Specifications

Beam Propagation
31
M
2
(M Squared)
Factor
The actual laser beams differ somewhat from the ideal Gaussian
profile shown in Figure 13. To handle the handle the deviation from
the ideal case, the factor M
2
or K has been developed and is often
quoted in laser specifications. For the ideal beam the M
2
factor is 1
and the factor increases as the beam deviates more from ideal
behavior. For a beam with an M
2
factor of 1.2, the beam is actually
√ 1.2 = 1.1 larger than an ideal Gaussian beam. It basically relates to
the factor by which the beam diameter is different from ideal. As
will be shown in the later examples, it has practical use to determine
the beam size at various locations in a beam delivery system. Note
that the M
2
= 1/K and is also in common use.
Beam Propagation
As a laser beam propagates away from it narrowest point or beam
waist, it will increase in size in a very predictable fashion. To calcu-
late the beam size at a specific location, one must know the size of
the beam waist and its location. Thus the beam diameter, D at a
distance Z away from the beam waist with a beam diameter of D
0
follows the equation:
The factor Θ is the beam divergence. The beam divergence depends
on some basic properties of the beam including the wavelength, λ
and the beam waist size D
0
. The relationship for the beam diver-
gence at full angle, then is:
Often the beam divergence is a value included in the specifications
of a laser. If a calculation is being made of the divergence, the units
of the wavelength and the beam waist diameter must be the same. As
an example a laser operating at a wavelength of 10.6 µ, a 7 mm
beam waist diameter, and an M
2
of 1.2 the calculated divergence is
as follows:
Θ = 4 x 0.0106 mm x 1.2 / (3.14 x 7) = 0.0023 rad = 2.3 mrad
Now calculating the beam diameter for the same laser as above at
2 meters from the beam waist;
D = √(49 mm
2
+ 0.0023
2
x 2000mm
2
)
D= √(49 mm
2
+ 5.29 x 10
-6
x 4 x 10
6
mm
2
) = 8.4 mm
DD
o
2
Θ
2
Z
2
+=
Θ
4λM
2
πD
o
--------------=