1. Jim Burge, Bill Anderson, Scott Benjamin,
Development of optimal grinding and polishing tools for aspheric surfaces
Marty Valente
Jim Burge, Bill Anderson, Scott Benjamin,
Myung Cho, Koby Smith
Optical Sciences Center
University of Arizona

2. Fabrication of spherical surfaces
Spheres are natural and easy, as long as you use
large stiff tools
good supports
smart polishing strokes,
The tool and the part tend to wear to form mating spheres
The tool always fits the surface, giving rapid convergence, excellent surface
Measurement is easy - interferometer, spherometer
The process itself results in a spherical surface

3. The largest lens (in the world?)
1.8-m diameter test plate for measuring the MMT wide field secondary mirror
Both sides spherical, concave side requires high accuracy
Polished at OSC to achieve slope spec of 0.01 waves/cm
Now has computer generated hologram on concave surface
polishing
handling
Final figure

4. Grinding and polishing aspheric surfaces
The process by itself does not converge to make the correct shape
It is necessary to set up an accurate test, and work the surface based on the measurement
The laps generally do not fit the aspheric surface so
there is no tendency towards the correct shape
special attention must be paid to the surface finish

5. General lap misfit for aspheres
Circular lap radius a
off center by b
Aspheric departure is dominated by lowest modes, with P-V deviation from vertex fit of
The grinding and polishing tool must always accommodate the misfit, as the tool is rotated and stroked over the part

6. Mechanisms for working aspheric surfaces
Lapping relies on two mechanisms for correcting shape errors:
Directed figuring : rubbing more, or pressing harder on the high spots
Natural smoothing : Using a stiff lap, small scales bumps are naturally worn down
Optimal tooling will take natural smoothing as far as possible to maximize efficiency

7. Control of small scales by smoothing

8. Stiffness of lap
Grinding and polishing of aspheres, always fighting between two issues:
#1) Desire to use large, rigid laps for passive smoothing
#2) Requirement that the lap conform to the asphere
usually leads to small tools or flexible tools at the expense of #1
for small parts, it can be economic to use small tools under computer control
Optimal lap, controlled to fit the asphere, and very stiff to figure errors
Next best thing, lap is compliant in modes necessary to fit the asphere, yet remains stiff to figure errors

9. Large tool for aspheres - stressed lap
Used at Steward Observatory Mirror Lab for f/1 mirrors
1.2-m, 60 cm, and 30 cm stressed laps are in operation
bent by actuators as the lap is moved,
NC shape changes every msec so it always fits desired asphere
bends up to 1 mm
6.5-m f/ nm rms

10. Active vs. passive laps
The actively controlled stressed lap works extremely well, yet requires significant initial investment and maintenance.
Is it possible to design a lap that is naturally compliant to the modes required to fit the asphere, yet remains stiff enough for natural smoothing?

11. The magic of rings
If the lap is shaped like a ring, power and coma terms go away

12. Bending required for ring tool - Astigmatism

13. Astigmatic bending
Rings bend easily in astigmatism if the cross section is compliant in torsion
Use geometry to make rings stiff locally in one direction
Analogy, bandsaw blade.
Totally compliant for astigmatism
Very rigid over scales of few inches
So a lap made from thin rings will fit the asphere! Power and coma are taken up by rigid body motion and astigmatism is easily bent in

14. An important detail for the rings - coning
Cylindrical rings, pressure aligned for maximum stiffness for near-flat surfaces only
For curved surfaces, tilted interface causes torsion, too compliant!
Solution: Tilt the beam, rings then become sections of a cone, rather than a cylinder
Cross-section of ring
Flexible, incompressible joint
Grinding or polishing pad
Polishing pressure

15. Ring tool design
We need to use nested rings to get sufficient polishing area
The rings can be faced with either grinding or polishing pads
The cross sectional height and width of the rings are chosen using finite element modeling to determine the stiffness.

16. Calculation of ring geometry
Using finite element modeling, we created an empirical model of ring stiffness as function of cross section geometry
Then, for each ring in the nested set:
The aspheric departure is calculated to determine the amount of astigmatic bending required for each ring, at the end of its stroke.
Choose ring cross section to allow the ring to bend by the required amount, forced by pressure variations small compared to the nominal polishing pressure.

17. Software for designing ring tools
User enters parameters for asphere and desired tool size and stroke
Software calculates the corresponding ring geometry

18. Attachments of rings
Allow vertical motion using guide rods in linear bearings
Allow rotation using spherical joint
Constrain lateral motion
Lateral force near polishing surface to minimize moments
Supply drive force in circumferential direction
Apply force using weights
Designed for fabrication ease
weight
Teflon bearing
Support frame
Teflon bearing
spring
ring
guide rod
ball joint

19. Grinding interface Use pads, small enough to always fit asphere
Stiff attachment to ring
Pivot on ball bearing
Held on by silicone
Grinding surface - metal
Polishing surface - urethane
Design for fabrication ease
Maintenance is important
guide rod
ring
ball joint
ball bearing
RTV
Aluminum pad with seat for bearing
grinding/polishing surface

20. Prototype ring tool
Working 40 cm f/0.5 asphere

21. Preliminary results from ring tool
The tool is basically well-behaved
not problems at edge
no problems with chatter
Fits the aspheric surface
Good smoothing achieved
Initial Ronchigram, after aspherizing with full size compliant tool
Ronchigram, after 4 hr run with the prototype ring tool

22. Ring tool frame Connects drive pin to rings
Has bearings for guide rods
Frame “floats” on rings using soft springs
Drive torque and lateral forces taken at hub
Lifting eyes are used to hoist frame
3-m tool for 4-m f/0.5 paraboloid

23. Membrane laps
Smaller laps can achieve a good compromise between desired stiffness for smoothing and compliance to fit the asphere using laps faced with membranes
Pin, connecting to polishing machine
Rigid tool
Compliant interface (CC neoprene foam)
Membrane
Grinding or polishing pads

24. Membrane stiffness
Finite element used to establish modal stiffness of membrane
Solve for membrane thickness for a given tool, stroke, and membrane material
Membrane stiffness goes as t3
Stiffness to ripples on the surface goes as L-4
L is the period of the ripple
Membranes with the correct curves can be made by
direct machining
hot-forming plastic sheets onto surface
casting, layup on surface

25. Analysis using modal decomposition
Displacement
Pressure
Required Pressure Distribution
Membrane Tool’s
Aspheric Misfit
The modal stiffness was calculated using a finite element model
Note that the dominant pressure variations are at the edge of the tool

26. Software for designing membrane tools
User enters parameters for asphere, desired tool size and stroke
Given membrane, software calculates pressure distribution under lap
or given allowable pressure distribution, software calculates membrane thickness

27. Experience with membranes
Initial Ronchigram
Tool made by hot forming plastic sheet, faced with grinding pads
After 5 hours directed figuring with membrane tool
For f/0.5 convex asphere
(tested in transmission)

28. Conclusion
There is much activity and interest at the University of Arizona in the area of fabrication of aspheric surfaces.
Stay tuned! Things develop very quickly