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4.5
Fraction of PIB
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Real Beam 1
Real Beam 2
D E Ideal Gaussian Beam
B
A
F
C
0 1 2 3 4 5 6 7 8 9 /D
Figure 4.6: The vertical beam quality of real beam 1 is given by square
root of the ratio of segment AC to segment AB and the horizontal beam
quality of real beam 2 is given by the ratio of segment DF to segment DE8
Section 4.2: M2 Factor
The beam quality of a laser is characterized by the M2 factor, comparing
the true shape of the beam to that of an ideal Gaussian beam. The ISO
Standard 11146 defi nes the M2 factor as3:
In Equation 4.2, w₀ is the beam waist, θ is the divergence angle of the
laser, and λ is the lasing wavelength (Figure 4.4). As defi ned in Section 2:
Gaussian Beam Propagation, the divergence angle of a Gaussian beam is
determined by the following equation:
Inserting the found divergence angle into Equation 4.2 simplifi es the
equation for the M2 factor of a Gaussian beam:
Therefore, an M2 factor of 1 corresponds to a diff raction-limited Gaussian
beam. Larger M2 factors (greater than 1) correspond to deviations
from an ideal Gaussian beam. A value less than 1 cannot be achieved.
The M2 factor of a Hermite-Gaussian mode is given by (2n + 1) in the x
direction and (2m + 1) in the y direction,4 For example, TEM13 has an M2
factor of 3 in the x direction and of 7 in the y direction. A typical Helium
Neon laser has an M2 factor between 1 and 1,1.
Along with a laser beam’s optical power, the M2 factor determines the
radiance of the beam. The M2 factor can also be used to approximate
the radius of a beam as it propagates by replacing the wavelength of a
laser with the wavelength multiplied by the M2 factor found in all of the
equations in Section 2: Gaussian Beam Propagation,2 (pages 8-11)
The M2 factor is important because it represents how well a laser beam
can be focused for a given divergence. Lower M2 factors correspond
with a tighter focus, a more effi cient use of the power within the beam,
and a higher potential eff ective power of the laser.
Measuring M2 is not as simple as measuring the beam profi le at a single
plane on the laser axis. ISO 11146 dictates that fi ve beam radius measurements
must be taken at diff erent positions along the optical axis in both
the near fi eld and the far fi eld,5 It is possible to make a beam look like an
ideal Gaussian at one specifi c plane without it containing any of the TEM00
mode at all (Figure 4.5). Even though the cross-section at that specifi c plane
will look like a perfect Gaussian distribution, the beam will propagate very
diff erently than a Gaussian beam and will have a greater divergence angle,
6 Multiple radius measurements at diff erent planes will quickly reveal
the diff erence between this beam and a true Gaussian beam.
One disadvantage to characterizing beams with the M2 factor compared
to other beam quality parameters is that it places more emphasis on the
"wings", or the lower-power density parts of the beam furthest away
from the center, making it occasionally better suited for academic settings
rather than industrial applications.
Section 4.3: Beam Parameter Product
The beam parameter product (BPP) is another metric used to evaluate
the quality of a laser beam and is defi ned as the product of the beam radius
at the beam waist and the half-angle beam divergence. It is typically
reported in mm mrad, and is related to the M2 factor by:
Since the BPP is directly proportional to the M2 factor, a larger beam
parameter product corresponds with a worse beam. The minimum value
of the BPP is λ/π and this can only occur for an ideal Gaussian beam.
w (z)
z
wo
Figure 4.4: Illustration of the divergence angle and beam waist of a
laser beam
4.2
4.3
4.4
Superposition of Hermite-
Gaussian resonator modes
TEM00 = 0%
TEM01 = 44%
TEM10 = 17%
TEM11 = 19%
TEM20 = 11%
TEM21 = 6%
Figure 4.5: This beam cross-section appears Gaussian at a specifi c
plane even though it does not contain any of the TEM00 mode, illustrating
the importance of making several intensity measurements along the
laser axis to determine a laser’s M2 factor