The best methods for making second-moment beam-width
measurements with CCD cameras are described and compared to ISO recommendations.
In the past quarter-century, CCD cameras have become popular tools to
image and quantify laser beams. During this period, both camera technology and
the analytical methods used to perform the most basic beam measurements have
seen many changes.
About 10 years into this evolution, the International
Organization for Standardization (ISO) published ISO standards 11145 and 11146.
ISO 11145 created a set of definitions for various laser beam terms, symbols,
and units of measurement. ISO 11146 described a method for measuring beam
widths, divergence, and the beam quality factor (㎡). This second standard
necessitated measuring laser beam widths according to the so-called
second-moment method, in which energy vs. distance from the centroid of the beam
is integrated to obtain a properly weighted beam width.
During these
early years, a number of different beam-width measurement methods were adopted
by the industry. Typically, these beam-width definitions were unique to various
laser manufacturers, system integrators, and end users. Those involved in the
development of laser-beam-analysis instruments would often incorporate
algorithms to meet customer needs even some that seemed dubious. Within a few
years, measuring beam widths using the second-moment method became most
significant. There already existed a number of second-moment "equivalent"
methods. Most common among these equivalent methods are:
ㆍ
13.5% of peak ㆍ 86.5% of total
power/energy ㆍ Knife edge based on set high/low clip levels
times a correction factor
However, these methods are only "correct" when
applied to Gaussian single-mode (TEM00) beams. Laser beams consisting of higher
modes are not as accurately measured with these methods. The advantage of these
equivalent methods is that they are compatible with CCD cameras. This means that
they are not highly susceptible to the main limitations of CCD cameras, which
are:
The output from early cameras was interlaced RS-170 or
CCIR monochrome analog video that required a well-designed video
digitizer/frame-grabber interface. The need for a frame grabber has been a
factor in the evolution of today's megapixel progressive scan digital cameras
with USB 3, FireWire, or Gig-E serial interface. Included in these improvements
are specific controls for adjusting the black level, gain, and other camera
features, plus higher signal-to-noise ratios that allow for 12- and 14-bit
outputs that digitize well into the video noise.
These improvements have
helped to make it easier for a beam analyzer to directly compute second-moment
beam widths. Two image-processing techniques are needed to achieve accurate
results.
First, the video black level needs to be computed with a high
degree of accuracy to establish a black baseline with preserved positive and
negative noise components. ISO/TR 11146-3:2004(E), section 3.3, describes how to
establish an accurate black-level baseline. This method was also developed by
Spiricon and was patented in the mid-1990s. Averaging a large number of frames
can resolve a baseline to per-pixel fractional counts and thus help to eliminate
fractional offset errors that are inherent in the quantization process.
Second, the region on the imager that contains the beam image needs to
be isolated from regions outside of the beam to remove the effects of small and
random fluctuations in the black baseline computed in the first step above. ISO
11146-1:2005(E), section 7.2, instructs that the integration integrals should be
carried out over an area limited to 3x the beam widths in the x and y
directions. This second requirement is due to the nature of the second-moment
integral, which contains a multiplying factor. This factor is the square of the
distance that a pixel lies from the centroid. A small positive or negative bias
over many thousands of pixels located at large distances from the beam centroid
can significantly impact the computed beam results. Likewise, small stray rays
in these distant regions can also have a dramatic impact on the final
results.
This spatial limiting process effectively apertures the beam
from the regions where the beam isn't. In the mid-1990s, before ISO addressed
this subject, we experimented with how to size this isolating aperture and
eventually incorporated it into our software as an auto-aperture feature. Our
results differed from the ISO recommendation; the balance of this article will
discuss how we came to this
conclusion.
Aperturing TEM00 beams is natural starting
point
All beam-analysis
algorithm studies naturally begin with the Gaussian TEM00 beam. It is the
easiest to model and good real-world near-TEM00 beams are available from
high-quality helium-neon (HeNe) lasers. In one method developed by us to quickly
assess the performance of a beam-measurement algorithm, a beam's intensity was
adjusted so that its peak was just below camera saturation. The beam intensity
was then reduced to see how well the beam-width measurement would track as the
signal-to-noise ratio degraded. Obviously, the measurement should begin at a
high accuracy and then remain reasonably stable as the peak intensity
drops.
Figure 1a shows a plot of a normalized x-axis second-moment (D4σX)
beam width vs. peak intensity for a HeNe laser without the use of an isolating
aperture. The camera is a high-quality 2 Mpixel CCD with a signal-to-noise ratio
of 61 dB root-mean-squared (dBrms). The baseline is normalized; however, some
small positive offset is present and causing the results to be more than 10%
larger than the actual beam size. In this case the offset is positive, but could
just as likely be negative. The positive offset is likely caused by the camera
imager still warming up. This points to the need not only for aperturing, but
also for good temperature stability when making long-term measurements.
Figure 1b is a plot of a
normalized D4σX beam width for a modeled TEM00 beam with both a 2x and a 3x
aperture. The model is replicating a 12-bit camera with a signal-to-noise ratio
of 60 dBrms. These specifications are typical for a modern good-quality CCD
camera. Both positive and negative noise is preserved and a +½ count positive
offset has been added to the baseline to simulate a typical camera's short-term
fluctuation. The fluctuations can go both positive and negative, but these plots
will only show a positive shift. Flipping the data around the 100% axis would
show how things look with a -½ count bias.
A plot of a modeled TEM01*
(㎡=2, donut) beam with both a 2x and a 3x aperture is shown in Fig. 1c. It can
be seen in Figs. 1b and 1c that the 2x aperture yields a more-accurate result
compared to the 3x aperture. However, both the 2x and 3x apertures yield
reasonably good accuracy (under +5%) with intensity as low as 15% of peak. This
indicates that either aperture size would perform well as long as the dark field
near the beam is clear of stray background reflections or pump
glow.
Figure 2a is a plot of a modeled higher-order Laguerre TEM24 beam
(㎡=9). The 3x aperture is now growing quite large compared to the actual area of
the beam. The accuracy is now down to 5% at 50% intensity for the 3x aperture,
while the 2x shows good accuracy all the way down to 10% intensity.
Figures 2b and 2c show two
different Laguerre beams, both with an ㎡=16. They are configured as TEM55 and
TEM47, respectively. Beam accuracy for the 3x aperture now begins to degrade,
even at quite-high beam intensities. The 3x aperture leaves very large no-signal
regions around the beam profiles. These enclosed no-signal regions can yield
large measurement fluctuations as the camera baseline noise bounces in the
background and invites added errors for any stray light in these
areas.
Figure 3 shows a collection of other modes with both 2x and a 3x
apertures. The larger 3x empty regions around many of the higher-order modes are
an invitation for offset errors and stray light to degrade accuracy. In many
cases, even a 2x aperture can invite trouble, although always less trouble than
a 3x aperture.
Three additional modes with 2x and 3x
aperture FIGURE 3. Three additional modes with 2x and 3x apertures are shown:
Laguerre TEM10 (a), Laguerre TEM21 (b), and Hermite TEM11 (c).
We chose to only model
beams with individual modes for simplicity. In reality, few real lasers contain
just a pure higher mode. It has been demonstrated that beams build higher modes
in a more or less systematic way based on the design of the lasing cavity. Thus,
as the higher modes increase, the structures become even more complex than what
is demonstrated above. As the modal complexity increases, the beam widths
increase relative to the area the beam covers. As a result, any multiplied
aperture increases even faster.
2x aperture yields better
results
What we have looked for in this article is a simple rule
that can be applied for isolating and accurately calculating the second-moment
beam widths of laser beams imaged on CCD cameras. While ISO recommends a 3x
isolating region, we have found that a 2x aperture yields better results over a
wider range of real laser-beam measurement conditions.
There are other
approaches that could be used to customize an isolating aperture based on the
unique nature of the beam being measured. Other techniques have been discussed,
both in and out of print. Some require complex iterative processes to find the
"best" size, independent of any multiplication rule. We have experimented with
some iterative processes in the past and find that they consume more
computational power and do not yield significantly superior results, which isn't
to say that perhaps a better mousetrap-or rather "beam trap"-doesn't
exist.
Today's laser-beam analyzer programs are loaded with features and
are thus large and complex pieces of software. Many features, all attempting to
run in real time, compete for processing time. Frequently, third-party
applications are also running and demanding processor overhead. As a result, the
notion of keeping things simple is compelling.
REFERENCES
1. C. B. Roundy, "Techniques for
accurately measuring laser beam width with commercial CCD cameras," Proc. SPIE,
3405, ROMOPTO '97: Fifth Conference on Optics, 1045 (Jul. 2, 1998); http://dx.doi.org/10.1117/12.312711.
2. 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" (Jan. 15, 2005).
3. ISO/TR 11146-3:2004(E),
"Lasers and laser-related equipment-Test methods for laser beam widths,
divergence angles and beam propagation ratios-Part 3: Intrinsic and geometrical
laser beam classification, propagation and details of test methods" (Feb. 1,
2004).
4. ISO/TR 11146-3:2004/Cor.1:2005(E), "Lasers and laser-related
equipment-Test methods for laser beam widths, divergence angles and beam
propagation ratios-Part 3: Intrinsic and geometrical laser beam classification,
propagation and details of test methods" (Feb. 15, 2005).
5. A. E.
Siegman, "How to (maybe) measure laser beam quality," Proc. DLAI, paper MQ1
(1998).
6. A. E. Siegman, M. W. Sasnett, and T. F. Johnston, IEEE J.
Quantum Electron., 27, 4, 1098?1104 (Apr. 1991).
Gregory E. Slobodzian
is director of engineering (retired) at Ophir-Spiricon, North Logan, UT; e-mail:
gslobodzian@msn.com; www.ophiropt.com.
