Abstract

Tolerance analysis shows that an efficient M 2 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 M 2.

© 1998 Optical Society of America

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References

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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.

1993 (1)

1991 (1)

1990 (1)

T. F. Johnston, “M2 concept characterizes beam quality,” Laser Focus, 173–183 (May1990).

1966 (1)

Bélanger, P. A.

Hall, D. R.

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.

Jackson, P. E.

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.

Johnston, T. F.

T. F. Johnston, “M2 concept characterizes beam quality,” Laser Focus, 173–183 (May1990).

Kogelnik, H.

Li, T.

Margenau, H.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 500–502.

Murphy, G. M.

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 500–502.

Nemes, G.

O’Shea, D. C.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), pp. 235–237.

Sasnett, M. W.

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.

Serna, J.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Section 17.6.

Silfvast, W. T.

W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996), pp. 340–342.

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis (University Science, Mill Valley, Calif., 1982).

Appl. Opt. (1)

Laser Focus (1)

T. F. Johnston, “M2 concept characterizes beam quality,” Laser Focus, 173–183 (May1990).

Opt. Lett. (2)

Other (11)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Section 17.6.

W. T. Silfvast, Laser Fundamentals (Cambridge U. Press, Cambridge, UK, 1996), pp. 340–342.

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.

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.

D. C. O’Shea, Elements of Modern Optical Design (Wiley, New York, 1985), pp. 235–237.

“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.

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.

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.

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.

J. R. Taylor, An Introduction to Error Analysis (University Science, Mill Valley, Calif., 1982).

H. Margenau, G. M. Murphy, The Mathematics of Physics and Chemistry (Van Nostrand, New York, 1943), pp. 500–502.

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Figures (7)

Fig. 1
Fig. 1

Illustration of the three common types of beam asymmetry possible for a multimode beam. The window insets show the wave-front curvatures along the beam path. For the astigmatic beam there are two points with cylindrical wave fronts but none where the wave front is plane.

Fig. 2
Fig. 2

Beams of (a) constant divergence and (b) constant waist diameter illustrate the consequence of M 2 ≠ 1. The beam must be sampled both near and far from the waist to distinguish between these possibilities. The figures are drawn with values appropriate to the λ = 2.1-μm beam in the text.

Fig. 3
Fig. 3

(a) Definitions of the quantities in the transformation of a Gaussian beam through a lens. The principal planes of the lens are H1 and H2. (b) Illustration of the result that measurement of the beam diameter at the focal plane on the output side of the lens yields the beam divergence of the input beam, regardless of the location of the input waist.

Fig. 4
Fig. 4

Profiles of the 2.1-μm beam taken in the focal plane of the lens for different apertures translated across the beam. (a) The transmission (percent) past a knife edge, the 100% level is 2 J. (b) The transmitted energy (millijoules) through a pinhole. (c) The transmitted energy (millijoules) through a slit.

Fig. 5
Fig. 5

Fractional change in beam diameter p as a function of normalized propagation distance η from the waist. Cuts made to locate the waist in the shaded regions benefit from a fractional change of 80% or more of the maximum. This requires a minimum of a Rayleigh range of access to the beam around the waist.

Fig. 6
Fig. 6

Four-cuts method. The propagation plot of the beam behind the inserted lens is shown, and the circled numbers over the plot indicate the order of the cuts made to locate the waist. The propagation distance z match of the diameter matching the first diameter at z 1 gives the waist location as halfway between these equal diameters.

Fig. 7
Fig. 7

Graphic analysis of the auxiliary beam propagation data. The chords giving the Rayleigh ranges for the X- and Y-plane beams are drawn at heights on the plot (diameters) larger than the waist by √2.

Tables (4)

Tables Icon

Table 1 Conversions between Diameter Definitions for the First Pulse 2.1-μm Beam

Tables Icon

Table 2 Knife-Edge Diameters and Waist Locations Measured in 2 Space for the First Pulse 2.1-μm Beam

Tables Icon

Table 3 First Pulse Beam Constants Measured in 2 Space and Transformed to 1 Space

Tables Icon

Table 4 Weighted Least-Squares Curve Fit, Auxiliary Beam Y-Axis Data

Equations (39)

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2 w 0 = 2 W 0 / M ,
2 W z = 2 W 0 1 + z - z 0 2 / z R 2 1 / 2 .
z R = π w 0 2 λ = π W 0 2 M 2 λ .
Θ = 4 λ / π M 2 2 W 0
θ = 4 λ / π 1 2 w 0 .
z R = 2 W 0 / Θ .
M 2 = Θ / θ n .
Γ = f 2 / z 02 - f 2 + z R 2 2 ,
2 W 01 = Γ 1 / 2 2 W 02 ,   z R 1 = Γ z R 2 , z 01 - f = Γ z 02 - f .
Θ 1 = 2 W f / f .
2 W f = 2 W 02 1 + f - z 02 2 / z R 2 2 1 / 2 = 2 W 02 f z R 2 1 Γ 1 / 2 = 2 W 01 f z R 2 1 Γ = 2 W 01 f z R 1 = Θ 1 f ,
η = z - z 0 / z R .
η 0 2 g ,
p 1 W d W d η = η 1 + η 2 .
z match = z 2 + W 1 - W 2 W 3 - W 2 z 3 - z 2 ,
z 4 = z 0 = z 1 + z match 2 .
M 2 = π W 0 2 / λ z R .
2 W 0 = 2 W 4 1 + η 0 2 1 / 2 .
A = A 0 + a ,   B = B 0 + b ,
d i = y i - f x i ,   A ,   B .
d i = F i - a   f x i ,   A 0 ,   B 0 A - b   f x i ,   A 0 ,   B 0 B
F i = y i - f x i ,   A 0 ,   B 0 .
u i = f x i ,   A 0 ,   B 0 A ,   v i = f x i ,   A 0 ,   B 0 B
d i = F i - au i - bv i .
  c i d i d i a = 0 ,     c i d i d i b = 0 .
- 2     c i F i - au i - bv i u i = 0 , - 2     c i F i - au i - bv i v i = 0
Pa + Qb = R ,   Qa + Sb = T ,
P =   c i u i 2 ,   Q =   c i u i v i ,   R =   c i F i u i , S =   c i v i 2 ,   T =   c i F i v i .
a = RS - QT PS - Q 2 ,   b = PT - QR PS - Q 2 ,
s =   c i d i 2 /   c i 1 / 2 =   c i F i 2 - aR - bT   c i 1 / 2 .
2 W z = 4 W 0 2 + Θ 2 z - z 0 2 1 / 2 ,
2 W 0 = 2 W 0 + a ,   Θ = Θ + b ,
u i = W W 0 = 2 W 0 2 W i ,   v i = 2 W Θ = Θ z i - z 0 2 2 W i .
P =   2 W 0 2 2 W i 4 ,   Q =   2 W 0 Θ z i - z 0 2 2 W i 4 ,
R =   2 W 0 F i 2 W i 3 ,   S =   Θ 2 z i - z 0 4 2 W i 4 ,
T =   F i Θ z i - z 0 2 2 W i 3 .
2 W i = 2 W 0   1 + z i - z 0 2 / z R 2 1 / 2 ,
F i = 2 W i - 2 W i .
2 W Y =   1 / 2 W i   1 / 2 W i 2 = 3.81   mm

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