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Applied Optics

Applied Optics


  • Vol. 37, Iss. 21 — Jul. 20, 1998
  • pp: 4840–4850

Beam propagation (M 2) measurement made as easy as it gets: the four-cuts method

Thomas F. Johnston, Jr.  »View Author Affiliations

Applied Optics, Vol. 37, Issue 21, pp. 4840-4850 (1998)

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Tolerance analysis shows that an efficient M2 measurement plan is a first cut (beam diameter measurement) at 0.5–2.0 Rayleigh ranges to one side of the waist, which is matched by interpolation between second and third cuts to the opposite side. The waist is measured by a fourth cut halfway between the matched diameters, yielding an easy two-parameter curve fit for M2.

© 1998 Optical Society of America

OCIS Codes
(120.4800) Instrumentation, measurement, and metrology : Optical standards and testing
(140.3430) Lasers and laser optics : Laser theory
(350.5500) Other areas of optics : Propagation

Original Manuscript: November 3, 1997
Revised Manuscript: March 10, 1998
Published: July 20, 1998

Thomas F. Johnston, "Beam propagation (M2) measurement made as easy as it gets: the four-cuts method," Appl. Opt. 37, 4840-4850 (1998)

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Section 17.6.
  2. T. F. Johnston, “M2 concept characterizes beam quality,” Laser Focus, 173–183 (May1990).
  3. W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996), pp. 340–342.
  4. The author was the chief engineer for the development of the ModeMaster beam propagation analyzer, a product of Coherent, Inc., Instruments Group, 2303 Lindbergh St., Auburn, Calif., 95602.
  5. M. W. Sasnett, “Propagation of multimode laser beams—the M2 factor,” in The Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, New York, 1989), Chap. 9, pp. 132–142.
  6. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1566 (1966). [CrossRef] [PubMed]
  7. P. A. Bélanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991). [CrossRef]
  8. J. Serna, G. Nemes, “Decoupling of coherent Gaussian beams with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993). [CrossRef] [PubMed]
  9. D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), pp. 235–237.
  10. “Test methods for laser beam parameters: beam widths, divergence angle, and beam propagation factor,” ISO/TC 172/SC9/WG1, ISO/DIS 11146, available from Deutsches Institut für Normung, Pforzheim, Germany.
  11. To drive home this point and show that it is nothing new (though often overlooked), the author calls this the Stonehenge Effect, after Fred Hoyle ’s convincing interpretation of stone placements made in Chap. 4 of his book On Stonehenge (Freeman, San Francisco, Calif., 1977). Hoyle shows that 5000 years ago the builders of this ancient monument understood that, to locate a null, you must look away from the null point. The monument was built to locate the exact day of the summer solstice, the day the position of the rising Sun in its northward march along the horizon reversed direction and turned southward. If the sighting stone had been placed directly to mark the position of the turnaround, there would have been an ambiguity of at least a day in locating the solstice because the Sun’s motion per day is nil at that time. Instead, the sighting stone obscured a span of the horizon and was of a width and placement that the exact day of the solstice was marked as the day midway between the days of disappearance and reappearance of the Sun on the south side of the stone. This resolved the ambiguity in the sightings on the trio of days around the solstice when the Sun emerged on the opposite (north) side of the stone.
  12. In support of the adoption of the ISO beam characterization standard (Ref. 10), Coherent, Inc. has stated that it will grant royalty-free license to its patent (U.S. patent 5,267,012) on the use of a lens to form an auxiliary waist for M2 measurement of an astigmatic beam.
  13. In the Coherent ModeMaster instrument, in the focus pass to determine M2, cuts are made at 260 points along the auxiliary beam propagation path in each independent plane.
  14. J. R. Taylor, An Introduction to Error Analysis (University Science, Mill Valley, Calif., 1982).
  15. H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 500–502.

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