Measurement of Gaussian laser beam radius using theknife-edge technique: improvement on data analysis

Marcos A. C. de Araújo,1 Rubens Silva,1,2 Emerson de Lima,1 Daniel 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, Brazil3Centro 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-measureddata. We have numerically calculated a new set of coefficients, which makes the approximated functionsuitable for a full experimental range, considerably improving the accuracy of the measurement of aradius of a focused Gaussian laser beam. © 2009 Optical Society of America

OCIS codes: 140.3295, 070.2580, 000.4430, 120.3940.

1. Introduction

The accurate measurement of the waist of a laserbeam near the focus of a lens is very important inmany applications [1], for instance, in a Z scan [2]and thermal lens spectrometry [3]. Many techniqueshave been developed with this purpose, such as theslit scan technique [4,5] and the pinhole technique[6]; but among the most used is the knife-edgetechnique [7–9]. The knife-edge technique is a beamprofiling method that allows for quick, inexpensive,and accurate determination of beam parameters.The knife-edge technique has been widely used fordecades and is considered a standard technique forGaussian laser beam characterization [10]. In thistechnique a knife edge moves perpendicular to thedirection of propagation of the laser beam, and thetotal transmitted power is measured as a functionof the knife-edge position. A typical experimentalsetup is shown in Fig. 1. The knife-edge technique

requires a sharp edge (typically a razor blade), atranslation stage with a micrometer, and a powermeter or an energy meter when working with pulses.

In our discussion we consider a radially symmetricGaussian laser beam with intensity described by

Iðx; yÞ ¼ I0 exp��ðx� x0Þ2 þ ðy� y0Þ2

w2

�; ð1Þ

where I0 is the peak intensity at the center of thebeam, located at ðx0; y0Þ, x and y are the transverseCartesian coordinates of any point with respect to anorigin conveniently chosen at the beginning of an ex-periment, and w is the beam radius, measured at aposition where the intensity decreases to 1=e timesits maximum value I0. Equation (1) is not the onlyway to express the intensity of a Gaussian laserbeam. Some authors prefer to define the beam radiusat a position where the electric field amplitude dropsto 1=e, while the intensity drops to 1=e2 times themaximum value. Our choice in the definition of theintensity follows the choice made by Khosrofianand Garetz [9].

0003-6935/09/020393-04$15.00/0© 2009 Optical Society of America

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With the knife-edge initially blocking the laserbeam, the micrometer can be adjusted in appropriateincrements, and the normalized transmitted poweris obtained by the integral

PN ¼Rx�∞

R∞

�∞

Iðx0; yÞdydx0R∞

�∞

R∞

�∞

Iðx0; yÞdydx0 ; ð2Þ

which gives

PNðxÞ ¼12

�1þ erf

�x� x0w

��; 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 atthe detection position; therefore, diffraction effectsmay be neglected. The large-area photodiode maybe substituted by a small-area photodiode coupledto an integrating sphere [8].

2. Data Analysis

The error function in Eq. (3) is not an analytical func-tion and its use in fitting experimental data is not apractical procedure. One approach in data analysis isto work with the derivative of Eq. (3) [7,11,12], whichis analytical and is given by

dPNðxÞdx

¼ 1ffiffiffiπpwexp

��ðx� x0Þ2

w2

�: ð4Þ

But the process of taking derivatives of experimentaldata with fluctuations results in amplification of thefluctuations and, consequently, an increase in theerrors. To overcome this problem, Khosrofian andGaretz [9] suggested a substitution of PNðxÞ by ananalytical function, which approximately representsPNðxÞ, to fit the experimental data. This fittingfunction is given by

f ðsÞ ¼ 11þ exp½pðsÞ� ; ð5Þ

where

pðsÞ ¼Xmi¼0

aisi; ð6Þ

s ¼ffiffiffi2

p ðx� x0Þw

: ð7Þ

For practical reasons Khosrofian and Garetz limitedthe polynomial pðsÞ to the third order term, so that

f ðsÞ ¼ 1

1þ expða0 þ a1sþ a2s2 þ a3s3Þ: ð8Þ

Using data from tabulated normal distributionfunction and least-square analysis, the polynomialcoefficients were determined as

a0 ¼ �6:71387 × 10�3; a1 ¼ �1:55115;

a2 ¼ �5:13306 × 10�2;

a3 ¼ �5:49164 × 10�2:

Although this fitting function has been used fordecades 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 verifiedthat the fitting function presents a very good adjust-ment for f ðsÞ > 0:5 but fails to adjust for f ðsÞ < 0:5.This result is a consequence of the procedure thathas been employed to fit f ðsÞ to the data points be-cause the parameters that define f ðsÞ have been de-termined from tabulated normal data with positivearguments only. To extend the procedure to includenegative arguments of f ðsÞ, Khosrofian and Garetz[9] assumed that f ð�sÞ ¼ 1� f ðsÞ. But since f ðsÞ con-tains pðsÞ, which is a polynomial that includes termsof even powers of s, this assumption is not valid.Considering the symmetry of the error function,the fitting function f ðsÞ must contains only termsof odd powers of s. In fact, a fitting of f ðsÞ to the exactdata, given by Eq. (3), shows that a0 and a2

Fig. 1. Simplified scheme for the measurement of laser beamradius using the knife-edge technique. The gray color area repre-sents the shadow caused by the knife edge.

Fig. 2. (Color online) Comparison of the data obtained fromEq. (3) with f ðsÞ defined by Eq. (8).

394 APPLIED OPTICS / Vol. 48, No. 2 / 10 January 2009

numerically converge to zero and the new nonnulladjusted coefficients, up to the third order, are givenby

a1 ¼ �1:597106847; a3 ¼ �7:0924013 × 10�2:

We thus may write Eq. (8) as

f ðsÞ ¼ 1

1þ expða1sþ a3s3Þ: ð9Þ

To arrive at these new coefficients we have gener-ated a set of points directly from Eq. (3) with x0 ¼ 0and w ¼ 1 by using Maple 10, and with the help ofOrigin 7.5, we fit the data set with Eq. (8). The fittingprocedure was to keep x0 and w fixed, while allowingthe coefficients to vary. The result is shown in Fig. 3.By fitting the same simulated data set with f ðsÞ

given by Eq. (8) with the old coefficients, the obtainedvalues for x0 and 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 centerposition, relative to the beam radius, of about 1.3%.These differencesmay represent a serious problem inhigh accuracy experiments. For example, since thelaser intensity is inversely proportional to the squareof the radius, an overestimation of about 7.6% of thelaser intensity will result, if Eq. (8) is used, as thefitting function. On the other hand, an estimationof the error in w and x0 give values in a range of10�7

–10�8 when fitting Eq. (9) to the exact function,given by Eq. (3). With these results we may say thatEq. (9) is not only a good approximation for our par-ticular problem, but it may also be useful in manynumerical problems in different fields of science in-volving the error function. As an example of the useof analytical expressions for the error function in an-other physical problem, we may refer to the work ofVan Halen [15], which was used to calculate the elec-tric field and potential distribution in semiconductorjunctions with a Gaussian doping profile.

The inclusion of the fifth order term in the polyno-mial pðsÞ will further improve the accuracy but is notworth doing in an analysis of the knife-edge techni-que data, where the experimental fluctuations dom-inate the errors in the data analysis. However, sincethe focus of our discussion is on the improvement ofdata analysis and the possible use of this fittingfunction in different kinds of problems, we extendour discussion to analyze the behavior of f ðsÞ whenthe fifth order term is included. The first annotationabout the inclusion of the fifth order term a5 in thepolynomial pðsÞ is that it will require a recalculationof all the coefficients; therefore a1 and a3 will change.The new calculated coefficients are given by

a1 ¼ �1:5954086; a3 ¼ �7:3638857 × 10�2;

a5 ¼ þ6:4121343 × 10�4:

To verify how close the approximated functions arefrom the exact function PNðxÞ, we have plotted thedifferences between f ðsÞ and PNðxÞ for ðx� x0Þ=wranging from -4.0 to 4.0, covering the full range of in-terest. In Fig. 4(a), f ðsÞ, given by Eq. (8), was used intwo different ways: with the parameters w ¼ 1:0 andx0 ¼ 0:0 (solid line), and w ¼ 0:9612 and x0 ¼ 0:0132(dashed line), obtained when one tries to fit PNðxÞwith f ðsÞ. In Fig. 4(b), the differences are calculatedwith f ðsÞ given by Eq. (9) in two ways: where only thecoefficients a1 and a3 are considered (solid line), andwhen the new set of coefficients that includes a5 isconsidered (dashed line).

By analyzing the curves shown in Fig. 4, we mayconclude that the approximated function f ðsÞ definedby Eq. (9) is, on average, two orders of magnitude clo-ser to the exact function PNðxÞ than that defined byEq. (8). When the fifth order term is included in thepolynomial pðsÞ, the approximation is even better,

Fig. 3. (Color online) Fitting the data obtained from Eq. (3) withf ðsÞ defined by Eq. (9).

Fig. 4. (Color online) Differences between f ðsÞ and PNðxÞ. (a) f ðsÞis given by Eq. (8) with the parameters w ¼ 1:0 and x0 ¼ 0:0 (solidline) andw ¼ 0:9612 and x0 ¼ 0:0132 (dashed line). (b) f ðsÞ is givenby Eq. (9) when only the coefficients a1 and a3 are considered(solid line), and when the new set of coefficients that includesa5 is considered (dashed line).

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making the biggest difference to be about 2 × 10�5 inthe full range of interest.

3. Analysis of Experimental Data

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 shownin Fig. 1. In our experiment a He–Ne laser withan output power of 10mW was focused by a 25 cmfocal length lens. A razor blade was mounted ontop of a motorized translation stage made by New-port (model M-UTM150PP.1) with a resolution of0:1 μm. The translation stage position was controlledby a computer while the total transmitted laserpower was measured by an Ophir NOVA power me-ter. The analog output signal of the power meter wassent to the computer through a National Instru-ments USB-6000 acquisition card. We set the speedof the translation stage at 0:5mm=s and the acquisi-tion rate at 100 samples=s. The experimental data,taken at a position near the focus of the lens, isshown in Fig. 5, where we also show a fitting ofthe experimental data with Eq. (9). The same fittingwas done with Eq. (8) and, although both equationsgive rise to curves that apparently are representativeof the experimental data, they result in different va-lues for the laser beam radius. After analyzing 10scans, fitting each data set with Eq. (9), we arrivedat the mean value w ¼ 36:60� 0:06 μm. A result3.8% lower than this is obtained if one tries to fitthe 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 nowcompare the position of the beam center, given bythe two fitting functions, we find a difference, rela-tive to the radius, of 1.2% between the results. Sincethe type of errors introduced by the use of Eq. (8) issystematic, past results on laser beam radius may becorrected 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=e2 times

the maximum value, one needs to multiply w byffiffiffi2

pto arrive at the desired value.

4. Conclusions

We have shown that a modified sigmoidal function,based on the Khosrofian and Garetz function, withnew coefficients is needed for correct laser beamcharacterization in the knife-edge technique. Wehave found these new coefficients and showed thatthe new function fits the experimental data very welland improves the accuracy of the results.

We thank the financial support from the BrazilianagenciesFinanciadoradeEstudoseProjetos (FINEP),Conselho Nacional de Desenvolvimento Científico eTecnológico (CNPq), and Coordenação de Aperfeiçoa-mento de Pessoal de Nível Superior (CAPES).

References

1. S. Nemoto, “Determination of waist parameters of a Gaussianbeam,” Appl. Opt. 25, 3859–3863 (1986).

2. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W.Van Stryland, “Sensitive measurement of optical nonlineari-ties using a single beam,” IEEE J. Quantum Electron. 26,760–769 (1990).

3. M. L. Baesso, J. Shen, and R. D. Snook, ”Mode-mismatchedthermal lens determination of temperature coefficient of opti-cal path length in soda lime glass at different wavelengths, J.Appl. Phys. 75, 3732–3737 (1994).

4. R. L. McCally, “Measurement of Gaussian beam parameters,”Appl. Opt. 23, 2227 (1984).

5. P. B. Chapple, “Beam waist and M2 measurement using afinite slit,” Opt. Eng. 33 2461–2466 (1994).

6. P. J. Shayler, “Laser beam distribution in the focal region,”Appl. Opt. 17, 2673–2674 (1978).

7. J.A.Arnaud,W.M.Hubbard,G.D.Mandeville,B.delaClavière,E. A. Franke, and J. M. Franke, “Technique for fast measure-ment of Gaussian laser beam parameters,” Appl. Opt. 10,2775–2776 (1971).

8. D. R. Skinner and R. E. Whitcher, “Measurement of the radiusof a high-power laser beam near the focus of a lens,” J. Phys. E5, 237–238 (1972).

9. J.M.KhosrofianandB.A.Garetz, “Measurement of aGaussianlaser beamdiameter through the direct inversion of knife-edgedata,” Appl. Opt. 22, 3406–3410 (1983).

10. D. Wright, P. Greve, J. Fleischer, and L. Austin, “Laser beamwidth, divergence and beam propagation factor: an interna-tional standardization approach,” Opt. Quantum Electron.24, S993–S1000 (1992).

11. M. Mauck, “Knife-edge profiling of Q-switched Nd:YAG laserbeam and waist,” Appl. Opt. 18, 599–600 (1979).

12. R. M. O’Connell and R. A. Vogel, “Abel inversion of knife-edgedata from radially symmetric pulsed laser beams,” Appl. Opt.26, 2528–2532 (1987).

13. Z. A. Talib and W. M. M. Yunus, “Measuring Gaussian laserbeam diameter using piezoelectric detection,” Meas. Sci.Technol. 4, 22–25 (1993).

14. L. Bachmann, D. M. Zezell, and E. P. Maldonado, “Determina-tion of beam width and quality for pulsed lasers using theknife-edge method,” Instrum. Sci. Technol. 31, 47–52 (2003).

15. P. Van Halen, “Accurate analytical approximations for errorfunction and its integral,” Electron. Lett. 25, 561–563(1989).

Fig. 5. (Color online) Fitting of the experimental data usingEq. (9). A similar curve is obtained by using Eq. (8), but withthe adjusted laser beam radius 3:8% lower.

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