OSA's Digital Library

Applied Optics

Applied Optics


  • Vol. 38, Iss. 25 — Sep. 1, 1999
  • pp: 5272–5281

Evaluation of the Modal Structure of Light Beams Composed of Incoherent Mixtures of Hermite-Gaussian Modes

Massimo Santarsiero, Franco Gori, Riccardo Borghi, and Giorgio Guattari  »View Author Affiliations

Applied Optics, Vol. 38, Issue 25, pp. 5272-5281 (1999)

View Full Text Article

Acrobat PDF (153 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



A new, to our knowledge, technique for determining the modal content of partially coherent beams that are made up of an incoherent superposition of Hermite–Gaussian modes is studied. The algorithm makes use of the intensity profile of the beam at an arbitrarily chosen transverse plane. Analytical derivations are presented for a Gaussian Schell-model source and flat-topped beams, as well as an analysis of their performances in the presence of experimental errors and noise. Numerical simulations are performed to test the accuracy and the stability of the recovery algorithm.

© 1999 Optical Society of America

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.4070) Coherence and statistical optics : Modes
(140.3460) Lasers and laser optics : Lasers
(140.4780) Lasers and laser optics : Optical resonators

Massimo Santarsiero, Franco Gori, Riccardo Borghi, and Giorgio Guattari, "Evaluation of the Modal Structure of Light Beams Composed of Incoherent Mixtures of Hermite-Gaussian Modes," Appl. Opt. 38, 5272-5281 (1999)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
  3. When two or more modes oscillate at the same frequency (i.e., in the case of degeneracy), the hypothesis of total noncorrelation has to be removed, and the treatment becomes more complex. See, for example, Ref. 2.
  4. J. Turunen, E. Tervonen, and A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
  5. E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
  6. B. Lü, B. Zhang, B. Cai, and C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed-mode laser beams behaving like Gaussian–Schell model beams,” Opt. Commun. 101, 49–52 (1993).
  7. A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
  8. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, and L. Zeni, “Transverse-mode analysis of a laser beam by near- and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995).
  9. R. Borghi and M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
  10. F. Gori, M. Santarsiero, and G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
  11. M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
  12. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
  13. R. Gase, T. Gase, and K. Blüthner, “Complex wave-field reconstruction by means of the Page distribution function,” Opt. Lett. 20, 2045–2047 (1995).
  14. G. Iaconis and I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
  15. J. Tu and S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
  16. F. Gori, “Shape-invariant propagation of the cross-spectral density,” in Coherence and Quantum Optics, L. Mandel and E. Wolf, eds. (Plenum, New York, 1984), p. 363.
  17. F. Gori, M. Santarsiero, R. Borghi, and G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
  18. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  19. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
  20. E. Collett and E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
  21. P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductors lasers,” Opt. Commun. 33, 265–270 (1980).
  22. E. G. Johnson, Jr., “Direct measurements of the spatial mode of a laser pulse: theory,” Appl. Opt. 25, 2967–2975 (1986).
  23. A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
  24. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  25. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
  26. A. Starikov and E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
  27. S. De Silvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
  28. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
  29. C. J. R. Sheppard and S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
  30. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
  31. M. Santarsiero, D. Aiello, R. Borghi, and S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
  32. R. Borghi, M. Santarsiero, and S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
  33. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  34. R. Borghi and M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
  35. R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).
  36. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.
  37. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

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