In
laser science
, the parameter
M
2
, also known as the
beam propagation ratio
or
beam quality factor
is a measure of
laser beam quality
. It represents the degree of variation of a beam from an ideal
Gaussian beam
.
[1]
It is calculated from the ratio of the
beam parameter product
(BPP) of the beam to that of a Gaussian beam with the same
wavelength
. It relates the
beam divergence
of a laser beam to the minimum focussed spot size that can be achieved. For a
single mode
TEM
00
(Gaussian) laser beam, M
2
is exactly one. Unlike the beam parameter product, M
2
is unitless and does not vary with wavelength.
The M
2
value for a laser beam is widely used in the laser industry as a specification, and its method of measurement is regulated as an ISO standard.
[2]
Measurement
[
edit
]
There are several ways to define the width of a beam. When measuring the beam parameter product and M
2
, one uses the
D4σ or "second moment" width
of the beam to determine both the radius of the beam's waist and the divergence in the far field.
[1]
M
2
can be measured by placing an
array detector
or
scanning-slit profiler
at multiple positions within the beam after focusing it with a
lens
of high optical quality and known
focal length
. To properly obtain M
2
, the following steps must be followed:
[3]
- Measure the D4σ widths at 5 axial positions near the beam waist (the location where the beam is narrowest).
- Measure the D4σ widths at 5 axial positions at least one
Rayleigh length
away from the waist.
- Fit the 10 measured data points to
where
is half of the
beam width, and
is the location of the beam waist with width
.
[4]
Fitting the 10 data points yields M
2
,
, and
. Siegman showed that all beam profiles ? Gaussian,
flat-top
,
TEMxy
, or any shape ? must follow the equation above provided that the beam radius uses the D4σ definition of the beam width.
[1]
Using other definitions of beam width does not work.
In principle, one could use a single measurement at the waist to obtain the waist diameter, a single measurement in the far field to obtain the divergence, and then use these to calculate the M
2
. The procedure above gives a more accurate result in practice, however.
Utility
[
edit
]
M
2
is useful because it reflects how well a
collimated
laser beam can be focused to a small spot, or how well a divergent laser source can be collimated. It is a better guide to
beam quality
than Gaussian appearance because there are many cases in which a beam can
look
Gaussian, yet have an M
2
value far from unity.
[1]
Likewise, a beam intensity profile can appear very "un-Gaussian", yet have an M
2
value close to unity.
The quality of a beam is important for many applications. In
fiber-optic communications
beams with an M
2
close to 1 are required for coupling to
single-mode optical fiber
.
M
2
determines how tightly a
collimated
beam of a given diameter can be focused: the diameter of the focal spot varies as M
2
, and the
irradiance
scales as 1/M
4
. For a given
laser cavity
, the output beam diameter (collimated or focused) scales as M, and the irradiance as 1/M
2
. This is very important in
laser machining
and
laser welding
, which depend on high
fluence
at the weld location.
Generally, M
2
increases as a laser's output power increases. It is difficult to obtain excellent beam quality and high average power at the same time due to
thermal lensing
in the
laser gain medium
.
Multi-mode beam propagation
[
edit
]
Real laser beams are often non-Gaussian, being multi-mode or mixed-mode. Multi-mode beam propagation is often modeled by considering a so-called "embedded" Gaussian, whose beam waist is M times smaller than that of the multimode beam. The
diameter
of the multimode beam is then M times that of the embedded Gaussian beam everywhere, and the divergence is M times greater, but the wavefront curvature is the same. The multimode beam has M
2
times the beam area but 1/M
2
less beam intensity than the embedded beam. This holds true for any given optical system, and thus the minimum (focussed) spot size or beam waist of a multi-mode laser beam is M times the embedded Gaussian beam waist.
See also
[
edit
]
References
[
edit
]
- ^
a
b
c
d
Siegman, A. E. (October 1997).
"How to (Maybe) Measure Laser Beam Quality"
(PDF)
. Archived from
the original
(PDF)
on June 4, 2011
. Retrieved
Feb 8,
2009
.
Tutorial presentation at the Optical Society of America Annual Meeting, Long Beach, California
- ^
"Lasers and laser-related equipment ? Test methods for laser beam widths, divergence angles and beam propagation ratios". ISO Standard.
11146
. 2005.
- ^
ISO 11146-1:2005(E), "Lasers and laser-related equipment ? Test methods for laser beam widths, divergence angles and beam propagation ratios ? Part 1: Stigmatic and simple astigmatic beams".
- ^
See Siegman (1997), p. 9. There is a typo in the equation on page 3. Correct form comes from equations on page 9.