Optical Design Work for a Laser-Fiber Scanned Image Source for the Crusader Helmet Janet Crossman-Bosworth Research Engineer – Optical Design Human Interface Technology Laboratory University of Washington January 16, 2003

Introduction Three optical designs will be presented. First Design - Low frequency fiber resonance input Second Design - High frequency fiber resonance input Third Design -

First Design

Goals for First Design Point Source Re-imaging Circular Scan 19mm Screen ≤ 20µm RMS Spot Sizes 532nm Wavelength Low frequency fiber resonance input (2.5kHz) from endoscope prototype Axial Length of System < 100mm

Fiber Input for First Design Plotted fiber tip positions from model data for endoscope prototype (linear mode shape) Optical Node Length * = 4.5mm average Maximum fiber tip displacement = ± 2mm N.A. = 0.11 (single mode fiber) * Optical Node Length = The distance between the fiber tip and the position along the axis from which the light appears to emanate.

Optical Layout for First Design File Name: HMD – I All Custom Lenses Display Diameter = 18mm Lens System Length = 23mm (fiber tip to screen) Estimated Weight = 0.5g

At the Screen (Image Plane) File Name: HMD – I Central Region: RMS Spot Diameter = 31.19µm 32 spots/mm 64 resolvable spots/mm approx.* Peripheral Region: RMS Spot Diameter = 303.05µm 3 spots/mm 6 resolvable spots/mm approx. * We have been able to resolve approximately twice as many spots/mm as that calculated from the RMS spot diameter.

Resolution: Diffraction Limited Example Rayleigh Criterion: The maximum illumination of one diffraction pattern coincides with the first dark ring of the other diffraction pattern. Separation = 1.22 λ (F/#) (This is also called the Airy Disk Radius.) Sparrow Criterion: There is no minimum between the maxima from the two diffraction patterns. Separation = λ (F/#) Our measurements use a criterion between that of Rayleigh and Sparrow.

Summary for First Design Fiber tip displacements of ± 2mm do not occur for video rate frequencies. The first design will not work for video rates. There is not sufficient resolution in the periphery of the first design.

Second Design

Goals for Second Design ≤ 15µm RMS Spot Size 2mm to 4mm Optical Node Length Maximum Fiber Tip Displacement = ± 1mm (Representative of higher frequency systems) Axial Length of System < 80mm

Fiber Input for Second Design Simplified model (Not actual measurements) 4mm Optical Node Length Maximum fiber tip displacement = ± 1mm across a spherical curve N.A. = 0.11 (single mode fiber)

Optical Layout for Second Design File Name: HMD – ZK3e All Custom Lenses Display Diameter = 20mm Lens System Length = 52mm (fiber tip to screen) Estimated Weight = 1.0g

At the Screen (Image Plane) File Name: HMD – ZK3e Central Region: RMS Spot Diameter = 61.99µm 16 spots/mm 32 resolvable spots/mm approx. Peripheral Region: RMS Spot Diameter = 107.30µm 9 spots/mm 18 resolvable spots/mm approx.

Summary for Second Design The required field of view has been achieved. The illumination across the field of view is more uniform. A spot size of ≤ 15µm is not achievable across a 19mm field of view, using a 0.11 N.A. fiber with a maximum displacement of ± 1mm, according to the Optical Invariant*. * For more information about the Optical Invariant, see Appendix A.

Third Design

Goals for Third Design 50µm RMS Spot Size Increase the fiber N.A. to 0.4 or 0.5 50µm RMS Spot Size 0.95mm Optical Node Length Maximum Fiber Tip Displacement = ± 0.5mm (Representative of higher frequency systems) Axial Length of Lens System < 80mm 5 Lenses or Less All Commercial Lenses to Reduce Cost

Fiber Input for Third Design Simplified model (not actual measurements) 0.95mm Optical Node Length Flat object plane using a Noliac ring bender Maximum fiber tip displacement = ± 0.5mm across a flat plane N.A. = 0.4 (custom fiber)

Third Design – Prototype Design File Name: HMD – ZZH1c4 1 Custom Lens, 4 Commercial Lenses, and 1 Fiber Optic Taper Display Diameter = 20mm (at large end of 2x magnification fiber optic taper) Intermediate Image Plane Diameter = 10mm (at small end of taper) System Length = 69mm (fiber tip to taper) + 19mm (taper) = 88mm Estimated Weight = 6g (lenses) + 16g (taper) = 22g

Fiber Optic Taper Schott Fiber Optic Taper 2x Magnification Large end diameter = 20mm Small end diameter = 10mm Taper Length = 19mm Fiber diameter at large end = 6µm Estimated Weight = 16g

Image at Small End of Taper File Name: HMD – ZZH1c4 Central Region: Airy Disk Diameter = 15.65µm (Diffraction Limited) 64 spots/mm 128 resolvable spots/mm approx. Mid-Peripheral Region: RMS Spot Diameter = 25.87µm 39 spots/mm 78 resolvable spots/mm approx. Peripheral Region: Airy Disk Diameter = 22.43µm 45 spots/mm 90 resolvable spots/mm approx.

Image at Large End of Taper File Name: HMD – ZZH1c4 Central Region: Spot Diameter = 31.30µm 32 spots/mm 64 resolvable spots/mm approx. Mid-Peripheral Region: Spot Diameter = 51.74µm 19 spots/mm 39 resolvable spots/mm approx. Peripheral Region: Spot Diameter = 44.86µm 22 spots/mm 45 resolvable spots/mm approx. A design goal of 50µm diameter spots yields 20 spots/mm and approximately 40 resolvable spots/mm.

Tolerance Analysis of the Third Design (Tolerance Analysis for the Intermediate Image Plane) 40 tolerances were used which, each by themselves, would allow no more than a 100m RMS spot diameter at the intermediate image plane for any field point, but with a 1% minimum tolerance on all tolerances except the decenters and tilts. 10 Radius of Curvature Tolerances 5 Spacing Tolerances 5 Center Thickness Tolerances 10 Decenter Tolerances, ranging from 0.05mm to 0.20mm 10 Tilt Tolerances, which were either 0.6 or 1.0 The optical design program uses the final spacing to the intermediate image plane to adjust the back focus during tolerancing.

Tolerance Analysis (continued) Results: A Monte Carlo tolerance analysis was run, which simulates the effect of all the tolerance errors simultaneously. The mean RMS spot diameter was 134µm. This translates to approximately 15 resolvable spots/mm. After being magnified by the 2x taper, there would be approximately 7 resolvable spots/mm. This design is highly sensitive to tolerance errors. Very tight tolerances are required to maintain intended design performance.

Third Design with Curved Source Vignetting High field curvature Peripheral RMS spot size diameters = 1.022mm

Third Design with IR Source Wavelength = 1.31µm RMS spot size diameters = 2.7mm to 3.2mm Nearly parallel light impinges upon the screen. Distance between last lens and taper = 35mm (A beamsplitter could be placed here.) Light from 2 object points Light from 11 object points

Summary for Third Design The image meets the 50µm spot diameter goal, except in the mid-periphery where the spot diameter is approximately 52 microns. The system exceeds the 80mm length goal by 8mm. Only 5 lenses were used. 1 custom lens was needed. Tight tolerances are required for this design. A flat image source is required for this design. A beamsplitter could be used with this design for IR light. Will the crosshatching of the taper be visible?

Conclusions The original goal was a 19mm screen with 809 resolvable spots, or approximately 43 resolvable spots/mm. The third design very nearly meets this original goal across the field of view. The first and second designs do not. An analysis of the optical invariant is needed to determine what characteristics are needed in the optical fiber. Methods to increase the fiber N.A. and increase the fiber tip displacement for a standard fiber are known here at the HIT Lab.

Conclusions (continued) Fiber scanners are being designed and fabricated to meet these optical specifications. Large fiber tip displacements at high resonant frequencies are difficult to achieve. Just as there is an optical invariant, there may also be an invariant for resonant fiber scanning. Designs are limited to geometrical size limitations of the Crusader Helmet. (i.e. 20mm flat screen & 100mm length)

Possibilities for Future Design Work Use Other Fiber Input Characteristics Further Aberration Control Circular or Rectilinear Scan Gradient Index Optics Diffractive Optics Doublets and/or Triplets Most or All Custom Lenses No Fiber Optic Taper No Field Flattening Eight or More Lenses

Appendix A Optical Invariant

Optical Invariant Optical Invariant = ypnu – ynup y & yp = Axial & Principal Ray Heights u & up = Axial & Principal Ray Angles n = Index of Refraction

Optical Invariant at Object & Image Surfaces ypnu – ynup = yp’n’u’ – y’n’up’ y = y’ = 0 and n = n’ = 1 So ypu = yp’u’ yp’ represents half of screen diameter = -9.5mm u’ represents the angle needed to produce an Airy Disk diameter of 15 µm. u’ = -2.48 º ypu = (-9.5)(-2.48) = Optical Invariant

ypu = (-9.5)(-2.48) = Optical Invariant yp represents maximum fiber displacement u represents axial ray angle from fiber tip An unmodified fiber may have a Numerical Aperture (N.A.) of 0.11, where N.A. = sin u If N.A. = 0.11, then u = 6.32°, and yp = 3.73mm If yp = 1, then u = 23.56°, and N.A. = 0.40

The End