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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: pco@fisica.ufpb.br

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

OCIS codes:

140.3295, 070.2580, 000.4430, 120.3940.

1.

The accurate measurement of the waist of a laser

beam near the focus of a lens is very important in

many applications [1], for instance, in a Z scan [2]

and thermal lens spectrometry [3]. Many techniques

have been developed with this purpose, such as the

slit scan technique [4,5] and the pinhole technique

[6]; 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 [10]. 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

Introduction

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

w2

?

;

ð1Þ

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 [9].

0003-6935/09/020393-04$15.00/0

© 2009 Optical Society of America

10 January 2009 / Vol. 48, No. 2 / APPLIED OPTICS393

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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

Rx

?∞

which gives

PN¼

?∞

R∞

R∞

?∞Iðx0;yÞdydx0

R∞

?∞Iðx0;yÞdydx0;

ð2Þ

PNðxÞ ¼1

2

?

1 þ erf

?x ? x0

w

??

;

ð3Þ

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 [8].

2.

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

Data Analysis

dPNðxÞ

dx

¼

1ffiffiffiπ

pwexp

?

?ðx ? x0Þ2

w2

?

:

ð4Þ

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 [9] suggested a substitution of PNðxÞ by an

analytical function, which approximately represents

PNðxÞ, to fit the experimental data. This fitting

function is given by

fðsÞ ¼

1

1 þ exp½pðsÞ?;

ð5Þ

where

pðsÞ ¼

X

m

i¼0

aisi;

ð6Þ

s ¼

ffiffiffi

2

p

ðx ? x0Þ

w

:

ð7Þ

For practical reasons Khosrofian and Garetz limited

the polynomial pðsÞ to the third order term, so that

fðsÞ ¼

1

1 þ expða0þ a1s þ a2s2þ a3s3Þ:

Using data from tabulated normal distribution

function and least-square analysis, the polynomial

coefficients were determined as

ð8Þ

a0¼ ?6:71387 × 10?3;

a1¼ ?1:55115;

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

[9] 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

Fig. 1.

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

Fig. 2.

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

Page 3

numerically converge to zero and the new nonnull

adjusted coefficients, up to the third order, are given

by

a1¼ ?1:597106847;

We thus may write Eq. (8) as

a3¼ ?7:0924013 × 10?2:

fðsÞ ¼

1

1 þ expða1s þ a3s3Þ:

ð9Þ

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 [15], 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

a1¼ ?1:5954086;

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,

Fig. 3.

fðsÞ defined by Eq. (9).

(Color online) Fitting the data obtained from Eq. (3) with

Fig. 4.

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

Page 4

making the biggest difference to be about 2 × 10?5in

the full range of interest.

3.

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

ffiffiffi

4.Conclusions

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

agenciesFinanciadoradeEstudoseProjetos(FINEP),

Conselho Nacional de Desenvolvimento Científico e

Tecnológico (CNPq), and Coordenação de Aperfeiçoa-

mento de Pessoal de Nível Superior (CAPES).

2

p

to arrive at the desired value.

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Fig. 5.

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