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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 19, Iss. 3 — Mar. 1, 2002
  • pp: 497–504

Modal decomposition of partially coherent beams using the ambiguity function

H. Laabs, B. Eppich, and H. Weber  »View Author Affiliations

JOSA A, Vol. 19, Issue 3, pp. 497-504 (2002)

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Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite–Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite–Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.

© 2002 Optical Society of America

OCIS Codes
(030.0030) Coherence and statistical optics : Coherence and statistical optics
(030.4070) Coherence and statistical optics : Modes
(140.0140) Lasers and laser optics : Lasers and laser optics
(140.3460) Lasers and laser optics : Lasers

Original Manuscript: June 15, 2001
Revised Manuscript: August 27, 2001
Manuscript Accepted: August 27, 2001
Published: March 1, 2002

H. Laabs, B. Eppich, and H. Weber, "Modal decomposition of partially coherent beams using the ambiguity function," J. Opt. Soc. Am. A 19, 497-504 (2002)

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  1. “Lasers and laser-related equipment—Test methods for laser beam parameters—Beam widths, divergence angle and beam propagation factor,” 9 (1999).
  2. “Optics and optical instruments—Lasers and laser-related equipment—Test methods for laser beam power (energy) density distribution,” 9 (2000).
  3. “Lasers and laser-related equipment—Test methods for laser beam parameters—Polarization,” 9 (1999).
  4. V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.
  5. B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996). [CrossRef]
  6. B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity,” Opt. Laser Technol. 30, 337–240 (1998). [CrossRef]
  7. “Lasers and laser-related equipment—Test methods for laser beam parameters—Beam positional stability,” 9 (1999).
  8. “Optics and optical instruments–Lasers and laser-related equipment–Test methods for laser beam parameters: phase distribution,” (1999).
  9. H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1050 (1992). [CrossRef]
  10. E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements—experimental results,” Appl. Phys. B 49, 409–414 (1989). [CrossRef]
  11. D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995). [CrossRef] [PubMed]
  12. M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994). [CrossRef] [PubMed]
  13. C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996). [CrossRef] [PubMed]
  14. C. C. Cheng, M. G. Raymer, H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000). [CrossRef]
  15. B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000). [CrossRef]
  16. J. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998). [CrossRef]
  17. J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997). [CrossRef]
  18. B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.
  19. B. Eppich, N. Reng, “Measurement of the Wigner distribution function based in the inverse Radon transformation,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 261–268 (1995). [CrossRef]
  20. K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984). [CrossRef]
  21. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979). [CrossRef]
  22. M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981). [CrossRef]
  23. H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.
  24. A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from multimode stable-cavity laser,” IEEE J. Quantum Electron. 19, 1212–1217 (1993). [CrossRef]
  25. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, L. Zeni, “Transverse mode analysis of a laser beam by near- and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995). [CrossRef] [PubMed]
  26. Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999). [CrossRef]
  27. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998). [CrossRef]
  28. A. Liesenhoff, F. Ruehl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995). [CrossRef]
  29. P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980). [CrossRef]
  30. L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995). [CrossRef]
  31. C. M. Warnky, B. L. Anderson, C. A. Klein, “Determining spatial modes of lasers with spatial coherence measurements,” Appl. Opt. 39, 6109–6117 (2000). [CrossRef]
  32. J. B. McManus, P. L. Kebian, M. S. Zahniser, “Astigmatic mirror multipath absorption cells for long-path-length spectroscopy,” Appl. Opt. 34, 3336–3348 (1995). [CrossRef] [PubMed]
  33. J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999). [CrossRef]
  34. M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999). [CrossRef]
  35. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis of partial coherent beams with Hermite–Gaussian beams,” Opt. Lett. 23, 989–991 (1998). [CrossRef]
  36. X. Xue, H. Wei, A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite–Gaussian functions,” J. Opt. Soc. Am. A 17, 1086–1091 (2000). [CrossRef]
  37. H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).
  38. H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999). [CrossRef]
  39. I. S. Gradshteyn, I. M. Ryzhik, Tables of Series and Products (Academic, San Diego, Calif., 1980).
  40. P. Toft, “The Radon transform—theory and implementation,” Ph.D. dissertation (Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark, 1996).
  41. H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996). [CrossRef]
  42. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990). [CrossRef]
  43. H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica 12, 405ff (1946). [CrossRef]
  44. R. Simon, G. S. Agarwal, “Wigner representation of Laguerre–Gaussian beams,” Opt. Lett. 25, 1313–1315 (2000). [CrossRef]

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