Tutorial: Calculation of Image Rotation for a Scanning Optical System

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Tutorial Calculation of Image Rotation for a
Scanning Optical System

• Sergio Guevara
• OPTI 521 Distance Learning
• College of Optical Science
• University of Arizona

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Brief Overview of Scanning Optical Systems

• Uses
• Image Scanning
• IR Imaging Systems
• Laser Scanning
• Laser Printers
• Laser Projectors
• Microvisions SHOW Projector
• Artifacts
• Image Distortion
• Image Rotation

Figure from US Patent 4,106,845
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Steps to Analyze Image Rotation

• Sketch the system including
• Mirror locations
• Mirrors axis of rotation.
• Input beam.
• Image plane.
• Choose a global coordinate system.
• Define all objects in the global coordinate
  system.
• Define mirror matrices for mirrors in global
  coordinate system.
• Define axis of rotation for mirrors in global
  coordinate system.
• Find image for non-rotated case.
• Add rotation to mirrors.
• Find image in rotated case.
• Project rotated and non-rotated images onto image
  plane.
• Compare the projected images of the rotated case
  with the non-rotated case to determine rotation
  of image.

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Sketch the System
Top View
Top View
Figure from US Patent 4,106,845
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Sketch the System
Front View
Right View
Figure from US Patent 4,106,845
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Choose Global Coordinate System
Top View
Right View
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Define Input Beam Image Planein Global
Coordinates

• Input Beam is Parallel to Z-axis.
• Image Plane lies on the X-Z plane.

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Define Mirrors in Global Coordinates

• Find Mirrors Normal Unit Vectors
• Mirror 1 normal unit vector
• Mirror 2 normal unit vector
• Create Mirror Matrices
• Mirror 1 Matrix
• Mirror 2 Matrix

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Define Rotation of Mirrorsin Global Coordinates

• Mirror Rotation
• Rotation Matrices
• Mirror 1 Rotation Matrix(where a is the angle
  of rotation)
• Mirror 2 Rotation Matrix (where ß is the angle
  of rotation)

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System Model

• Input thru Mirror
• For no image rotation
• Projection onto Image Plane

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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

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Calculations
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Questions?
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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 dij
  is the Kronecker delta, and eijk is the
  permutation symbol.

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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
• Euler Parameters
• http//mathworld.wolfram.com/EulerParameters.html
• Line onto Plane Projection
• http//www.euclideanspace.com/maths/geometry/eleme
  nts/plane/lineOnPlane/index.htm
• http//www.euclideanspace.com/maths/geometry/eleme
  nts/line/projections/index.htm.