'Wavefront Correctors By Diamond Turning', Applied Optics Vol. 25, Meinel et.al. succeeded in achieving satisfactory results while machining a phase corrector plate using a piezoelectric actuator.
In the traditional machining realm, Tsao and Tomizuka’s research, titled 'Adaptive and Repetitive Digital Control Algorithms for Noncircular Machining' presented at the Proceedings of American Control Conference, Atlanta, 1988, enabled the successful machining of noncircular cylindrical parts.
This article details how nonaxisymmetric optical surfaces, which are offaxis segments of larger optics, can be singlepoint diamond turned. These segments can be deployed as offaxis mirrors, or as segments making up a larger onaxis mirror. The 10 m Keck telescope — a joint project of the University of California and the California Institute of Technology — is an example of the latter.
This article explains the segment geometry, as well providing an example for showing some of the major implementation details.
NonAxisymmetric Geometry
A general conic of revolution was the parent optic considered in this project. This was selected since a closedform description of the offaxis segment surface is known and can be formed such that it can be implemented realtime by a high speed controller. It is important to understand that implementation details need to be considered at all steps while developing this machining system.
The surface of an offaxis conic segment in cylindrical coordinates is descibed by the following equation:

(1) 
where z_{3} is the segment sagitta, ρ is the radius from segment center, f is the angular position around the segment and the d_{i}s are functions of the parent optic and the distance of the segments from the parent axis.
A comprehensive development of this equation, and a complete surface description, have been given by Gerchman in 'A Description Of OffAxis Conic Surfaces for NonAxisymmetric Surface Generation', SPIE Proceedings Vol. 1266.
Implementation
The machining of the nonaxisymmetric segment surface is achieved when the the X and Z axes’ movements are coordinated with the tool servo, specified here as the Z' axis. Since the segment rotation is performed by the machine spindle, based on the measured spindle position, tool servo motion is generated. The proper segment surface can be obtained by separating the Z and Z' motions so that Equation 1 describes the resulting surface. This can be achieved by separating the equation into one part which is φ and ρdependent, and another which is only ρdependent.
A suitable method to make this division, such that Z' motion is minimized, is precalculating X and Z motions at each ρ value. By determining the maximum and minimum values of z_{3}(φ) at each ρ and choosing a point half way between them as a reference, it is possible to establish a baseline motion for the machine slides.
The value at each ρ is subtracted from z_{3}(ρ,φ) to find tool servo motion. Using this method, it is possible to determine tool servo motion with the following expression:

(2) 
One instance of tool motion that depends on the use of this optimal baseline is shown in Figure 1, where tool servo motion is plotted as a function of the angular position φ. The solid line denotes motion at ρ_{max}, dashedline motion at 3/4ρ, the dashdot line at 1/2ρ, and the dotted line at 1/4ρ.
This example is for cutting a parabloid’s offaxis segment. It must be noted that the motion appears like a decaying sinusoid. However, as shown by the dotted line, this is not so towards the segment’s center.
Figure 1. Tool servo motion with optimum baseline subtracted. Solid line is tool motion at pmAX.
Machining Setup
Using a device called the Fast Tool Servo (FTS), machining experiments were done on a Rank Pneumo AS G2500. This is a high bandwidth, short range, tool motion device based on a piezoelectric actuator. For position feedback, in order to compensate for actuator nonlinearities, there is an integral capacitance gage used for position feedback. The present device has a 20 µm range and a bandwidth higher than 1 KHz.
In order to develop the complete machining system, there was a need for two more components. The first component was a high voltage amplifier (HVA), for which the input is a low voltage control signal and the output is a high voltage, high current drive signal for the piezoelectric actuator.
The second component was a highspeed controller that combines all the components and produces the actuator’s control signal. The components of the experimental setup and the interconnections are shown in Figure 2.
Figure 2. Diagram of experimental setup
Results
Several nonaxisymmetric surfaces were created to demonstrate the feasibility of this type of machining and for demonstrating some possible error sources.
For instance, the tilting of a flat surface, with reference to the plane perpendicular to the spindle rotation axis, was fabricated by producing a decaying, once per revolution sinusoidal signal for the tool servo. Figure 3 shows an interferogram of this tilted flat with a normal reference flat around its periphery. The tiled inner flat was such that the distance between its lowest and highest point was about 5 µm. The tilted surface diameter was 20 mm.
Figure 3. Interferogram of tilted flat surface machined with tool servo at a spindle speed of 480 rpm.
While machining this tilted surface under less than ideal conditions, two surface errors were apparent. Initial machining attempts were performed with no feedback from the capacitance gage and as a result, a nonsymmetric surface error was seen.
Furthermore, the Xaxis position was not initially used, but estimated based on X starting and federate position. This produced a comalike surface error  a once per revolution type error. Both these errors show how important it is to have a good control system that has enough inputs.
Conclusions
The validity of the concept has been proved by initial experiments using a diamond turning tool for machining nonaxisymmetric optical surfaces. The experiments have also shown the significance of key components that make a system for this kind of machining.
These components include a high voltage amplifier, a high bandwidth tool servo, a highspeed control system with sufficient number of inputs, and efficient software to generate the appropriate control signal.
Further research is required for providing a system capable of machining offaxis segments of aconicoid surfaces. Ongoing research focuses on the development of a method using Zernike Polynomials for defining a more general nonaxisymmetric surface and a control system using this description.
This information has been sourced, reviewed and adapted from materials provided by Precitech.
For more information on this source, please visit Precitech.