▲

BEAM-PROFILE ANALYSIS

Comparing M2 of an

actual laser beam

Propagation factor quantifies
to a pure TEM00 laser beam performance
Gaussian beam allows

beam-propagation
Carlos Roundy
characteristics to be

accurately predicted.
d 0 focal-spot
diameter with
M 2 > 1, d 0 = M 2d 00

D 00 or D 0
Input multimode
beam of wavelength λ,
width D 00 or D 0, Lens
any emerging applications of lasers

M
divergence θ00 or θ0
require extremely high TEM00 Gaussian
mode quality. These range from scientif- d 00 focal-spot
diameter of
ic experiments—in which the beam single-mode laser
must be focused to a very high intensity
(or irradiance) for nonlinear processes—to indus-
f f
trial processes in which a beam must be focused Focal
to the smallest possible spot for such applications length
as drilling holes in stainless steel.
In the past, a Gaussian fit to the beam profile has FIGURE 1. Two equivalent input beams that are equal in all other respects will focus to two
been used to evaluate how close the beam is to different waist sizes if the M2 of each beam is different.
TEM00. It was shown more than 10 years ago, how-
ever, that a multimode combination of beams can have a nearly d0 = M2 4λf/πD0
perfect Gaussian shape.1–3 Thus, a Gaussian fit can deceive a
user into assuming propagation properties of a laser beam that where λ is the wavelength, f is the focal length of the lens, and
will not exist in practice. The Gaussian fit then becomes not D0 is the waist width of the input multimode laser beam at the
only a meaningless measurement, but also one that is decep- focal length of the lens. This equation shows that the focused
tive—giving the user a false sense of laser performance. spot size d0 is M2 times larger than it would be for a pure
If a Gaussian fit does not adequately provide the mode TEM00 Gaussian beam of the same input width D0. Thus, for a
characteristics of a laser beam, what measurement does? beam of M2 = 2, the focused spot size is two times larger than
The answer is the beam-propagation factor M 2 . With would be obtained with a TEM00 beam. The irradiance, which
today’s beam-profile-analysis equipment and software, it is is proportional to the beam width squared, would be only one-
easy to identify this parameter, which quantitatively com- fourth of that achieved with a pure TEM00 Gaussian beam.
pares the propagation characteristics of the real beam to
those of a pure TEM00 Gaussian beam. For a given input
beam width and lens focal length, this comparison allows
the exact focused spot size to be predicted, as well as the
irradiance of a focused spot, the Rayleigh range over which
the beam is relatively collimated, and the far-field diver-
z
gence of the beam. While the M2 concept has been known
for many years, the popularity of making this measurement
has only recently been catching on in both scientific and
industrial communities.
SPIRICON INC.

The theory of M 2 Artificial Artificial

A common use of the M2 concept is in determining the size of waist region far field
a focused spot when a focusing lens is used (see Fig. 1), as
FIGURE 2. The ISO standard defines the method required to accurately
CARLOS ROUNDY is president of Spiricon Inc., 2600 N. Main St., Logan, UT measure M2, which is based on a fixed-position lens and multiple
84341-5740; e-mail: president@spiricon.com. beam-width measurements made through the waist.

Reprinted with revisions to format, from the December 1999 edition of LASER FOCUS WORLD
Copyright 1999 by PennWell Corporation
 ▲
BEAM-PROFILE ANALYSIS

The focused spot size and irradiance With the beam irradiance one-half that ty of making an accurate measurement.
of a laser beam have profound effects in of a Gaussian beam, the hole may not be It cannot be determined with a single
both science and industry. In science, drilled to the expected depth. calculation, as would be possible with a
nonlinear processes are typically pro- In both scientific and industrial Gaussian fit, for example.
portional to the irradiance squared or cases, it is essential for the user to The International Organization for
cubed. Thus, a beam with an M2 of 1.4 know what to expect from the process. Standardization (ISO; Geneva, Switzer-
(and an irradiance of one-half that of the While experimentation is often used to land) committee has defined a method-
Gaussian) would have a nonlinear out- obtain this information, by knowing ology that provides for reliable measure-
put of 0.25 to 0.125 of a beam with an the M2 of the laser beam, a scientist or ment of M2 so that this parameter can be
M2 of 1, all other characteristics being production manager can make accurate used with confidence by anyone making
equal. predictions. the measurement.4 The method involves
In the industrial process of very fine placing a lens of a known focal length in
hole drilling, a beam with an M2 of 1.4 Measurement of M 2 a laser beam, then making a series of
would drill holes 1.4 times larger than One reason the concept of M2 has not measurements through the focused
would a beam that was pure TEM00. been particularly popular is the difficul- waist of the beam (see Fig. 2). Measure-
ments that are usually essential include
a b the width of the spot at the smallest
focus, the position of the spot at focus,
the width of the beam at the focal length
of the lens—which may not be the same
place as the smallest spot—and the
divergence of the beam beyond focus.
Typically, a more-reliable and consistent
measurement is obtainable when the
equipment end user makes a series of
measurements and then performs a
curve fit to the measured data to calcu-
late the M2 parameters from the fit.
Examples illustrating the contrast
c d between Gaussian fit and M2 are shown
in Figs. 3 and 4. In Fig. 3, a computer-
generated beam is shown that is com-
posed of several modes.3,5 The beam
shape provides an almost perfect
Gaussian fit at 0.97, yet the M2 of the
beam is 3.3.

Equipment perspectives
Certain steps are essential to making a
reliable and consistent measurement of
M2. The first is to perform the measure-
e f ment as specified in the ISO standard.
That is, the lens must be stationary and
the sensor moved through the waist of
the beam. In some cases, the user finds
it easier to hold the sensor stationary,
and move the lens in the incoming
beam (see Fig. 5 on p. 122). This method
is typically reliable when the input
SPIRICON INC.

beam is well collimated over the range

of motion of the focusing lens. If, how-
ever, the beam is either diverging or
converging over the travel length of the
FIGURE 3. Simulated composite laser beam (a; M2 = 3.3) composed of 0% TEM00, 16% TEM*10, 44% lens, then the M2 measurement can be
TEM*01, 20%TEM*11, 12% TEM*20, and 8% TEM*21 appears Gaussian (fit = 0.97) even though it is incorrect and very misleading.
composed entirely of higher-mode laser beams The two white cross-section profiles are that of the A second part of the ISO definition is
beam and of the Gaussian fit, which are seen to be almost indistinguishable (b is TEM*1,0, c is TEM*0,1, that the width of the laser beam must
d is TEM*1,1, e is TEM*2,0, and f is TEM*2,1). be measured by the Second Moment
method. Spiricon Inc. (Logan, UT) soft- ly large beams or wavelengths incom-
Lens motion
ware uses proprietary algorithms to cal- patible with the optics and cameras of Fixed
culate the Second Moment width, commercial instruments, a user typical- detector plane
which is difficult to do because of non- ly has to make the M 2 measurement
laser background signal and off-axis manually. In this case, software is avail-
laser light. While there are many other able that provides a detailed step-by-
definitions of laser-beam width, includ- step process for making consistent mea-
ing the 1/e2 points and 14% of peak, surements and calculating M2. Input
laser beam
none reliably provides an accurate eval-
uation of M2. Some suggestions have Commercial application
been made to enable other types of The concept of M 2 measurement FIGURE 5. Alternative M2 measurement method,
measurements to approximate the Sec- enables both laser manufacturers and which involves moving the lens instead of the detec-
ond Moment width. 6 Nevertheless, laser users to have greater confidence in tor, can be reliable if the input beam is well collimat-
only the Second Moment beam-width their ability to predict the performance ed over the range of motion of the focusing lens.

▼
measurement conforms to the laser- of the laser beam. In some cases, OEM
beam-propagation equation and is, users are requiring laser manufacturers readily available for M2 measurement,
therefore, the only measurement that to measure the M2 on each laser shipped which makes it much easier for the end
provides reliable and consistent mea- and hold a very tight specification on user to accurately evaluate this impor-
surements of M2. this parameter. tant parameter. Hence, experiments
There are several commercial instru- Commercial instrumentation is now come much closer to meeting the expec-
ments for measuring the M2 of laser tations of the laser scientist, and indus-
beams. Some use the stationary-detec- trial users are much better able to pre-
tor/moving-lens method described dict what a laser will do in a given
here. Some users are satisfied with application. ❏
these instruments, while others report
the results to be inconsistent. As noted, REFERENCES
the difference may be whether or not 1. M. W. Sasnett, The Physics and Technology of
the beam is measured through a colli- Laser Resonators, A. Hilger and D. Hall, eds.,
p.136 (1989).
mated section. Some manufacturers 2. A. E. Siegman, Lasers, University Science
report that they have special algo- Books, Sausalito, CA (1986).
rithms to correct for errors in the case 3. A. E. Siegman, OSA Trends in Optics and Pho-
tonics 17, 184 (1988).
of high-divergence beams. Others pro- 4. Test methods for laser beam parameters: Beam
vide instrumentation that holds the widths, divergence angle and beam propaga-
tion factor, Document ISO/11146, International
lens stationary and moves the detector, Organization for Standardization; available from
as mandated in the ISO standard. Spiri- FIGURE 4. Real multimode laser beam has M2 of ANSI at www.ansi.org (Nov. 1993).
con provides the fixed-lens type of 4.8 but its Gaussian fit is 0.91, so in an industrial 5. R. D. Jones, Laser Measurements Short
Course, NIST, Boulder, CO (1992–1999).
instrumentation. application this beam would not perform as well 6. T. F. Johnston Jr., Appl. Opt. 37(21), 4840
For unusual lasers that have extreme- as one with a smaller M2 parameter. (1998).