On the Formalization of Cardinal Points of Optical Systems

  • Umair SiddiqueEmail author
  • Sofiène Tahar
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 346)


Optical systems are widely used in safety critical applications such as aerospace, telecommunication and biomedical systems. The verification of such systems is usually performed by informal techniques (e.g., numerical simulation and paper-and-pencil based proofs) which may result in erroneous designs. Interactive theorem proving has the potential to verify complex optical designs with better accuracy and soundness. However, existing formalizations of optics theories do not provide the facility to analyze optical imaging properties which are used to characterize the behavior of objects under observation (e.g., cancer cells, human eye or commercial camera lenses). In this paper, we present the formalization of cardinal points which are the most fundamental requirement to model imaging properties. We also present the formal verification of the cardinal points for an arbitrary optical system consisting of any number of optical components. In order to demonstrate the usefulness of our formalization, we present the formal analysis of an optical instrument used to compensate the ametropia of an eye.


Theorem Proving HOL Light Optical Systems Cardinal Points 


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  1. 1.
    The Hubble Space Telescope Optical System Failure Report. Technical report, NASA (1990)Google Scholar
  2. 2.
    Care of the Patient with Myopia: Optometric Clinical Practice Guideline. American Optometric Association (2010)Google Scholar
  3. 3.
  4. 4.
    Afshar, S.K., Siddique, U., Mahmoud, M.Y., Aravantinos, V., Seddiki, O., Hasan, O., Tahar, S.: Formal Analysis of Optical Systems. Mathematics in Computer Science 8(1), 39–70 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Baier, C., Katoen, J.P.: Principles of Model Checking. MIT Press (2008)Google Scholar
  6. 6.
    Bass, M., DeCusatis, C., Enoch, J., Lakshminarayanan, V., Li, G., MacDonald, C., Mahajan, V., Van Stryland, E.: Handbook of Optics: Geometrical and Physical Optics, Polarized Light, Components and Instruments. McGraw-Hill Education (2009)Google Scholar
  7. 7.
    Cheng, Q., Cui, T.J., Zhang, C.: Waves in Planar Waveguide Containing Chiral Nihility Metamaterial. Optics and Communication 274, 317–321 (2007)CrossRefGoogle Scholar
  8. 8.
    Griffiths, D.J.: Introduction to Quantum Mechanics. Pearson Prentice Hall (2005)Google Scholar
  9. 9.
    Hales, T.C.: Introduction to the Flyspeck Project. In: Mathematics, Algorithms, Proofs, Dagstuhl Seminar Proceedings, vol. 05021 (2005)Google Scholar
  10. 10.
    Harris, W.F.: Pascal’s ring, cardinal points, and refractive compensation. Vision Research 51(14), 1679 (2011)CrossRefGoogle Scholar
  11. 11.
    Harrison, J.: HOL Light: A Tutorial Introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)Google Scholar
  12. 12.
    Harrison, J.: Handbook of Practical Logic and Automated Reasoning. Cambridge University Press (2009)Google Scholar
  13. 13.
    Harrison, J.: The HOL Light Theory of Euclidean Space. Journal of Automated Reasoning 50(2), 173–190 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Hasan, O., Khan Afshar, S., Tahar, S.: Formal Analysis of Optical Waveguides in HOL. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 228–243. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  15. 15.
    Hodgson, N., Weber, H.: Optical Resonators: Fundamentals, Advanced Concepts, Applications. Springer Series in Optical Sciences. Springer (2005)Google Scholar
  16. 16.
    Juhasz, T., Djotyan, G., Loesel, F.H., Kurtz, R.M., Horvath, C., Bille, J.F., Mourou, G.: Applications of Femtosecond Lasers in Corneal Surgery. Laser Physics 10(2), 495–500 (2011)Google Scholar
  17. 17.
    Yousri Mahmoud, M., Tahar, S.: On the Quantum Formalization of Coherent Light in HOL. In: Badger, J.M., Rozier, K.Y. (eds.) NFM 2014. LNCS, vol. 8430, pp. 128–142. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  18. 18.
    Naqvi, A.: Comments on Waves in Planar Waveguide Containing Chiral Nihility Metamaterial. Optics and Communication 284, 215–216 (2011)CrossRefGoogle Scholar
  19. 19.
    Radiant-Zemax (2014),
  20. 20.
    Saleh, B.E.A., Teich, M.C.: Fundamentals of Photonics. Wiley (2007)Google Scholar
  21. 21.
    Siddique, U., Aravantinos, V., Tahar, S.: A New Approach for the Verification of Optical Systems. In: Optical System Alignment, Tolerancing, and Verification VII. SPIE, vol. 8844, p. 88440G–88440G–14 (2013)Google Scholar
  22. 22.
    Siddique, U., Aravantinos, V., Tahar, S.: Formal Stability Analysis of Optical Resonators. In: Brat, G., Rungta, N., Venet, A. (eds.) NFM 2013. LNCS, vol. 7871, pp. 368–382. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  23. 23.
    Siddique, U., Aravantinos, V., Tahar, S.: On the Formal Analysis of Geometrical Optics in HOL. In: Ida, T., Fleuriot, J. (eds.) ADG 2012. LNCS, vol. 7993, pp. 161–180. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  24. 24.
    Siddique, U., Tahar, S.: Formal Reasoning about Geometrical Optics, Hardware Verification Group, Concordia University, QC, Canada (2014),
  25. 25.
    Siddique, U., Tahar, S.: Towards Ray Optics Formalization of Optical Imaging Systems. In: Conference on Information Reuse and Integration, pp. 378–385. IEEE (2014)Google Scholar
  26. 26.
    Träger, F.: Handbook of Lasers and Optics. Springer (2007)Google Scholar
  27. 27.
    Woodcock, J., Larsen, P.G., Bicarregui, J., Fitzgerald, J.: Formal Methods: Practice and Experience. ACM Computing Survey 41(4), 1–19 (2009)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringConcordia UniversityMontrealCanada

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