1. Image Quality Degradation due to Lens Surface Polishing Irregularity
Dave Stephenson
Madison, CT
June 1, 2009

2. Outline Motivation Spatial frequency regimes Performance measures
Background
Problem statement
Spatial frequency regimes
Performance measures
Empirical study: MTF degradation
Summary & References

3. Applications & Benefits of Lenses with Aspheric Surfaces
Consumer, medical, industrial applications
Fewer elements
Reduced cost & weight
Improved light transmission
Improved imaging performance
Multiple spheres combine to act aspheric

4. Polishing Techniques Aspheric lens surfaces Spherical lens surfaces
Computer controlled dwell via “hit” map
Polishing pads are smaller than the surface and can cause localized slope errors
Spherical lens surfaces
Traditionally the pad or lap is larger than the surface, so small-pad polishing errors don’t normally occur
But small-pad techniques are now being employed for spherical surfaces, not just aspheric

5. Full-Contact Polishing Errors (Figure)
Cylindrical
Non-rotationally symmetric irregularity
Pattern may clock arbitrarily
Typically all that is toleranced
“B” in ISO /A(B/C)
Hole & roll
Rotationally symmetric irregularity
Hole typical in center; bump possible
Edge typically rolled down; a rolled up edge is also possible
“C” in ISO /A(B/C)

6. Small-Pad Polishing Errors (Mid-Spatial Frequency)
Radial spoke-like defect
Control with slope or PSD spec
Non-rotationally symmetric irregularity
Pattern may clock arbitrarily
Adds to “B” in ISO /A(B/C)
Concentric ring-like defect
Rotationally symmetric irregularity
Adds to “C” in ISO /A(B/C)

7. Questions & Issues
How do these polishing errors degrade imaging performance?
What simulation tools are available to explore design sensitivity for tolerancing?

8. Outline Motivation Spatial frequency regimes Performance measures
Empirical study: MTF degradation
Summary & References

9. Spatial Frequency Regimes
Mid-Spatial Frequency (MSF)
Typically 0.2 – 3.0 c/mm for ø25 mm
Low-to-Mid boundary
Zernike polynomial limit for figure
5 – 10 cycles per diameter
Measured with laser Fizeau interferometer
Mid-to-High boundary
The roughness definition sets it (RMS roughness is the square root of area under PSD curve)
Measured with white light Mirau interferometer or AFM
J.E. Harvey and A. Kotha, “Scattering effects from residual optical fabrication errors”, Proc. SPIE 2576, pp

10. Control MSF with a PSD spec
Peak in the PSD corresponds to the ripple freq
Can define a limit line & stay below it during polishing
PSDlimit = 5x104 * freq-1.55 in units of A2µm shown here
Peak is above the limit at 0.3 c/mm (3.3 mm period)

11. Outline Motivation Spatial frequency regimes Performance measure: MTF
Empirical study: MTF degradation
Summary & References

12. Modulation Transfer Function (MTF)
Linear system frequency domain analysis
Reflectivity alters amplitude of complex object field
Height modifies phase of complex object field
Optics low-pass filter as function of spatial frequency
Complex image field is linear superposition of filtered complex-valued components, frequency-by-frequency
Image-to-object ratio is the Optical Transfer Function
MTF is modulus of the complex-valued OTF
MTF(f) = | OTF(f) | = | ImageField(f) / ObjectField(f) |

13. MTF Example Image MTF decreases with increasing frequency
Even perfect optics will low-pass filter the high frequency content
P. de Groot, “Instrument transfer function in interferometry”, FRINGE /12/2005.

14. Outline Motivation Spatial frequency regimes Performance measure: MTF
Empirical study: MTF degradation
Summary & References

15. Example Double Gauss Lens
~0.32 mm
depth-of-focus (DoF)
for 40% MTF
at 0.5 Nyquist
6-elements, 12 polished surfaces
Well balanced nominal MTF
0.5 Nyquist
best focus
~0.75 MTF
at 0.5 Nyquist

16. Cylindrical irregularity First surface perturbed .048λ (30 nm) RMS
Nearly all figure error; little MSF error
Lens design software tools
All model irregularity as a cylinder
Other shapes require extra modeling
Little impact on DoF or peak MTF

17. Cylindrical irregularity Every surface perturbed .048λ (30 nm) RMS
6 of 12 perturbations shown above
Random clockings
DoF: reduced from 0.32 to 0.25 mm
Peak: drops from 0.75 to 0.70 MTF

18. Hole & roll irregularity First surface perturbed .048λ (30 nm) RMS
Nearly all figure error; little MSF error
Lens design software tools
Model as Zernike terms or with aspheric perturbation terms
Little impact on DoF or peak MTF

19. Hole & roll irregularity Every surface perturbed .048λ (30 nm) RMS
6 of 12 perturbations shown above
Random clockings
DoF: reduced from 0.32 to 0.17 mm
Peak: drops from 0.75 to 0.52 MTF

20. Spoke-like MSF irregularity First surface perturbed .048λ (30 nm) RMS
Nearly all MSF error; little figure error
Lens design software tools
Not practical to model with Zernike terms
No tools have a native perturbation like this
Little impact on peak MTF or DoF

21. Spoke-like MSF irregularity Every surface perturbed .048λ (30 nm) RMS
6 of 12 perturbations shown above
Random clockings
DoF: reduced from 0.32 to 0.17 mm
Peak: drops from 0.75 to 0.48 MTF

22. Ring-like MSF irregularity First surface perturbed .048λ (30 nm) RMS
Nearly all MSF error; little figure error
Lens design software tools
Not practical to model with Zernike terms
CodeV has native perturbation to model
Significant impact on peak MTF & DoF

23. Ring-like MSF irregularity Every surface perturbed .048λ (30 nm) RMS
6 of 12 perturbations shown above
Random clockings
Bad: When on inner elements (0.45 peak)
Worse: When on outer elements (0.30 peak)
Worst: When on all elements (DoF  zero)

24. Outline Motivation Spatial frequency regimes Performance measure: MTF
Empirical study: MTF degradation
Summary & References

25. Summary Figure errors from traditional full-contact polishing
Significant mid-spatial frequency (MSF) content is unlikely
Cylinder, and hole & roll, typically result
Hole & roll is the worst (similar impact to spoke-like MSF)
If using small pad polishing techniques, MSF is important
Consider for both spherical and aspheric surfaces
Spoke-like MSF is less troublesome for the Double Gauss lens
Ring-like concentric MSF is the worst for the Double Gauss lens
Commercial “¼ wave P-V, λ/20 RMS” may not be adequate
Tolerance slope or PSD using ISO to limit MSF content
Tolerance analyses done with lens design software commonly only explore sensitivity to cylindrical irregularity
CodeV is now able to tolerance ring-like MSF

26. References
D. Aikens, J. E. DeGroote, and R. N. Youngworth, "Specification and Control of Mid-Spatial Frequency Wavefront Errors in Optical Systems," in Optical Fabrication and Testing, OSA Technical Digest (CD) (Optical Society of America, 2008), paper OTuA1.
J. Rogers, “Slope error tolerances for optical surfaces”, SPIE Technical Digest TD0404, (invited paper), SPIE Optifab Conference, Rochester NY May 2007.
P. de Groot, “Instrument transfer function in interferometry”, FRINGE /12/2005.
R. N. Youngworth & B. D. Stone, “Simple estimates for the effects of mid-spatial frequency surface errors on image quality”, Applied Optics 39(13), pp (2000).