Higher-Order Logic Formalization of Geometrical Optics · Umair Siddique Higher-Order Logic...
71
Umair Siddique Higher-Order Logic Formalization of Geometrical Optics Umair Siddique Vincent Aravantinos Sofiene Tahar Hardware Verification Group (HVG) Department of Electrical and Computer Engineering Concordia University, Canada Automated Deduction in Geometry – 2012 Edinburgh, UK September 17-19, 2012 ADG-2012 9/21/2012 1
Transcript of Higher-Order Logic Formalization of Geometrical Optics · Umair Siddique Higher-Order Logic...
Umair Siddique
Higher-Order Logic Formalization of Geometrical Optics
Umair Siddique Vincent Aravantinos Sofiene Tahar
Hardware Verification Group (HVG) Department of Electrical and Computer Engineering
Concordia University, Canada
Automated Deduction in Geometry – 2012 Edinburgh, UK
September 17-19, 2012
ADG-2012 9/21/2012 1
Umair Siddique
Motivation
Geometrical optics
Higher-order logic formalization
Application: stability analysis of optical resonator
Conclusions & future work
Outline
9/21/2012 2
Umair Siddique
Motivation
9/21/2012 3
Umair Siddique
Motivation
Real optical system complicated!!
9/21/2012 4
Video camera by Olympus
Umair Siddique
Motivation
Geometrical optics
Higher-order logic formalization
Application: stability analysis of optical resonator
Conclusions & future work
Outline
9/21/2012 5
Umair Siddique
Hierarchy of optical approaches
9/21/2012 6
Umair Siddique
Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
9/21/2012 7
Umair Siddique
Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
9/21/2012 8
Umair Siddique
Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
Wave optics: single scalar wave
9/21/2012 9
Umair Siddique
Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
Wave optics: single scalar wave
Geometrical optics: light as ray
9/21/2012 10
Umair Siddique
Light is diffraction-free ray
Optical medium (refractive index)
Paraxial approximation (small angles)
Fermat's principle of least time
Postulates of geometrical optics
9/21/2012 11
Umair Siddique
Sequence of optical components
What is an optical system?
C1 C2 C3 Cn
9/21/2012 12
Umair Siddique
Sequence of optical components
What is an optical system?
C1 C2 C3 Cn
9/21/2012 13
Umair Siddique
Optical interfaces
9/21/2012 14
Umair Siddique
Plane interface transmitted reflected
Optical interfaces
9/21/2012 15
Umair Siddique
Spherical interface transmitted reflected
Optical interfaces (contd.)
9/21/2012 16
Umair Siddique
Optical component characterization
9/21/2012 17
Umair Siddique
Optical component characterization
9/21/2012 18
Umair Siddique
Optical component characterization
9/21/2012 19
Umair Siddique
Optical system characterization
9/21/2012 20
Umair Siddique
Motivation
Geometrical optics
Higher-order logic formalization
Application: stability analysis of optical resonator
Conclusions & future work
Future work
Outline
9/21/2012 21
Umair Siddique
Formalization is two-fold - optical geometry and physical parameters - physical behavior of a ray
Higher-order logic formalization
9/21/2012 22
Umair Siddique
Formalization is two-fold - optical geometry and physical parameters - physical behavior of a ray
HOL Light theorem prover
Higher-order logic formalization
9/21/2012 23
Umair Siddique
Modeling of free space
Definition: Free Space
9/21/2012 24
Umair Siddique
Modeling of free space
Definition: Free Space
9/21/2012 25
Definition: Valid Free Space
Umair Siddique
Modeling of interfaces
Definition: Optical Interface
9/21/2012 26
Plane Spherical
Umair Siddique
Modeling of interfaces
Definition: Optical Interface
9/21/2012 27
Plane Spherical
Umair Siddique
Modeling of interfaces
Definition: Optical Interface
9/21/2012 28
Plane Spherical
Umair Siddique
Modeling of interfaces
Definition: Optical Interface
9/21/2012 29
Plane Spherical
Umair Siddique
Modeling of interfaces (contd.)
9/21/2012 30
Definition: Valid Free Space and Valid Optical Interface
Umair Siddique
Modeling of interfaces (contd.)
9/21/2012 31
Definition: Valid Free Space and Valid Optical Interface
Umair Siddique
Modeling of optical system
Definition: Optical System
9/21/2012 32
Umair Siddique
Modeling of optical system
Definition: Optical System
9/21/2012 33
Umair Siddique
Modeling of optical system
Definition: Optical System
9/21/2012 34
Umair Siddique
Modeling of optical system
Definition: Optical System
9/21/2012 35
Umair Siddique
Modeling of optical system
Definition: Optical System
9/21/2012 36
Definition: Valid Optical System
Umair Siddique
Modeling of a ray
Definition: Ray
9/21/2012 37
Umair Siddique
Modeling of a ray
Definition: Ray
9/21/2012 38
Umair Siddique
Modeling of a ray
Definition: Ray
9/21/2012 39
(𝑦1, 1)
(𝑦0, 0)
Umair Siddique
Modeling of a Ray
Definition: Ray
(𝑦2, 2)
9/21/2012 40
(𝑦1, 1)
(𝑦0, 0)
Umair Siddique
Behavior of ray
Definition: Behavior of a Ray in Free Space
9/21/2012 41
Umair Siddique
Behavior of ray (contd.)
9/21/2012 42
Definition: Behavior of a Ray at Given Interface
Umair Siddique
Behavior of ray (contd.)
Definition: Behavior of a Ray at Given Interface
9/21/2012 43
Umair Siddique
Behavior of ray (contd.)
Definition: Behavior of a Ray at Given Interface
9/21/2012 44
Umair Siddique
Properties of optical system
9/21/2012 45
Umair Siddique
Properties of optical system
Theorem: Matrix for Free Space
9/21/2012 46
Umair Siddique
Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
9/21/2012 47
Umair Siddique
Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
9/21/2012 48
Umair Siddique
Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
9/21/2012 49
Umair Siddique
Properties of optical system (contd.)
Theorem: Matrix relations for (Spherical Interface)
9/21/2012 50
Umair Siddique
Ray-transfer-matrix for optical system
9/21/2012 51
Umair Siddique
Ray-transfer-matrix for optical system
Theorem: Matrix for Optical System
9/21/2012 52
Umair Siddique
Ray-transfer-matrix for optical system
Theorem: Matrix for Optical System
9/21/2012 53
Umair Siddique
Motivation
Geometrical Optics
Higher-Order Logic Formalization
Application: stability analysis of optical resonator
Conclusions & future work
Outline
9/21/2012 54
Umair Siddique
Optical resonator
Applications
9/21/2012 55
Umair Siddique
Optical resonator
Some practical uses --Lasers --Biological sensing and Imaging --Optical transmission
Applications
9/21/2012 56
Umair Siddique
Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
9/21/2012 57
Umair Siddique
Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
9/21/2012 58
Stable resonator
Umair Siddique
Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
9/21/2012 59
Stable resonator Unstable resonator
Umair Siddique
Formal analysis is two fold:
- modelling using formalized library of components
- conditions for stability
Stability analysis of resonator
9/21/2012 60
Umair Siddique
Analysis of Z-Cavity
9/21/2012 61
Umair Siddique
Analysis of Z-Cavity
9/21/2012 62
Umair Siddique
Analysis of Z-Cavity
System Composition
9/21/2012 63
Umair Siddique
Analysis of Z-Cavity
System Composition
9/21/2012 64
Umair Siddique
Analysis of Z-Cavity
System composition
9/21/2012 65
Umair Siddique
Analysis of Z-Cavity
System composition
9/21/2012 66
Umair Siddique
Analysis of Z-Cavity
System composition
9/21/2012 67
Umair Siddique
Analysis of Z-Cavity
Theorem: Z-Cavity Matrix
9/21/2012 68
Umair Siddique
New application of formal methods
Formalization of geometrical optics
Verification of important properties
Formal analysis of optical resonator
Conclusions
9/21/2012 69
Umair Siddique
Formal verification of Snell’s law
Formalization of paraxial approximation
Extending existing library of components
Misaligned optical systems
Future work
9/21/2012 70
