OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 5 — May. 1, 2000
  • pp: 892–902

Electromagnetic degrees of freedom of an optical system

Rafael Piestun and David A. B. Miller  »View Author Affiliations

JOSA A, Vol. 17, Issue 5, pp. 892-902 (2000)

View Full Text Article

Enhanced HTML    Acrobat PDF (231 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We present a rigorous electromagnetic formalism for defining, evaluating, and optimizing the degrees of freedom of an optical system. The analysis is valid for the delivery of information with electromagnetic waves under arbitrary boundary conditions communicating between domains in three-dimensional space. We show that, although in principle there is an infinity of degrees of freedom, the effective number is finite owing to the presence of noise. This is in agreement with the restricted classical theories that showed this property for specific optical systems and within the scalar and paraxial approximations. We further show that the best transmitting and receiving functions are the solutions of well-defined eigenvalue equations. The present approach is useful for understanding and designing modern optical systems for which the previous approaches are not applicable, as well as for application in inverse and synthesis problems.

© 2000 Optical Society of America

OCIS Codes
(030.4070) Coherence and statistical optics : Modes
(100.3190) Image processing : Inverse problems
(110.6880) Imaging systems : Three-dimensional image acquisition
(180.0180) Microscopy : Microscopy
(260.1960) Physical optics : Diffraction theory
(260.2110) Physical optics : Electromagnetic optics
(350.5730) Other areas of optics : Resolution
(350.7420) Other areas of optics : Waves

Original Manuscript: June 17, 1999
Revised Manuscript: December 14, 1999
Manuscript Accepted: February 9, 2000
Published: May 1, 2000

Rafael Piestun and David A. B. Miller, "Electromagnetic degrees of freedom of an optical system," J. Opt. Soc. Am. A 17, 892-902 (2000)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. G. Toraldo di Francia, “Resolving power and information,” J. Opt. Soc. Am. 45, 497–501 (1955). [CrossRef]
  2. D. Gabor, “Light and information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 109–153.
  3. G. Toraldo di Francia, “Degrees of freedom of an image,” J. Opt. Soc. Am. 59, 799–804 (1969). [CrossRef] [PubMed]
  4. D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. 23, 1645–1647 (1998). [CrossRef]
  5. D. A. B. Miller, “Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. 39, 1681–1699 (2000). [CrossRef]
  6. D. Gabor, “Theory of information,” J. IEE 93, 429–153 (1946).
  7. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  8. H. Wolter, “On basic analogies and principal differences between optical and electronic information,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1961), Vol. 1, pp. 157–210.
  9. D. Slepian, “On bandwidth,” Proc. IEEE 64, 292–300 (1976). [CrossRef]
  10. D. Slepian, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. I,” Bell Syst. Tech. J. 40, 43–64 (1961). [CrossRef]
  11. H. J. Landau, H. O. Pollack, “Prolate spheroidal wave functions, Fourier analysis and uncertainty. II,” Bell Syst. Tech. J. 40, 65–84 (1961). [CrossRef]
  12. B. R. Frieden, “Evaluation, design, and extrapolation methods for optical signals, based on use of the prolate functions,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1971), Vol. 9, pp. 313–407.
  13. F. Gori, L. Ronchi, “Degrees of freedom for scatterers with circular cross section,” J. Opt. Soc. Am. 71, 250–258 (1981). [CrossRef]
  14. F. Gori, G. Guattari, “Shannon number and degrees of freedom of an image,” Opt. Commun. 7, 163–165 (1973). [CrossRef]
  15. W. Lukosz, “Optical systems with resolving powers exceeding the classical limit,” J. Opt. Soc. Am. 56, 1463–1472 (1966). [CrossRef]
  16. A. W. Lohmann, “The space–bandwidth product applied to spatial filtering and holography,” (IBM San Jose Research Laboratory, San Jose, Calif., 1967), pp. 1–23.
  17. M. Bendinelli, A. Consortini, L. Ronchi, B. Roy Frieden, “Degrees of freedom, and eigenfunctions, for the noisy image,” J. Opt. Soc. Am. 64, 1498–1502 (1974). [CrossRef]
  18. R. Barakat, “Shannon numbers of diffraction images,” Opt. Commun. 6, 391–394 (1982). [CrossRef]
  19. G. Newsam, R. Barakat, “Essential dimension as a well-defined number of degrees of freedom of finite-convolution operators appearing in optics,” J. Opt. Soc. Am. A 2, 2040–2045 (1985). [CrossRef]
  20. E. R. Pike, J. G. McWhirter, M. Bertero, C. de Mol, “Generalised information theory for inverse problems in signal processing,” IEE Proc. F, 131, 660–670 (1984).
  21. E. Wolf, “New theory of partial coherence in the space–frequency domain. Part I: spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982). [CrossRef]
  22. A. Starikov, “Effective number of degrees of freedom of par-tially coherent sources,” J. Opt. Soc. Am. 72, 1538–1544 (1982). [CrossRef]
  23. S. A. Basinger, E. Michielssen, D. J. Brady, “Degrees of freedom of polychromatic images,” J. Opt. Soc. Am. A 12, 704–714 (1995). [CrossRef]
  24. I. J. Cox, C. J. R. Sheppard, “Information capacity and resolution in an optical system,” J. Opt. Soc. Am. A 3, 1152–1158 (1986). [CrossRef]
  25. O. M. Bucci, G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag. 37, 918–926 (1989). [CrossRef]
  26. R. Pierri, F. Soldovieri, “On the information content of the radiated fields in the near zone over bounded domains,” Inverse Probl. 14, 321–337 (1998). [CrossRef]
  27. A. W. Lohmann, R. G. Dorsch, D. Mendlovic, Z. Zalevsky, C. Ferreira, “Space–bandwidth product of optical signals and systems,” J. Opt. Soc. Am. A 13, 470–473 (1996). [CrossRef]
  28. R. Harrington, Time Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  29. L. Felsen, N. Markuvitz, Radiation and Scattering of Waves (IEEE Press, Piscataway, N.J., 1994).
  30. R. Courant, D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1961).
  31. Note that by Lebesgue’s (dominated convergence) theorem32 this is possible if there exists an integrable majorant function m(r) such that ‖ΣibiaTi(r′)‖⩽m(r) for all i. This is satisfied, for example, if the basis functions aTi are bounded almost everywhere.
  32. See, for example, D. Porter, D. S. G. Stirling, Integral Equations (Cambridge U. Press, Cambridge, UK, 1990).
  33. R. Piestun, J. Shamir, “Control of wave-front propagation with diffractive elements,” Opt. Lett. 19, 771–773 (1994). [CrossRef] [PubMed]
  34. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996). [CrossRef]
  35. R. Piestun, D. A. B. Miller, “Ill-posedness of the three-dimensional wave-field synthesis problem,” presented at the Annual Meeting of the Optical Society of America, Santa Clara, California, September 26–October 1, 1999, paper WLL27.
  36. See, for example, R. Kress, Linear Integral Equations (Springer-Verlag, Berlin, 1989).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited