1. Tutorial: Calculation of Image Rotation for a Scanning Optical System Sergio Guevara OPTI 521 – Distance Learning College of Optical Science University of Arizona

2. Brief Overview of Scanning Optical Systems  Uses: Image Scanning  IR Imaging Systems Laser Scanning  Laser Printers  Laser Projectors Microvision’s SHOW Projector  Artifacts Image Distortion Image Rotation Figure from US Patent #4,106,845

3. Steps to Analyze Image Rotation 1. Sketch the system including: a) Mirror locations b) Mirrors’ axis of rotation. c) Input beam. d) Image plane. 2. Choose a global coordinate system. 3. Define all objects in the global coordinate system. a) Define mirror matrices for mirrors in global coordinate system. b) Define axis of rotation for mirrors in global coordinate system. 4. Find image for non-rotated case. 5. Add rotation to mirrors. 6. Find image in rotated case. 7. Project rotated and non-rotated images onto image plane. 8. Compare the projected images of the rotated case with the non-rotated case to determine rotation of image.

4. Sketch the System Figure from US Patent #4,106,845 Top View

5. Sketch the System Figure from US Patent #4,106,845 Front ViewRight View

6. Choose Global Coordinate System Top ViewRight View

7. Define Input Beam & Image Plane in Global Coordinates  Input Beam is Parallel to Z-axis.  Image Plane lies on the X-Z plane.

8. Define Mirrors in Global Coordinates  Find Mirror’s Normal Unit Vectors Mirror 1 normal unit vector: Mirror 2 normal unit vector:  Create Mirror Matrices Mirror 1 Matrix: Mirror 2 Matrix:

9. Define Rotation of Mirrors in Global Coordinates  Mirror Rotation  Rotation Matrices Mirror 1 Rotation Matrix: (where α is the angle of rotation) Mirror 2 Rotation Matrix: (where β is the angle of rotation)

10. System Model  Input thru Mirror For no image rotation  Projection onto Image Plane

11. Calculation of Rotation  Is a comparison of the initial image with no mirror rotation to an image with mirror rotation.  Use the definition of the cross product:  Rotation Equation:

12. Calculations

13. Questions?

14. Rotation Matrices via Euler Parameters  Euler Parameters where the axis of rotation is a unit vector,, and the angle of rotation about that axis is,.  Rotation Matrix in Einstein Notation where δ ij is the Kronecker delta, and ε ijk is the permutation symbol.

15. Further Resources  How to perform mirror rotations in Zemax http://www.zemax.com/kb/articles/25/1/How-To-Model- a-Scanning-Mirror/Page1.html http://www.zemax.com/kb/articles/25/1/How-To-Model- a-Scanning-Mirror/Page1.html  Euler Parameters http://mathworld.wolfram.com/EulerParameters.html  Line onto Plane Projection http://www.euclideanspace.com/maths/geometry/elements /plane/lineOnPlane/index.htm http://www.euclideanspace.com/maths/geometry/elements /plane/lineOnPlane/index.htm http://www.euclideanspace.com/maths/geometry/elements /line/projections/index.htm. http://www.euclideanspace.com/maths/geometry/elements /line/projections/index.htm