Focal Spot Cross Section
Diffraction Limited Lens
As-Built Lens
1
0.7
0.5
0.4
0.3
0.2
0.1
Emitting Area
Fast axis
Slow axis
Figure 4.8: The geometry of laser diodes causes them to produce
elliptical beams with two different divergence angles
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BPP is commonly used to characterize fiber or semiconductor lasers
with large M2 factors, as well as diode-laser fiber-coupled systems for
determining the quantity of light that can be coupled into the fiber.
Section 4.4: Power in the Bucket
Power in the bucket (PIB) is another metric for defining beam quality,
and is often used in high power laser systems and materials processing
applications. PIB describes the laser power integrated over a specified
“bucket,” most often a spot of a specific radius at the surface being processed.
While this is a seemingly simple concept, the bucket shape in
the far field must be well defined and comparisons to ideal scenarios
depend on the specification of the ideal near field laser beam shape.
While there is no industry standard for the exact definition of PIB, it is
most often reported as either a vertical or horizontal beam quality:7
Similarly to the M2 factor and BPP, a lower PIB value corresponds with
a higher quality beam. The PIB can be visualized by plotting the fraction
of power in the defined “bucket” as a function of λ/D, where D is the
diameter of the near-field beam (Figure 4.6). The vertical beam quality is
the square root of the ratio of the fraction of power in the bucket of an
ideal Gaussian beam to that of the real beam at a given λ/D, which corresponds
to the vertical dimension of the plot. Similarly, the horizontal
beam quality is the ratio of the λ/D value of an ideal Gaussian beam to
that of the real beam at a given fraction of power in the bucket, which
corresponds to the horizontal dimension of the plot.
Section 4.5: Strehl Ratio
Just as the M2 factor compares the real performance of a laser to an
ideal beam, the Strehl ratio of an optical system or component compares
its real performance with an ideal version. The Strehl ratio of focusing
optics, including spherical and aspheric lenses, is the ratio of maximum
focal spot irradiance of the actual optic from a point source to the ideal
maximum irradiance from a theoretical diffraction-limited optic (Figure
4.7).8 A Strehl ratio of 1 would indicate that an optic is perfect and aberration
free. Common industry practice considers a lens “diffractionlimited”
when the Strehl ratio is greater than 0.8.
An optic’s Strehl ratio is approximately related to RMS transmitted wavefront
error by Equation 4.8, where S is the optic’s Strehl ratio and σ is
optic’s RMS wavefront error in waves.8 This approximation is valid when
the wavefront error is <0.2 waves.
For information on how an optic’s surface irregularity affects its Strehl
ratio see Aspheric Lens Irregularity and Strehl Ratio (pages 22-23).
Section 4.6: Circular vs. Elliptical Beams
When considering the shape of a laser beam, determining whether the
laser produces a circular or elliptical beam is important. Semiconductor
laser diodes emit elliptically-shaped beams with different divergence
angles in the x and y directions because of the rectangular shape of
its active region (Figure 4.8). Diffraction is greater for small apertures,
therefore the shorter dimension of the active region will produce a more
divergent beam and result in an astigmatic beam. The axis with the larger
divergence angle is defined as the fast axis, while the axis with the
smaller divergence angle is defined as the slow axis. Cylinder lenses are
often used to circularize elliptical beams (Figure 4.9). For more information
on cylinder lenses see Laser Beam Shaping (pages 16-19).
4.6
4.7
4.8
-2 -1.5 -1 -0.5 0 0.5 1 1.5
Relative
Irradiance
at y = 0m
X-Pos.
(m)
0.9
0.8
0.6
0
Figure 4.7: This lens has a Strehl ratio of 0.826, which is considered
diffraction limited because it is greater than 0.8
f2
f1
Cladding Layer
Active Layer
Cladding Layer
Figure 4.9: Cylinder lenses are often used to circularize an elliptical
beam by acting on the fast and slow axes separately
References
1. Paschotta, Rüdiger. Encyclopedia of Laser Physics and Technology,
RP Photonics, October 2017, www.rp-photonics.com/encyclopedia.html.
2. Paschotta Rüdiger. Field Guide to Lasers. SPIE Press, 2008.
3. International Organization for Standardization. (2005). Lasers and laser-
related equipment – Test methods for laser beam widths, divergence angles
and beam propagation ratios (ISO 11146).
4. A. E. Siegman, “New developments in laser resonators”, Proc. SPIE 1224,
2 (1990)
5. International Organization for Standardization. (2005). Lasers and laser-
related equipment — Test methods for laser beam widths, divergence
angles and beam propagation ratios — Part 1: Stigmatic and simple
astigmatic beams (ISO 11146-1:2005).
6. A. Siegman, “’Non-Gaussian’ Beam”, OSA Annual Meeting, Long Beach,
CA (1997)
7. Strehl, Karl W. A. “Theory of the telescope due to the diffraction of light,”
Leipzig, 1894.
8. Mahajan, Virendra N. "Strehl ratio for primary aberrations in terms of
their aberration variance." JOSA 73.6 (1983): 860-861.
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