Table of Contents

Preface

Acronyms and Abbreviations

1 Introduction

1.1 Brief History

1.2 The Laws of Reflection and Refraction

     1.2.1 Law of reflection

     1.2.2 Law of refraction

     1.2.3 Intersection of a ray and a plane mirror

1.3 Scan Field and Scan Patterns

     1.3.1 Modeling mirror-scanning devices

     1.3.2 Elementary concepts of scan field distribution

     1.3.3 Optical distortions in scan patterns

     1.3.4 Resolution of laser scanners

Bibliography and Links

Part I: Mirror Scanning Devices with One Axis of Mirror Rotation

2 One-Mirror and One-Axis Scanning Devices

2.1 Single-Facet Model Scanner

     2.1.1 Scanning geometry and the acceptance angle of the ray of incidence

     2.1.2 Single-origin scanning

     2.1.3 Line scan and its application to produce multi-point star scan patterns

     2.1.4 Expansion and rotation of the elliptical scanning spot during scanning

     2.1.5 Effect of input offset

2.2 The Galvanometer-Based Scanner and the Conic-Section Scan Patterns

     2.2.1 Galvanometric scanner and conic-section scan patterns

     2.2.2 Paddle scanner

     2.2.3 The golf-club scanner

     2.2.4 Scan patterns on curved surfaces

2.3 Shaft Encoders and Digital Galvanometric Scanner

     2.3.1 Optical rotary encorders and mirror orientation control through real-time closed loop system

     2.3.2 Incremental and absolute encoders

     2.3.3 Digital Galvanometric scanners

     2.3.4 Optical, magnetic encoders and resolver

Bibliography and Links

3 Scan Field of Rotating Reflective Polygons

3.1 Ray-Tracing Equations for Regular Polygon Scanners

3.2 Structural Analysis of the Scan Field Produced by Regular Polygon Scanners

     3.2.1 Locus of the point of reflection

     3.2.2 Structure of scan field and degree of field distribution asymmetry

     3.2.3 Scanning geometry for symmetric scan field distributions

     3.2.4 Effect of vignetting and the scan duty cycle

     3.2.5 Utility rate of a polygon facet

     3.2.6 Scan pattern on plane surface and the f-theta scanning lens

3.3 Locus of the Scan Center and Depth of the Scan Field

     3.3.1 Graphic and analytic approaches for the locus of the scan center

     3.3.2 Displacement of scan center and depth of scan field

     3.3.3 Scanning with a convergent beam and the locus of its focal point

     3.3.4 The inverted prismatic polygon

3.4 Pyramidal Polygon Scanners

     3.4.1 Regular pyramidal polygon scanners

     3.4.2 Inverted pyramidal polygon scanners

Bibliography and Links

4 Differential Geometry of the Ruled Surfaces Optically Produced by Mirror Scanning Devices

4.1 Ruled Surfaces Produced by Single-Mirror Scanners

     4.1.1 Differential geometry of surfaces and the advantages of differential approach

     4.1.2 Fundamentals of the theory of ruled surfaces optically produced by mirror scanning devices

4.2 Ruled Surfaces Produced by Optical Scanning Systems with a Finite Number of Free Parameters in Their Scanning Geometries Specifications

     4.2.1 Ruled fourth-order surface optically produced by a rotating prismatic polygon scanner with two free parameters in scanning geometry specification

     4.2.2 Ruled surface produced by a Galvanometric scanner

     4.2.3 Ruled sixth-order surfaces produced by a rotating pyramidal polygon with three free parameters for scanning geometry specification

     4.2.4 Ruled fourth- and second-order surfaces optically produced by the pyramidal polygon scanners with one to two free parameters for scanning geometry specification

4.3 Main Features and Classification of the Ruled Surfaces Optically Produced by One-Mirror and One-Axis Scanning Systems

4.4 Optically Creation of Helicoids, Conoids and Hyperbolic Paraboloids by Scanning an Infinitely Long Line

     4.4.1 Optical creation of a scanning infinitely long line

     4.4.2 Optical creation of helicoids, conoids, and hyperbolic paraboloids

Bibliography and Links

Part II: Mirror Scanning Devices with Two Axes of Mirror Rotation

5 Two-Mirror and Two-Axis Scanning Systems of Different Configurations

5.1 Modeling the XY Scanning Systems

     5.1.1 Scanning geometries of the XY scanning systems

     5.1.2 Model scanner for the XY scanning systems in different configurations

5.2 Scan Patterns Produced by XY Scanning Systems

     5.2.1 XY scan patterns produced by systems in different configurations

     5.2.2 Raster scan patterns in the near- and far-regions of the scan fields

     5.2.3 Translation and rotation of the XY scan patterns

5.3 Optical Distortions in the XY Scan Patterns

     5.3.1 Pincushion distortion in the XY scan patterns

     5.3.2 Kinematics and distortions of scanning spots in raster scan patterns

     5.3.3 Expansion and rotation of elliptical and rectangular scanning spots in screens of different formats

5.4 Dynamic Focusing, z-Axis Compensation, and Three-Axis Scanning

5.5 Software Correction of Distortions in Digital Images Produced by Two-Mirror and Two-Axis Scanners

     5.5.1 Software correction of scanning errors in digital images

     5.5.2 Analytic method for inverse mapping of the digital images produced by Galvanometric XY scanners in different configurations

     5.5.3 Numerical method for inverse mapping of the digital image produced by Galvanometric XY scanners in dual-Galvo configuration

     5.5.4 Rectification of defocus error in the images produced by Galvanometric scanners

     5.5.5 Rectification of mirror mount offset error

Bibliography and Links

6 Gimbaled Mirror for Two-Dimensional Beam-Steering

6.1 High-Order Conic-Section Scan Patterns Produced by Turning a Gimbaled Mirror

     6.1.1 Ray deflection at a single-mirror turntable about a fixed point

     6.1.2 High-order conic scan fields produced by turning a gimbaled mirror

6.2 Scan Patterns Synthesis

     6.2.1 Preliminaries of scan patterns synthesis

     6.2.2 A straight line produced by turning a gimbaled mirror

     6.2.3 A circular scan pattern produced by turning a gimbaled mirror

     6.2.4 Raster scan pattern produced by gimbaled mirrors

6.3 Distortions in the Images Produced by Gimbaled Mirrors

     6.3.1 Effect of input offset

     6.3.2 Scan line broadening

     6.3.3 Square and non-square pixel distortions in the images on screens of different formats

6.4 Correspondence between mirror positions and points in a scan pattern

6.5 Methods for Scan Data Processing

     6.5.1 Graphic method for scan data processing

     6.5.2 Scan pattern scaling

     6.5.3 Projection images translation and rotation

     6.5.4 Images displayed on a tilted screen

6.6 Comparison of Gimbaled Mirror Beam-Sterling System and Galvanometric XY Scanner

     6.6.1 Dependence of the image size on the amplitude of mirror motion

     6.6.2 Comparison of optical distortions at the pixel level on screens of different formats

6.7 Gimbaled Mirror and MEMS Micro-Scanners

     6.7.1 Heliostat and a single mirror with two axes of rotaion

     6.7.2 MEMS micro-scanners

Bibliography and Links

Part III: Risley-Prism-Based Beam-Steering Systems

7 Exact and Approximate Solutions for Risley-Prism-Based Beam-Steering Systems in Different Configurations

7.1 Historical Introduction

7.2 Exact and Approximate Expressions for Risley-Prism-Based Beam-Steering Systems

     7.2.1 Combinations of prisms for beam steering

     7.2.2 Analytic ray-tracing of a ray though a pair of thick prisms

     7.2.3 Scan field distribution over the interior of the Risley prism scanning system

     7.2.4 Approximate expressions of scan patterns

7.3 Ray Deviation Power of Risley Prism

     7.3.1 Standard analytic expressions of ray-tracing results for Risley prism pairs in different configurations

     7.3.2 Ray deviation angle of Risley prism pair and total-internal-reflection-induced blind zone in the scan field

     7.3.3 Influence of total internal reflection on the power of ray deviation by Risley prism

Bibliography and Links

8 Forward and Inverse Solutions for Two-Element Risley-Prism-Based Beam-Steering Systems in Different Configurations

8.1 First-Order Graphical and Analytical Solutions for Target Tracking

     8.1.1 Vector-based graphics of the first-order inverse solution

     8.1.2 First-order approximate solutions to the inverse problem

8.2 Third-Order Approximation to the Inverse Problem

     8.2.1 Beam steering angle of a Risley prism pointer

     8.2.2 Two-step method for the inverse solution of Risley prism pointer

8.3 Closed-Form Analytic Inverse Solution of a Two-Element Risley Prism Pointers in different configurations

     8.3.1 Closed-form noniterative inverse solutions of two-element Risley prism pointers in different configurations

     8.3.2 Comparison of the predictions by theories with different degrees of accuracy

     8.3.3 Pointing stability and agility

     8.3.4 Symmetric and asymmetric Risley prism configurations and their influence on precise pointing

8.4 Generalization of the Inverse Solution from Precise Target Pointing to Highlight a Specific Pattern

     8.4.1 Control law of Risley prism pairs for steering a laser beam to highlight a specific pattern

     8.4.2 Straight line segment, circular and elliptical scan patterns produced by the Risley-prism-based beam-steering systems

8.5 Exact Analytic Solutions of Two-Element Risley Prism Pointers

     8.5.1 Locus of the point where the ray exits the pointer and the validity of the solution presented in section 8.3.1

     8.5.2 Exact solutions of two-element Risley prism pointers for tracking targets of any size

Bibliography and Links

9 Inverse Solutions for Three-Element Risley-Prism-Based Beam-Steering Systems in Different Configurations

9.1 Three-Element Risley Prism Pointer for Moving Targets Tracking

     9.1.1 Moving target indication and tracking

     9.1.2 Singularity functions of Risley-prism-based tracking systems

     9.1.3 Vector-based graphics of the first-order inverse solution for three-element Risley prism optical beam pointers

9.2 Closed Form Analytic Inverse Solutions to Three-Element Risley Prism Pointers

     9.2.1 Reduction theory of three-element Risley prism pointer based on a combination of the first two co-rotational prisms into a single equivalent prism

     9.2.2 Iterative solutions to three-element Risley prism pointers

9.3 Design Configuration and Software of Three-Element Risley Prism Laser Beam Pointer

     9.3.1 Opto-mechanical design

     9.3.2 Control law and software

Bibliography and Links

10. Error Sources and Their Influence on the Performance of Risley-Prism-Based Beam Steering Systems

10.1 First-Order Theory of Scan Pattern Distortions Produced by the Two Prisms with Slightly Different Characterization Parameters

     10.1.1 Rose-like scan patterns produced by prisms with slightly different rates of rotation

     10.1.2 Line segment scan patterns produced by prisms with slightly different in their powers of ray deviation

10.2 Effect of Prism Assembly Errors on Beam Pointing Accuracy of Risley Prism Pointer

     10.2.1 Non-parallelism in the prism pair and its influence on pointing accuracy

     10.2.2 Bearing rotational axis misalignment and its influence on the accuracy of beam pointing

Bibliography and Links

APPENDICES

A. Vector Algebra Preliminaries

B. Scanning Spot Distortions along a Curved Path in Conic-Section Scan Pattern Produced by Galvanometric Scanners

C. Scanner-Lens Configurations: Objective, Post-Objective, Pre-Objective Scanning and the Flat Field Scanning Lens

D. Derivation of the Expression for the Depth of Scan Field Produced by a Prismatic Polygon Scanner

E. Gaussian First and Second Differential Forms of the Ruled Surfaces Generated by Single-Mirror Scanning Devices of Different Configurations

F. Ruled Surfaces Optically Generated by Gimbaled Mirrors

G. One-Step Method for the Inverse Solution of Risley Prism Pair

H. Blind Zone and Control Singularities for a Risley Prism Pair

I. Direction Cosines of the Ray Emergent Form a Three-Element Risley Prism Pointer

J. Analytical Model for N-Element Risley Prism Pointers

Index

Preface

Scanning technology for optical and laser systems is used in the controlled deflection of optical and laser beam for information transfer, such as actively or passively detecting events in a given direction (e.g., lidar), detecting information from a given surface (e.g., bar-code scanning) or inducing a physical effect (e.g., photoconductivity or photomagnetism) on the surface during a flying spot scan. All of these scanning systems have been applied to a number of useful products that have a direct bearing on our life.

The scope of this book is restricted to mirror scanning systems and rotating wedge prism scanning systems. Here, "mirror" means a plane reflecting surface, and "wedge" means a thin prism with a right triangle cross-section. Over nearly 30 years, I have been interested in the mathematical analysis of scanning devices for optical and laser systems to yield results with higher accuracy than those obtained from geometrical construction of imaging of an object by a movable mirror or prism, which was a method to yield design data for scanning devices engineering when mathematical model is difficulty to obtain from open publications, in which the exact form of important equations may be considered as proprietary.

My analytical results have been summarized according to the format of original research papers, most of which were written solely by me and published in archival journals. After their publication, I usually received emails from readers with questions, friendly comments and reprints requests. This situation was different from my experience acquired after publishing theoretical papers on diffraction and coherence of light in optical physics, although many of them have high citation numbers over a longer period of time. Reader reactions to my research encouraged me to prioritize this book as a summary of the results I obtained in the past.

I appreciated the opportunity that SPIE gave me to collect my published and unpublished manuscripts together, along with detailed commentary and corrections in a book so that readers need not search through old journals. Readers of this book are assumed to have a foundation in vector operation and calculus, and a reasonable knowledge of elementary optics and laser. Detailed proofs that require long derivations are sometimes omitted but can be found in either the appendices or the cited references.

This book is divided into three parts, starting with an introductory chapter for the laws of reflection and refraction and the mathematical preliminaries of analytical raytracing (most expressions are in vector forms). Chapters 2 through 4 are the first part, which covers topics of mirror scanning devices with one axis of rotation for conic-sections scanning. Chapters 5 through 6 are the second part, which covers topics of mirror scanning devices with two axes of rotation, e.g., gimbaled mirror and Galvanometric scanners in cascade for two-dimensional scanning. Chapters 7 through 10 are in the third part to address Risley-prism-based beam-steering systems (i.e., rotating wedge prism scanning systems) with two to three elements for moving target searching and tracking.

Since each chapter in this book is an enlarged version of my technical paper focused on a specific topic as noticed by a single line about the main reference material of that topic at the beginning of each chapter, which may be helpful to readers who are interested in a specific scanning device and want to know its analytic model and computed results.

Writing a book is time-consuming and laborious; I doubt that many books have been written without some substantial help and encouragement by others. This book is no exception; therefore, I wish to express my appreciation to my friends at Symbol Technologies, Inc., especially Dr. Jerome Swartz and Dr. Satya Sharma.

I am obliged to SPIE Press, specifically, Dr. Eugene Arthurs, who initiated this project, and Mr. Scott McNeill, who helped bring it to fruition. I am also obliged to the Optical Society for permission to reproduce drawings from my papers published in the Journal of Optical Society of America A and Applied Optics. I thank my sister, Professor Xiaoyu Li; my niece, Lily Ji; and my friend, Earl O'Neil - without their encouragement and support, this book would not have been possible.

Yajun Li
October 2021