Advertisement

Mathematical Description of Abbe’s Theory of Image Formation in the Microscope Based on Diffraction

  • Barry R. MastersEmail author
Chapter
  • 54 Downloads
Part of the Springer Series in Optical Sciences book series (SSOS, volume 227)

Abstract

In Abbe’s 1873 seminal publication he promised another paper that would contain his mathematical analysis of image formation in the light microscope, a paper that never appeared because he died. After Abbe’s death his students and colleagues developed and published details of his theory of image formation in terms of Fourier optics (see Appendix A).

References

  1. Abbe, E. (1873a). Beiträge zur Theorie des Mikroskops und der mikroskopischen Wahrnehmung. Archiv für mikroskopische Anatomie, IX, 413–468.Google Scholar
  2. Abbe, E. (1873b). Über einen neuen Beleuchtungsapparat am Mikroskop. Archiv für mikroskopische Anatomie, IX, 469–480.Google Scholar
  3. Duffieux, P. M. (1946). L’intégrale de Fourier et ses applications à l’optique. Paris: Masson, Editeur. Republished in English as: Duffieux, P. M. (1970). The Fourier Transform and its applications to optics, second edition. New York: John Wiley and Sons.Google Scholar
  4. Goodman, J. W. (2017). Introduction to Fourier Optics. Fourth Edition. New York: W. H. Freeman and Company.Google Scholar
  5. Gross, H. (2005a). Wave Optics, in Handbook of Optical Systems: Fundamentals of Technical Optics, Volume 1, Chapter 12. Weinheim, FRG: Wiley-VCH Verlag GmbH & Co. KGaA.Google Scholar
  6. Gross, H. (2005b). Handbook of Optical Systems. Volume 6, Weinheim, FRG: Wiley-VCHGoogle Scholar
  7. Hopkins, H. H. (1951). The concept of partial coherence in optics. Proceedings of the Royal Society of London Series A, 208, 263–277.Google Scholar
  8. Hopkins, H. H. (1953). On the diffraction theory of optical images. Proceedings of the Royal Society of London Series A, 217, 408–432.Google Scholar
  9. Kӧhler, A. (1893). Ein neues Beleuchtungsverfahren für mikrophotographische Zwecke. Zeitschrift für wissenschaftliche Mikroskopie und für Mikroskopische Technik, 10, 433–440.Google Scholar
  10. Kӧhler, A. (1894). New method of illumination for photomicrographical purposes. Journal of the Royal Microscopical Society, 14, 261–262.Google Scholar
  11. Porter, A. B. (1906). On the diffraction theory of microscope vision. Philosophical Magazine, 6, 154–166.Google Scholar
  12. Reynolds, G. O., DeVelis, J. B., Parrent Jr., G. B., and Thompson, B. J. (1989). The New Physical Optics Notebook: Tutorials in Fourier Optics. Bellingham: SPIE Optical Engineering Press.Google Scholar
  13. Singer, W., Totzeck, M., and Gross, H. (2005). The Abbe Theory of Imaging, Handbook of Optical Systems: Physical Image Formation Volume 2, Chapter 21. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA.Google Scholar
  14. Van Cittert, P. H. (1934). Die Wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene. [The Probable vibrational distribution in one of the one light source directly or via a lens Illuminated plane]. Physica, 1, 201–210.Google Scholar
  15. Zernike, F. (1938). The concepts of degree of coherence and its application to optical problems. Physica, 5, 785–795.Google Scholar

Further Reading

  1. Adams, C. S., and Hughes, I. G. (2019). Optics f2f. Oxford: Oxford University Press.Google Scholar
  2. Abbe, E. (1874). A contribution to the theory of the microscope and the nature of microscopic vision. Translated into English by H. E. Fripp. Proceedings of the Bristol Naturalists Society, I, 202–258. Read before the Bristol Microscopical Society, December 16, 1974.Google Scholar
  3. Abbe, E. (1882a). The relation of aperture and power in the microscope. Journal of the Royal Microscopical Society, Section II, 300–309. [read before the Society on May 10, 1882], [Abbe wrote the paper in English].Google Scholar
  4. Abbe, E. (1882b). The relation of aperture and power in the microscope condenser. Journal of the Royal Microscopical Society, Section II, 460–473. [read before the Society on June 14, 1882], [Abbe wrote the paper in English].Google Scholar
  5. Abbe, E. (1889). On the effect of illumination by means of wide-angled cones of light. Journal of the Royal Microscopical Society, Series II, IX, 721–724.Google Scholar
  6. Abbe, E. (1989). Gesammelte Abhandlungen, I–IV. Hildesheim: Georg Olms Verlag. [Originally published in 1904, Jena: Verlag von Gustav Fischer].Google Scholar
  7. Abbe, E. (1883). The relation of aperture and power in the microscope condenser. Journal of the Royal Microscopical Society, Section II, 790–812. [read before the Society on June 14, 1882], [Abbe wrote the paper in English].Google Scholar
  8. Berek, M. (1929). XXI. On the extent to which real image formation can be obtained in the microscope. Journal of the Royal Microscopical Society, 49, 240–249.Google Scholar
  9. Born, M., and Wolf, E. (1999). Principles of Optics, 7th (expanded) edition. Cambridge: Cambridge University Press.Google Scholar
  10. Bratt, J. and Török, P. (2019). Imaging Optics. Cambridge: Cambridge University Press.Google Scholar
  11. Feffer, S. M. (1994). Microscopes to munitions: Ernst Abbe, Carl Zeiss, and the transformation of technical optics, 1850–1914. PhD Dissertation, University of California, Berkeley, 1994. Ann Arbor: UMI Dissertation Services.Google Scholar
  12. Gu, M. (2000). Advanced Optical Imaging Theory. Berlin: Springer.Google Scholar
  13. Kӧhler, H. (1981). On Abbe’s theory of image formation in the microscope. Journal of Modern Optics, 28, 1691–1701.Google Scholar
  14. Linfoot, E. H. (1964). Fourier Methods in Optical Image Evaluation. London: Focal Press, Ltd.Google Scholar
  15. Lummer, O., and Reiche, F. (1910). Die Lehre von der Bildentstehung im Mikroskop von Ernst Abbe. Braunschweig: Druck und Verlag von Friedrich Vieweg und Sohn.Google Scholar
  16. Mansuripur, M. (2009). Classical Optics and its Applications. Second Edition. Cambridge: Cambridge University Press.Google Scholar
  17. Martin, L. C. (1966). The Theory of the Microscope. New York: American Elsevier Publishing Company, Inc. and London: Blackie.Google Scholar
  18. Mertz, J. (2019). Introduction to Optical Microscopy, second edition. New York: Cambridge University Press.Google Scholar
  19. Michel, K. (1964). Die Grundzüge der Theorie des Mikroskops in elementarer Darstellung. 2. neubearbeitete Auflage. Stuttgart: Wissenschaftliche Verlagsgesellschaft M.B. H.Google Scholar
  20. Williams, C. S., and Beckland, O. A. (1989). Introduction to the Optical Transfer Function. New York: John Wiley & Sons.Google Scholar
  21. Volkmann, H. (1966). Ernst Abbe and His Work. Applied Optics, 5, 1720–1731.Google Scholar
  22. von Rohr, M. (1940). Ernst Abbe. Jena: Gustav Fischer.Google Scholar
  23. Zernike, F. (1934a). Beugungstheorie des Schneidenverfahrens und Seiner Verbesserten Form, der Phasenkontrastmethode. [Diffraction theory of the cutting process and its improved form of the phase contrast method] Physica, 1, 689–704.Google Scholar
  24. Zernike, F. (1934b). Diffraction theory of the knife-edge test and its improved form, the phase-contrast method. Monthly Notices of the Royal Astronomical Society, 94, 377–384.Google Scholar
  25. Zernike, F. (1936). Deutsches Reichspatent No. 636168 (September 1936).Google Scholar
  26. Zernike, F. (1942a). Phase contrast, a new method for the microscopic observation of transparent objects, Part 1. Physica, 9, 686–698.Google Scholar
  27. Zernike, F. (1942b). Phase contrast, a new method for the microscopic observation of transparent objects, Part 2. Physica, 9, 974–986.Google Scholar
  28. Zernike, F. (1948). Diffraction and optical image formation. Proceedings of the Physical Society, 61, 158–164.Google Scholar
  29. Zernike, F. (1950). Color phase-contrast microscopy: requirements and applications. Physica, 9, 974–986.Google Scholar
  30. Zernike, F. (1958). The wave theory of microscopic image formation. Appendix K. In: Concepts of Classical Optics, J. Strong, pp. 525–536. San Francisco: W. H. Freeman.Google Scholar
  31. Zernike, F., and Brinkman, H. C. (1935). Hypersphärische Funktionen und die in sphärischen Bereichen orthogonalen Polynome. Verh. Akad. Wet. Amst., (Proceedings Royal Academy Amsterdam), 38, 161–170.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Previously, Visiting Scientist Department of Biological EngineeringMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Previously, Visiting Scholar Department of the History of ScienceHarvard UniversityCambridgeUSA

Personalised recommendations