Measurement of Gaussian laser beam radius using the
knife-edge technique: improvement on data analysis
Marcos A. C. de Araújo,1Rubens Silva,1,2Emerson de Lima,1Daniel P. Pereira,1,3
and Paulo C. de Oliveira1,*
1Departamento de Física, Universidade Federal da Paraíba, João Pessoa 58051-970, Paraíba, Brazil
2Faculdade de Física, ICEN, Universidade Federal do Pará, Belém 66075-110, Pará, Brazil
3Centro Federal de Educação Tecnológica do Pará, Belém 66093-020, Pará, Brazil
*Corresponding author: firstname.lastname@example.org
Received 13 October 2008; accepted 17 November 2008;
posted 26 November 2008 (Doc. ID 102613); published 8 January 2009
We revisited the well known Khosrofian and Garetz inversion algorithm [Appl. Opt. 22, 3406–3410
(1983)] that was developed to analyze data obtained by the application of the traveling knife-edge tech-
nique. We have analyzed the approximated fitting function that was used for adjusting their experimen-
tal data and have found that it is not optimized to work with a full range of the experimentally-measured
data. We have numerically calculated a new set of coefficients, which makes the approximated function
suitable for a full experimental range, considerably improving the accuracy of the measurement of a
radius of a focused Gaussian laser beam.© 2009 Optical Society of America
140.3295, 070.2580, 000.4430, 120.3940.
The accurate measurement of the waist of a laser
beam near the focus of a lens is very important in
many applications , for instance, in a Z scan 
and thermal lens spectrometry . Many techniques
have been developed with this purpose, such as the
slit scan technique [4,5] and the pinhole technique
; but among the most used is the knife-edge
technique [7–9]. The knife-edge technique is a beam
profiling method that allows for quick, inexpensive,
and accurate determination of beam parameters.
The knife-edge technique has been widely used for
decades and is considered a standard technique for
Gaussian laser beam characterization . In this
technique a knife edge moves perpendicular to the
direction of propagation of the laser beam, and the
total transmitted power is measured as a function
of the knife-edge position. A typical experimental
setup is shown in Fig. 1. The knife-edge technique
requires a sharp edge (typically a razor blade), a
translation stage with a micrometer, and a power
meter or an energy meter when working with pulses.
In our discussion we consider a radially symmetric
Gaussian laser beam with intensity described by
Iðx;yÞ ¼ I0exp
?ðx ? x0Þ2þ ðy ? y0Þ2
where I0is the peak intensity at the center of the
beam, located at ðx0;y0Þ, x and y are the transverse
Cartesian coordinates of any point with respect to an
origin conveniently chosen at the beginning of an ex-
periment, and w is the beam radius, measured at a
position where the intensity decreases to 1=e times
its maximum value I0. Equation (1) is not the only
way to express the intensity of a Gaussian laser
beam. Some authors prefer to define the beam radius
at a position where the electric field amplitude drops
to 1=e, while the intensity drops to 1=e2times the
maximum value. Our choice in the definition of the
intensity follows the choice made by Khosrofian
and Garetz .
© 2009 Optical Society of America
10 January 2009 / Vol. 48, No. 2 / APPLIED OPTICS393
With the knife-edge initially blocking the laser
beam, the micrometer can be adjusted in appropriate
increments, and the normalized transmitted power
is obtained by the integral
1 þ erf
?x ? x0
where erf is the error function.
The area of the photodiode is considered to be lar-
ger than the area of the laser beam cross section at
the detection position; therefore, diffraction effects
may be neglected. The large-area photodiode may
be substituted by a small-area photodiode coupled
to an integrating sphere .
The error function in Eq. (3) is not an analytical func-
tion and its use in fitting experimental data is not a
practical procedure. One approach in data analysis is
to work with the derivative of Eq. (3) [7,11,12], which
is analytical and is given by
?ðx ? x0Þ2
But the process of taking derivatives of experimental
data with fluctuations results in amplification of the
fluctuations and, consequently, an increase in the
errors. To overcome this problem, Khosrofian and
Garetz  suggested a substitution of PNðxÞ by an
analytical function, which approximately represents
PNðxÞ, to fit the experimental data. This fitting
function is given by
1 þ exp½pðsÞ?;
ðx ? x0Þ
For practical reasons Khosrofian and Garetz limited
the polynomial pðsÞ to the third order term, so that
1 þ expða0þ a1s þ a2s2þ a3s3Þ:
Using data from tabulated normal distribution
function and least-square analysis, the polynomial
coefficients were determined as
a0¼ ?6:71387 × 10?3;
a2¼ ?5:13306 × 10?2;
a3¼ ?5:49164 × 10?2:
Although this fitting function has been used for
decades and referenced by many authors [13,14],
we decided to compare it with the exact function,
given by Eq. (3). The first step in the comparison pro-
cess was to plot the equations within a single gra-
phic. The result is shown in Fig. 2. We verified
that the fitting function presents a very good adjust-
ment for fðsÞ > 0:5 but fails to adjust for fðsÞ < 0:5.
This result is a consequence of the procedure that
has been employed to fit fðsÞ to the data points be-
cause the parameters that define fðsÞ have been de-
termined from tabulated normal data with positive
arguments only. To extend the procedure to include
negative arguments of fðsÞ, Khosrofian and Garetz
 assumed that fð?sÞ ¼ 1 ? fðsÞ. But since fðsÞ con-
tains pðsÞ, which is a polynomial that includes terms
of even powers of s, this assumption is not valid.
Considering the symmetry of the error function,
the fitting function fðsÞ must contains only terms
of odd powers of s. In fact, a fitting of fðsÞ to the exact
data, given by Eq. (3), shows that a0 and a2
radius using the knife-edge technique. The gray color area repre-
sents the shadow caused by the knife edge.
Simplified scheme for the measurement of laser beam
Eq. (3) with fðsÞ defined by Eq. (8).
(Color online) Comparison of the data obtained from
394APPLIED OPTICS / Vol. 48, No. 2 / 10 January 2009
numerically converge to zero and the new nonnull
adjusted coefficients, up to the third order, are given
We thus may write Eq. (8) as
a3¼ ?7:0924013 × 10?2:
1 þ expða1s þ a3s3Þ:
To arrive at these new coefficients we have gener-
ated a set of points directly from Eq. (3) with x0¼ 0
and w ¼ 1 by using Maple 10, and with the help of
Origin 7.5, we fit the data set with Eq. (8). The fitting
procedure was to keep x0and w fixed, while allowing
the coefficients to vary. The result is shown in Fig. 3.
By fitting the same simulated data set with fðsÞ
given by Eq. (8) with the old coefficients, the obtained
values for x0and w were 0:0132 and 0:9612, respec-
tively. This corresponds to a difference of about 3.9%
in the laser beam radius, and the error in the center
position, relative to the beam radius, of about 1.3%.
These differencesmay represent a serious problem in
high accuracy experiments. For example, since the
laser intensity is inversely proportional to the square
of the radius, an overestimation of about 7.6% of the
laser intensity will result, if Eq. (8) is used, as the
fitting function. On the other hand, an estimation
of the error in w and x0give values in a range of
10?7–10?8when fitting Eq. (9) to the exact function,
given by Eq. (3). With these results we may say that
Eq. (9) is not only a good approximation for our par-
ticular problem, but it may also be useful in many
numerical problems in different fields of science in-
volving the error function. As an example of the use
of analytical expressions for the error function in an-
other physical problem, we may refer to the work of
Van Halen , which was used to calculate the elec-
tric field and potential distribution in semiconductor
junctions with a Gaussian doping profile.
The inclusion of the fifth order term in the polyno-
mial pðsÞ will further improve the accuracy but is not
worth doing in an analysis of the knife-edge techni-
que data, where the experimental fluctuations dom-
inate the errors in the data analysis. However, since
the focus of our discussion is on the improvement of
data analysis and the possible use of this fitting
function in different kinds of problems, we extend
our discussion to analyze the behavior of fðsÞ when
the fifth order term is included. The first annotation
about the inclusion of the fifth order term a5in the
polynomial pðsÞ is that it will require a recalculation
of all the coefficients; therefore a1and a3will change.
The new calculated coefficients are given by
a3¼ ?7:3638857 × 10?2;
a5¼ þ6:4121343 × 10?4:
To verify how close the approximated functions are
from the exact function PNðxÞ, we have plotted the
differences between fðsÞ and PNðxÞ for ðx ? x0Þ=w
ranging from -4.0 to 4.0, covering the full range of in-
terest. In Fig. 4(a), fðsÞ, given by Eq. (8), was used in
two different ways: with the parameters w ¼ 1:0 and
x0¼ 0:0 (solid line), and w ¼ 0:9612 and x0¼ 0:0132
(dashed line), obtained when one tries to fit PNðxÞ
with fðsÞ. In Fig. 4(b), the differences are calculated
with fðsÞ given by Eq. (9) in two ways: where only the
coefficients a1and a3are considered (solid line), and
when the new set of coefficients that includes a5is
considered (dashed line).
By analyzing the curves shown in Fig. 4, we may
conclude that the approximated function fðsÞ defined
by Eq. (9) is, on average, two orders of magnitude clo-
ser to the exact function PNðxÞ than that defined by
Eq. (8). When the fifth order term is included in the
polynomial pðsÞ, the approximation is even better,
fðsÞ defined by Eq. (9).
(Color online) Fitting the data obtained from Eq. (3) with
is given by Eq. (8) with the parameters w ¼ 1:0 and x0¼ 0:0 (solid
line) and w ¼ 0:9612 and x0¼ 0:0132 (dashed line). (b) fðsÞ is given
by Eq. (9) when only the coefficients a1and a3are considered
(solid line), and when the new set of coefficients that includes
a5is considered (dashed line).
(Color online) Differences between fðsÞ and PNðxÞ. (a) fðsÞ
10 January 2009 / Vol. 48, No. 2 / APPLIED OPTICS395
making the biggest difference to be about 2 × 10?5in
the full range of interest.
To verify how the choice of the fitting function inter-
feres in the true experimental data analysis, we per-
formed a simple experiment using the setup shown
in Fig. 1. In our experiment a He–Ne laser with
an output power of 10mW was focused by a 25cm
focal length lens. A razor blade was mounted on
top of a motorized translation stage made by New-
port (model M-UTM150PP.1) with a resolution of
0:1μm. The translation stage position was controlled
by a computer while the total transmitted laser
power was measured by an Ophir NOVA power me-
ter. The analog output signal of the power meter was
sent to the computer through a National Instru-
ments USB-6000 acquisition card. We set the speed
of the translation stage at 0:5mm=s and the acquisi-
tion rate at 100samples=s. The experimental data,
taken at a position near the focus of the lens, is
shown in Fig. 5, where we also show a fitting of
the experimental data with Eq. (9). The same fitting
was done with Eq. (8) and, although both equations
give rise to curves that apparently are representative
of the experimental data, they result in different va-
lues for the laser beam radius. After analyzing 10
scans, fitting each data set with Eq. (9), we arrived
at the mean value w ¼ 36:60 ? 0:06μm. A result
3.8% lower than this is obtained if one tries to fit
the same experimental data with Eq. (8). This con-
firms the necessity of using the correct fitting func-
tion to analyze the experimental data. If we now
compare the position of the beam center, given by
the two fitting functions, we find a difference, rela-
tive to the radius, of 1.2% between the results. Since
the type of errors introduced by the use of Eq. (8) is
systematic, past results on laser beam radius may be
corrected by using a multiplying factor of 1:04.
If one defines the radius of the laser beam at a
position where the intensity drops to 1=e2times
Analysis of Experimental Data
the maximum value, one needs to multiply w by
We have shown that a modified sigmoidal function,
based on the Khosrofian and Garetz function, with
new coefficients is needed for correct laser beam
characterization in the knife-edge technique. We
have found these new coefficients and showed that
the new function fits the experimental data very well
and improves the accuracy of the results.
We thank the financial support from the Brazilian
Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), and Coordenação de Aperfeiçoa-
mento de Pessoal de Nível Superior (CAPES).
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Eq. (9). A similar curve is obtained by using Eq. (8), but with
the adjusted laser beam radius 3:8% lower.
(Color online) Fitting of the experimental data using
396APPLIED OPTICS / Vol. 48, No. 2 / 10 January 2009