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**Image Quality Degradation due to Lens Surface Polishing Irregularity**

Dave Stephenson Madison, CT June 1, 2009

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**Outline Motivation Spatial frequency regimes Performance measures**

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

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**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

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**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

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**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)

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**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)

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

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**Outline Motivation Spatial frequency regimes Performance measures**

Empirical study: MTF degradation Summary & References

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**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

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**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)

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**Outline Motivation Spatial frequency regimes Performance measure: MTF**

Empirical study: MTF degradation Summary & References

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**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) |

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**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.

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**Outline Motivation Spatial frequency regimes Performance measure: MTF**

Empirical study: MTF degradation Summary & References

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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)

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**Outline Motivation Spatial frequency regimes Performance measure: MTF**

Empirical study: MTF degradation Summary & References

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**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

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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).

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