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Section 6.3: Minimizing Beam
Diameter at a Distance
Although it may be unintuitive, increasing the diameter of a laser using
a beam expander may result in a smaller beam diameter far from
the laser aperture. A beam expander will increase the input laser beam
by a specific expansion power while decreasing the divergence by the
same expansion power, resulting in a smaller collimated beam at a large
distance.
The output beam diameter at a specific working distance (L) is a function
of DO, L, and θO (Figure 6.3).
Laser beam divergence is specified in terms of half angle, which is why
a factor of 2 is required in the second term of Equation 6.4. A beam
expander will increase the input beam and decrease the output divergence
by the Magnifying Power. Substituting Equation 6.3 into Equation
6.4 results in the following:
Section 6.4:
Minimizing Focused Spot Size
Spot size is typically defined as the radial distance from the center point
of the maximum irradiance to the point where the intensity drops to 1/
e2 of the initial value (Figure 6.4). The focused spot size of an ideal lens
can be calculated by using wavelength (λ), the focal length of the lens
(f ), the input beam diameter (DI), the refractive index of the lens (n), and
the beam’s M2 factor, which represents the degree of variation from an
ideal Gaussian beam.
Spot size is fundamentally determined by the combination of diffraction
and aberrations illustrated by red and blue, respectively, in Figure 6.5.
Generally, when focusing laser beams, spherical aberration is assumed
to be the only and dominant type of aberration, which is why Equation
6.6 only takes spherical aberration into account. In regard to diffraction,
the shorter the focal length, the smaller the spot size. More importantly,
the larger the input beam diameter the smaller the spot size.
By expanding the beam within the system, the input diameter (D) is increased
by a factor of MP, reducing the divergence by a factor of MP.
When the beam is focused down to a small spot, the spot is a factor of
MP smaller than that of an unexpanded beam for an ideal, diffractionlimited
spot. However, there is a tradeoff with spherical aberration because
it increases along with the input beam diameter.
Section 6.5: Compensating for Input
Laser Beam Variability
Most commercial lasers specify an output beam diameter of the laser at
the aperture with a tolerance that is often on the order of 10% or more.
For many laser applications, a specific beam diameter is required at the
end of the system. A variable beam expander can be inserted into the
system to compensate for variability between individual laser units, ensuring
that the final beam diameter is consistent for all systems.
D
L
θO
Figure 6.3: A laser’s input beam diameter and divergence can be used
to calculate the output beam diameter at a specific working distance
I (r)
r
diameter
I0
I0
e2
Figure 6.4: Spot size is usually measured at the point where the intensity
I (r ) drops to 1/e2 of the initial value I0
Input Beam Diameter (mm)
100
FOCUSED SPOT SIZE AS A FUNCTION OF INPUT BEAM SIZE
Focused Spot Size (μm)
Contribution due to
spherical aberration (m)
Contribution due to
diffraction (m)
1
90
80
70
60
50
40
30
20
10
0
2 3 4 5 6 7 8
Figure 6.5: At small input beam diameters, the focused spot size is diffraction
limited. As the input beam diameter increases, spherical aberration
starts to dominate the spot size
References
1. Greivenkamp, John E. Field Guide to Geometrical Optics. Vol. FG01.
Bellingham, WA: SPIE–The International Society for Optical Engineers,
2004.
2. Smith, Warren J. Modern Optical Engineering. 3rd ed. New York, NY:
McGraw-Hill Education, 2000.
6.4
6.5
6.6
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