OSA's Digital Library

Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 67, Iss. 9 — Sep. 1, 1977
  • pp: 1252–1261

The time-dependent physical spectrum of light

J. H. Eberly and K. Wódkiewicz  »View Author Affiliations

JOSA, Vol. 67, Issue 9, pp. 1252-1261 (1977)

View Full Text Article

Acrobat PDF (1305 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



We investigate the time-dependent spectrum of light from an observational point of view and define a time-dependent “physical spectrum” of light based on the counting rate of a photodetector. The tunable element, the filter, that allows observation of different spectral components of the light is shown to play an essential role in the time-dependent spectrum. Its bandwidth cannot be taken arbitrarily narrow. We establish the connection between our physical spectrum and other time-dependent spectra associated with Page, Lampard, Silverman, and Kolmogorov, as well as with the Wiener-Khintchine power spectrum. Also, we show the conditions under which these earlier definitions can be used as the first approximations to the complete physical spectrum, and give an expression for the correction terms.

© 1977 Optical Society of America

J. H. Eberly and K. Wódkiewicz, "The time-dependent physical spectrum of light," J. Opt. Soc. Am. 67, 1252-1261 (1977)

Sort:  Author  |  Journal  |  Reset


  1. A brief summary of Newton's first paper, in which he boldly proposed that white light should be understood simply as the composite of rays of all the different colors revealed by his prism, is given by L. N. Cooper in An Introduction to the Meaning and Structure of Physics (Harper & Row, New York (1968), Chap. 16.
  2. A description of the problems that jointly concerned these physicists, as well as a commentary on their approaches, and an excellent bibliography, have been provided by N. Wiener (Ref. 3 below).
  3. N. Wiener, Acta Math. 55, 117 (1930).
  4. A stationary process is one whose average values are time translation-invariant. For example, in second order, (〈V*(t+s)V(t)〉=〈V*(s)V(0)〉, for all t and s.
  5. A short development of the Wiener-Khintchine theorem, using both real functions and the corresponding positive and negative frequency functions, in the context of partially coherent light is given by M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Oxford, 1975), Sec. 10. 3. 2. Reference to more extensive treatments are also given there.
  6. C. H. Page, J. Appl. Phys. 23, 103 (1952).
  7. D. G. Lampard, J. Appl. Phys. 25, 803 (1954).
  8. R. A. Silverman, Proc. I.R.E. (Trans. Inf. Th.) 3, 182 (1957).
  9. J. B. Roberts, J. Sound Vib. 2, 336 (1965); see also 17, 299 (1971).
  10. M. B. Priestley, J. Sound Vib. 6, 86 (1967); 17, 51 (1971).
  11. W. D. Mark, J. Sound Vib. 11, 19 (1970).
  12. E. Garcia, H. Stark, and R. C. Barker, Appl. Opt 11, 1480 (1972).
  13. W. L. Kruer, Phys. Fluids 14, 2397 (1971).
  14. B. E. Grant, J. Sound Vib. 3, 407 (1966).
  15. T. K. Caughey and H. J. Stumpf, J. Appl. Mech. 28, 563 (1961).
  16. See L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), Chaps. 3 and 4, for an introduction to the theory of coherent optical transients, and a number of references to the original papers.
  17. F. Schuda, C. R. Stroud, Jr., M. Hercher, J. Phys. B 7, 198(L) (1974).
  18. F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett. 35, 1426 (1975), and R. E. Grove, F. Y. Wu, and S. Ezekiel, Phys. Rev. A 15, 227 (1977).
  19. H. Walther, in Laser Spectroscopy, edited by S. Haroche et al. (Springer, New York 1975), and W. Hartig, W. Rasmussen, R. Schieder, and H. Walther, Z. Phys. A 278, 205 (1976).
  20. B. R. Mollow, Phys. Rev. 188, 1969 (1969), and Phys. Rev. A 12, 1919 (1975).
  21. B. Renaud, R. M. Whitley, and C. R. Stroud Jr., J. Phys. B 10 19 (1977), and B. Renaud, Ph.D. Thesis. University of Rochester (1976).
  22. Our definition of V(r) differs by a factor of 2 from the definition adopted by Born and Wolf (Ref. 5). This explains the absence in our Eq. (1) of a factor of 4, when compared with Eq. (30), Sec. 10. 3. 2, of Born and Wolf.
  23. See, for example, R. J. Glauber in Quantum Optics and Electronics, edited by C. DeWitt, A. Blandin, and C. Cohen-Tannoudji (Gordon and Breach, New York, 1956), lecture IV, where a quantized-field approach is used. However, the same results follow from a semiclassical treatment: L. Mandel, E.C.G. Sudarshan, and E. Wolf, Proc. Phys. Soc. 84, 835 (1964).
  24. 1t should be pointed out that relation (5) may be taken to be valid even for nonstationary processes if we revise the fundamental first equation. If the left-hand side of (1) is replaced by [equation], then it is easy to see that (5) still follows. [See, for example G. R. Cooper and C. D. McGillem, Methods of Signal and System Analysis (Holt, Rinehart, and Winston, New York, 1967) Sec. 11-5]. However, such a revision does not lead to a time-dependent spectrum, nor does it avoid objections (b) and (c) following Eq. (6). In particular, the spectrum still depends on the future of the signal.
  25. A. N. Kolmogorov, Dokl. Acad. Nauk, SSSR 30, 229 (1941) and 32, 19 (1941). See also V. I. Tatarskii, Wave Propagation in a Turbulent Atmosphere (Nauka, Moscow, 1967), translated in The Effects of the Turbulent Atmosphere on Wave Propagation, edited by J. W. Strohbehnm (National Science Foundation, 1971), available from National Technical Information Service, U. S. Department of Commerce.
  26. 0ne can show that a change of gauge allows VD(t) to be interpreted, alternatively, as the electromagnetic vector poténtial, if desired,
  27. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975) Sec. 7. 6.
  28. See W. D. Mark, J. Sound Vib. 11, 19 (1970) Sec. 5, for a discussion closely similar to ours in its formal relationships. Because of this similarity we have adopted Mark's term "physical spectrum," However, Mark's window functions are not future truncated or even necessarily positive, as is our function J(t-t′), and therefore the meaning of Mark's physical spectrum is different from ours in some important ways.

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