Post on 30-Jun-2020
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
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Motivation
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Motivation
Real optical system complicated!!
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Video camera by Olympus
Umair Siddique
Motivation
Geometrical optics
Higher-order logic formalization
Application: stability analysis of optical resonator
Conclusions & future work
Outline
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Hierarchy of optical approaches
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Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
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Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
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Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
Wave optics: single scalar wave
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Hierarchy of optical approaches
Quantum optics: quantum electrodynamics
Electromagnetic optics: coupled vector fields
Wave optics: single scalar wave
Geometrical optics: light as ray
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Light is diffraction-free ray
Optical medium (refractive index)
Paraxial approximation (small angles)
Fermat's principle of least time
Postulates of geometrical optics
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Sequence of optical components
What is an optical system?
C1 C2 C3 Cn
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Sequence of optical components
What is an optical system?
C1 C2 C3 Cn
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Optical interfaces
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Plane interface transmitted reflected
Optical interfaces
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Spherical interface transmitted reflected
Optical interfaces (contd.)
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Optical component characterization
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Optical component characterization
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Optical component characterization
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Optical system characterization
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Motivation
Geometrical optics
Higher-order logic formalization
Application: stability analysis of optical resonator
Conclusions & future work
Future work
Outline
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Formalization is two-fold - optical geometry and physical parameters - physical behavior of a ray
Higher-order logic formalization
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Umair Siddique
Formalization is two-fold - optical geometry and physical parameters - physical behavior of a ray
HOL Light theorem prover
Higher-order logic formalization
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Modeling of free space
Definition: Free Space
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Modeling of free space
Definition: Free Space
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Definition: Valid Free Space
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Modeling of interfaces
Definition: Optical Interface
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Plane Spherical
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Modeling of interfaces
Definition: Optical Interface
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Plane Spherical
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Modeling of interfaces
Definition: Optical Interface
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Plane Spherical
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Modeling of interfaces
Definition: Optical Interface
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Plane Spherical
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Modeling of interfaces (contd.)
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Definition: Valid Free Space and Valid Optical Interface
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Modeling of interfaces (contd.)
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Definition: Valid Free Space and Valid Optical Interface
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Modeling of optical system
Definition: Optical System
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Modeling of optical system
Definition: Optical System
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Modeling of optical system
Definition: Optical System
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Modeling of optical system
Definition: Optical System
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Modeling of optical system
Definition: Optical System
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Definition: Valid Optical System
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Modeling of a ray
Definition: Ray
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Modeling of a ray
Definition: Ray
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Modeling of a ray
Definition: Ray
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(𝑦1, 1)
(𝑦0, 0)
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Modeling of a Ray
Definition: Ray
(𝑦2, 2)
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(𝑦1, 1)
(𝑦0, 0)
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Behavior of ray
Definition: Behavior of a Ray in Free Space
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Behavior of ray (contd.)
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Definition: Behavior of a Ray at Given Interface
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Behavior of ray (contd.)
Definition: Behavior of a Ray at Given Interface
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Behavior of ray (contd.)
Definition: Behavior of a Ray at Given Interface
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Properties of optical system
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Properties of optical system
Theorem: Matrix for Free Space
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Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
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Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
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Properties of optical system (contd.)
Theorem: Matrix relations for (Plane)
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Properties of optical system (contd.)
Theorem: Matrix relations for (Spherical Interface)
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Ray-transfer-matrix for optical system
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Ray-transfer-matrix for optical system
Theorem: Matrix for Optical System
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Ray-transfer-matrix for optical system
Theorem: Matrix for Optical System
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Motivation
Geometrical Optics
Higher-Order Logic Formalization
Application: stability analysis of optical resonator
Conclusions & future work
Outline
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Optical resonator
Applications
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Optical resonator
Some practical uses --Lasers --Biological sensing and Imaging --Optical transmission
Applications
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Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
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Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
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Stable resonator
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Stability means after N round trips, beam of light will be confined within the resonator
Stability analysis of resonator
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Stable resonator Unstable resonator
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Formal analysis is two fold:
- modelling using formalized library of components
- conditions for stability
Stability analysis of resonator
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Analysis of Z-Cavity
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Analysis of Z-Cavity
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Analysis of Z-Cavity
System Composition
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Analysis of Z-Cavity
System Composition
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Analysis of Z-Cavity
System composition
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Analysis of Z-Cavity
System composition
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Analysis of Z-Cavity
System composition
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Analysis of Z-Cavity
Theorem: Z-Cavity Matrix
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New application of formal methods
Formalization of geometrical optics
Verification of important properties
Formal analysis of optical resonator
Conclusions
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Formal verification of Snell’s law
Formalization of paraxial approximation
Extending existing library of components
Misaligned optical systems
Future work
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