OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 5 — May. 1, 1999
  • pp: 1049–1058

Effects of partial spatial coherence with uniform-intensity diffractive axicons

Ari T. Friberg and Sergei Yu. Popov  »View Author Affiliations

JOSA A, Vol. 16, Issue 5, pp. 1049-1058 (1999)

View Full Text Article

Enhanced HTML    Acrobat PDF (470 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Apodized annular-aperture logarithmic axicons that form uniform-intensity axial line images with coherent light are studied in Gaussian-correlated illumination. Diffractive assessment of the line-image distributions and spatial coherence properties involves a highly oscillating double two-dimensional integral. The on-axis behavior depends only on radial integrals that can be computed with special-purpose routines. We show that at all correlation levels the images at off-axis points can be evaluated by using a technique based on spline approximations. We also demonstrate that the method of stationary phase can be sequentially applied to the four-dimensional diffraction integral, yielding accurate three-dimensional closed-form results. The stationary-phase formulas find applications in fast image evaluation, in intensity balancing by varying the irradiance, and in designing axicon phase profiles. The results complement our earlier study of partially coherent axicon images by radiometric transport techniques.

© 1999 Optical Society of America

OCIS Codes
(030.5630) Coherence and statistical optics : Radiometry
(350.3950) Other areas of optics : Micro-optics
(350.5500) Other areas of optics : Propagation

Original Manuscript: September 3, 1998
Manuscript Accepted: November 13, 1998
Published: May 1, 1999

Ari T. Friberg and Sergei Yu. Popov, "Effects of partial spatial coherence with uniform-intensity diffractive axicons," J. Opt. Soc. Am. A 16, 1049-1058 (1999)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. L. M. Soroko, “Axicons and meso-optical imaging devices,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1989), Vol. 27, pp. 109–160.
  2. N. Davidson, A. A. Friesem, E. Hasnam, “Efficient formation of nondiffracting beams with uniform intensity along the propagation direction,” Opt. Commun. 88, 326–330 (1992). [CrossRef]
  3. V. P. Koronkevich, I. A. Mikhaltsova, E. G. Churin, Yu. I. Yurlov, “Lensacon,” Appl. Opt. 34, 5761–5772 (1995). [CrossRef] [PubMed]
  4. Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research & Development Treatises (Society of Photo-Optical Instrumentation Engineers Polish Chapter, Warsaw, 1997).
  5. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, S. Bara, “Nonparaxial design of generalized axicons,” Appl. Opt. 31, 5326–5330 (1992). [CrossRef] [PubMed]
  6. N. Davidson, A. A. Friesem, E. Hasman, “Holographic axilenses: high resolution and long focal depth,” Opt. Lett. 16, 523–525 (1991). [CrossRef] [PubMed]
  7. J. Sochacki, S. Bara, Z. Jaroszewicz, A. Kołodziejczyk, “Phase retardation of the uniform-intensity axilens,” Opt. Lett. 17, 7–9 (1992). [CrossRef] [PubMed]
  8. M. V. Perez, C. Gomez-Reino, J. M. Cuadrado, “Diffraction patterns and zone plates produced by thin linear axicons,” Opt. Acta 33, 1161–1176 (1986). [CrossRef]
  9. J. Turunen, A. Vasara, A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27, 3959–3962 (1988). [CrossRef] [PubMed]
  10. A. T. Friberg, “Stationary-phase analysis of generalized axicons,” J. Opt. Soc. Am. A 13, 743–750 (1996). [CrossRef]
  11. L. R. Staroński, J. Sochacki, Z. Jaroszewicz, A. Kołodziejczyk, “Lateral distribution and flow of energy in uniform-intensity axicons,” J. Opt. Soc. Am. A 9, 2091–2094 (1992). [CrossRef]
  12. J. Sochacki, Z. Jaroszewicz, L. R. Staroński, A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J. Opt. Soc. Am. A 10, 1765–1768 (1993). [CrossRef]
  13. Z. Jaroszewicz, J. Sochacki, A. Kołodziejczyk, L. R. Staroński, “Apodized annular-aperture logarithmic axicon: smoothness and uniformity of intensity distributions,” Opt. Lett. 18, 1893–1895 (1993). [CrossRef] [PubMed]
  14. A. J. Cox, J. D’Anna, “Constant-axial-intensity nondiffracting beam,” Opt. Lett. 17, 232–234 (1992). [CrossRef] [PubMed]
  15. R. M. Herman, T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31, 5913–5915 (1992). [CrossRef] [PubMed]
  16. S. Yu. Popov, A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537–548 (1998). [CrossRef]
  17. S. Yu. Popov, A. T. Friberg, M. Honkanen, J. Lautanen, J. Turunen, B. Schnabel, “Apodized annular-aperture diffractive axicons fabricated by continuous-path-control electron beam lithography,” Opt. Commun. 154, 359–367 (1998). [CrossRef]
  18. S. Yu. Popov, A. T. Friberg, “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639–1641 (1998). [CrossRef]
  19. A. T. Friberg, J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5, 713–720 (1988). [CrossRef]
  20. Q. He, J. Turunen, A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67, 245–250 (1988). [CrossRef]
  21. W. Wang, A. T. Friberg, E. Wolf, “Focusing of partially coherent light in systems of large Fresnel numbers,” J. Opt. Soc. Am. A 14, 491–497 (1997). [CrossRef]
  22. S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988). [CrossRef] [PubMed]
  23. R. Gase, “The multimode laser radiation as a Gaussian Schell-model beam,” J. Mod. Opt. 38, 1107–1115 (1991). [CrossRef]
  24. Special issue on laser beam quality, Opt. Quantum Electron. 24, S861–S1135 (1992).
  25. A. T. Friberg, ed., Selected Papers on Coherence and Radiometry, Vol. 69 of Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993).
  26. A. Giesen, M. Morin, eds., Proceedings of the 4th International Workshop on Laser Beam and Optics Characterization, Munich, June 16–18, 1997 (Institut für Strahlwerkzeuge, Stuttgart, 1997).
  27. A. T. Friberg, S. Yu. Popov, “Partially coherently illuminated uniform-intensity holographic axicons,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 224–227.
  28. W. T. Welford, “A vector raytracing equation for hologram lenses of arbitrary shape,” Opt. Commun. 14, 322–323 (1975). [CrossRef]
  29. R. C. Fairchild, J. R. Fienup, “Computer-originated aspheric holographic optical elements,” Opt. Eng. 21, 133–140 (1982). [CrossRef]
  30. D. Marcuse, Principles of Optical Fiber Measurements (Academic, New York, 1981), Sec. 4.7.
  31. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Sec. 3.1 and App. III.
  32. E. Wolf, “New theory of radiative energy transfer in free electromagnetic fields,” Phys. Rev. D 13, 869–886 (1976). [CrossRef]
  33. L. Mandel, E. Wolf, “Spectral coherence and the concept of cross-spectral purity,” J. Opt. Soc. Am. 66, 529–535 (1976). [CrossRef]
  34. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995), Chap. 4.
  35. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. AP-15, 187–188 (1967). [CrossRef]
  36. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982). [CrossRef]
  37. M. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Secs. 9.1, 9.6, and 9.7.
  38. A. Vasara, J. Turunen, A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6, 1748–1754 (1989). [CrossRef] [PubMed]
  39. H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
  40. J. Turunen, F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).
  41. E. Wolf, “Radiometric model for propagation of coherence,” Opt. Lett. 19, 2024–2026 (1994). [CrossRef] [PubMed]
  42. A. T. Friberg, S. Yu. Popov, “Radiometric description of intensity and coherence in generalized holographic axicon images,” Appl. Opt. 35, 3039–3046 (1996). [CrossRef] [PubMed]
  43. S. Yu. Popov, A. T. Friberg, “Linear axicons in partially coherent light,” Opt. Eng. 34, 2567–2573 (1995). [CrossRef]
  44. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). [CrossRef]
  45. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990). [CrossRef]
  46. C. De Boor, A Practical Guide to Splines (Springer-Verlag, New York, 1978).
  47. A. Erdelyi, Asymptotic Expansions (Dover, New York, 1956).
  48. T. Jannson, T. Aye, I. Tengara, D. A. Erwin, “Second-order radiometric ray tracing,” J. Opt. Soc. Am. A 13, 1448–1455 (1996). [CrossRef]
  49. E. Wolf, “The radiant intensity from planar sources of any state of coherence,” J. Opt. Soc. Am. 68, 1597–1605 (1978). [CrossRef]

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