April 1, 1994 / Vol. 19, No. 7 / OPTICS LETTERS 505

Method for direct measurement of the variance of laser beams

Yves Champagne, Claude Par6, and Pierre-Andr6 BMlanger

Jquipe Laser et Optique Guidde, D6partement de Physique, Centre d'Optique, Photonique et Laser,Universit6 Laval, Qudbec GlK 7P4, Canada

Received November 18, 1993

A method permitting direct measurement of the spatial variance along a given transverse direction of any laserbeam is described. The procedure relies on the use of a transmission filter whose local transmissivity variesspatially as an inverted parabola. The variance and the first moment of a transverse beam intensity distributionare readily obtained simply by measurement of the maximum power transmitted through the filter as this filteris moved across the beam's section.

The half-diameter of an arbitrary laser beam in-tensity distribution I(x, y) along one transverse di-rection, say, the x direction, can be defined by thequantity' Wx = 2J17, referred to here as the real-beam size. a-2 represents the spatial variance ofthe beam intensity distribution in a given transverseplane, that is,

Ox2 = (X2) -(x, (1)

where the second moment (x2) along the x directionis given by

(x 2 ) = Af 4 x 2 1(x, y)dxdy, (2)

with+x0 +00

Po = f f I(x, y)dxdy. (3)

The expression for the first moment (x) is similar toEq. (2), because the quadratic term x2 is simply re-placed by x. In the case of an ideal TEM0o Gaussianbeam, Wx(z) corresponds exactly to the Gaussian spotsize parameter wx(z). There are several benefits indefining transverse beam widths in this way. Thisdefinition is simple and does not make use of numer-ical criteria such as prefixed percentages of the totalbeam's power. In addition, the real-beam size ac-counts for the whole two-dimensional transverse in-tensity distribution, even if the width is defined infact for a single transverse direction. But the greatdeal of attention paid to this definition originatesfrom the simple quadratic propagation law for W 2(z)in free space,' valid for any laser beam independent ofits exact transverse shape. This definition forms thegroundwork of the method of moments2 from whichrecent developments have resulted in an elegant the-oretical body in which the well-known formulas gov-erning the propagation of Gaussian beam parametersin free space and through optical ABCD systems aregeneralized to non-Gaussian beams. These develop-ments rely mainly on the concepts of the beam quality

factor1 and of the effective radius of curvature 3- 5 ofthe beam's transverse phase distribution.

The measurement of the real-beam size in a giventransverse plane requires prior knowledge of thebeam transverse intensity distribution in that plane,obtained, for example, by use of a scanning slit ora CCD camera. Furthermore, a computer is neededfor carrying out preparation and numerical integra-tion of the recorded experimental data. As a result,the measurement of W, is more complicated and timeconsuming compared with most usual ways of mea-suring laser beam diameters, such as the variable-aperture method, the knife-edge technique, and thescanning slit.6

In this Letter we propose a method for measuringthe real-beam size of a beam intensity distributionthat requires neither measurement of intensity dis-tributions nor numerical integration of experimentaldata. The method is simple and relies primarily onthe use of a filter having a spatially varying transmis-sivity. Indeed, Eq. (2) for the second moment bearsa strong resemblance to the expression of the nor-malized total power carried by a laser beam after itstransmission through a filter having a parabolic vari-ation of its transmissivity along the x direction. Asdiscussed below, graded-reflectivity mirrors (GRM's),now available on a commercial basis, can act as suchparabolic transmission filters (PTF's).

The theoretical demonstration of the method startsfrom Fig. 1, which depicts a one-dimensional sym-metrical transmission profile having a maximumvalue Tmax at the center position XM. The refer-ence frame corresponds here to that in which theincident beam intensity distribution is defined. Thespacing between the minimum values Tmjn definesthe full width 2a of the PTF, and the transmissiv-ity is assumed to be null outside the parabolic re-gion. We suppose also that the PTF is mounted ona translation stage, permitting the displacement ofthe transmission profile along the x direction. Thetransmission profile shown in Fig. 1 varies accordingto

(Tmax - Tmin)IT(x) = Tmax .a (x - XM) (4

for XM - a ' x ' XM + a. The analysis simplifies

0146-9592/94/070505-03$6.00/0 © 1994 Optical Society of America

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506 OPTICS LETTERS / Vol. 19, No. 7 / April 1, 1994

T(x)

Tmin

0

XM , I >

xFig. 1. Transmission profile considered in the analysis.The transmissivity varies quadratically within the regionof width 2a and is assumed to be zero outside.

considerably if we assume at the outset that thewidth 2a of the transmission profile is much largerthan the beam's width, so that the fraction of thetotal beam's power falling outside the parabolic re-gion can be considered negligible. This assumptionallows one to write the following integral with un-bounded limits for the beam's power P after it passesthrough the PTF:

+xf +T

P = f f T(X)I(x, y)dxdy, (5)

where I(x, y) is the optical intensity distribution inci-dent upon the PTF. Substituting Eq. (4) into Eq. (5),one finds that

-r+X +w

P = Tmaxpo - (Tma 2- LTmin) x21(x, y)dxdya2o x

- 2XM f f xI(x, y)dxdy + XM2PO

In addition, the first moment of the beam intensitydistribution along the x direction is automaticallylocated.

As stated above, in this demonstration we assumedthat the transmission profile is much wider than thebeam. In a practical implementation of this method,the ratio a/Wi should not be too large, however, be-cause the accuracy of the measured values wouldthen become sensitive to possible small-scale de-partures of the transmission profile from a perfecttruncated parabola. In addition, the transmissionprofile seen by the optical beam would be relativelyflat, leading to a reduced sensitivity dPmax/dWx of themeasurement system. For a PTF of fixed width, thesensitivity is maximized by choosing values of Tmaxand Tmjni as different as possible.

On the other hand, one should be concerned aboutthe finite dimension of the parabolic transmissionprofile. Indeed, a ratio a/Wx too small means thata significant fraction of the beam's power falls out-side the region wherein T(x) varies quadratically.In this situation, the dependence of P/PO on XM isnot perfectly parabolic, so that Eq. (8) cannot beused directly for deducing W. from the measured ra-tio Pmax/PO. We have investigated the consequencesof the truncated transmission profile by simulat-ing numerically the transmission of various opti-cal intensity distributions through moving PTF'sfor a wide range of the ratio a/Wx. Figure 2 il-lustrates the calculated variations of the normalizedmaximum power Pmax/PO transmitted through thePTF versus a/W, for various intensity distributionsfalling on the PTF. These distributions (curves a,b, c, and d) correspond, respectively, to a Gaussianfunction, to Hermite-Gaussian functions of order 1and 4, and to a super-Gaussian function of order20, which closely approximates a rectangular inten-sity function. The dashed curve depicts the relationbetween Pmax/Po and a/W, given by Eq. (8), valid

where Po is given in Eq. (3).and (2) into Eq. (6), we cantransmitted power P variesPTF translation XM, because

1.By inserting Eqs. (1)

readily show that thequadratically with the

P_ (Tmax-Tm) 22P = Tmax - a2 m [x + (XM - (X))2]. (7)

The measurement of the parameters (x) and o-X2(and then Wx) does not require that the PTF betranslated over a long run along the x direction,because it suffices to search for the position XMthat maximizes the transmitted power. Indeed, onesees from Eq. (7) that P takes on a maximum valuewhen XM = (x). At this position, the real-beamsize is determined from the measured powerPmax P(XM = (x)) by the simple equation

2 4a2 I LWX2 = T _ )KTmax _ max* (8)

(Tmax Tmin) 0

The method is fast and simple since only two powermeasurements, Pmax and P0, are really needed.

00.8

0.6

0.4

0.2

0.

0.0 0.5 1.0 1.5 2.0

a/WxFig. 2. Normalized maximum power transmittedthrough the PTF, calculated as a function of the ratiobetween the transmission profile half-width a andthe real-beam size W,. The intensity distributionsare Gaussian (curve a), Hermite-Gaussian of order 1(curve b), Hermite-Gaussian of order 4 (curve c), andsuper-Gaussian of order 20 (curve d). The dashed curvedepicts the relation between Pmax/Po and a/W. given byEq. (8). Here Tmax = 0.9 and Tmin = 0.1.

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April 1, 1994 / Vol. 19, No. 7 / OPTICS LETTERS 507

Q)

0.5 1.0 1.5 2.0

aIWXFig. 3. Real-beam sizes estimated by reporting thevalues of Pmax/Po plotted in Fig. 2 in Eq. (8). Thesevalues are normalized to the exact result Wx for eachintensity distribution.

when a >> Wx. These curves hold for the typicalvalues Tmin = 0.1 and Tmax = 0.9. For small val-ues of a/Wi, the variations of Pmax/Po depend onthe exact intensity distribution. This dependenceweakens progressively as a/Wi increases, so that thecurves merge completely for values of a/W, largerthan -- 1.2. Excluding the Gaussian distribution, wesee that the remaining curves merge at a still lowervalue (0.90) for a/Wy. The somewhat distinct be-havior of the Gaussian intensity distribution resultsfrom its lower fractional power content within the re-gion -W. ' x • Wx. Because of the predominanceof laser beams having nearly Gaussian intensity pro-files, one may conclude from Fig. 2 that Eq. (8) canbe used for a wide variety of beam intensity profiles,provided that a/W, is kept larger than -1.2, corre-sponding to values of Pmax/Po higher than 0.76 (for thesituation in which Tmax = 0.9 and Tmin = 0.1). Theminimum ratio a/Wi leading to a complete merge ofthe curves of Fig. 2 remains roughly the same for alarge range of values of Tmax and Tmmn. The corre-sponding values of Pmax/Po are deduced by inversionof Eq. (8). Thus a useful feature of the procedure isthat the presence of clipping effects resulting froma narrow transmission profile can be detected fromthe measured ratio Pmax/P0. The variations of thereal-beam size West estimated from those of the ra-tio Pmax/Po reported in Eq. (8) are shown in Fig. 3.The estimated real-beam size is normalized to theexact value Wx for each intensity distribution. Theuse of a transmission profile having a width suchthat a/Wi = 1.2 underestimates the real-beam sizeof a Gaussian beam by less than 1%, and the er-ror becomes negligible for a/Wx 2 1.5. The mainconclusion that can be drawn from the curves ofFig. 3 is that the transmission profile need not besignificantly larger than the beam's size for accuratemeasurements of the real-beam size.

A successful implementation of this technique de-pends primarily on the availability of PTF's havingprecisely defined transmission profiles. GRM's arewell suited for demonstrating this method. Thesemirrors have attracted much attention during the

past few years owing to their advantages whenused as output couplers with unstable resonators.7 9

GRM's with reflectivity profiles varying accordingto common circularly symmetric functions, such asGaussian, super-Gaussian, and parabolic functions,are now routinely fabricated within stringent toler-ances. Special vacuum techniques allow for a widerange of values for the main parameters, such as thewidth of the reflectivity profile and the extremum val-ues of the reflectivity. Although most GRM designsare based on thin-film constructions and are thenwavelength specific, wideband GRM's made from ta-pered layers of absorbing material should be feasiblewith variations of the vacuum techniques discussedabove.

The transmissivity of a GRM used in a practicalimplementation of this method should ideally varyalong the x direction only. Unfortunately, conven-tional GRM's do not fulfill this requirement, owingto their circularly symmetric transmission profiles.When the x direction corresponds to one of the prin-cipal axes of the beam intensity distribution, one cancircumvent this difficulty by placing a cylindrical fo-cusing lens in front of the GRM, with the lens cur-vature along the y direction. The intensity distribu-tion falling upon the GRM would then appear as athin line parallel to the x axis, and the integrationalong the y direction, as required in Eq. (2), would beautomatically performed.

The method described above may find various ap-plications, such as the real-time monitoring of laserbeam widths for the optimization of laser cavityparameters and the fast measurement of the beam-propagation factor M 2 . An experimental investiga-tion of this method is planned.

This research was supported by the Fonds pour laFormation de Chercheurs et I'Aide a la Recherche(Qu6bec) and the Natural Science and EngineeringResearch Council (Ottawa). We acknowledge the ex-cellent help of Ga6tan Duplain of the National OpticsInstitute, Sainte-Foy, Qu6bec, for providing us within-depth information about GRM's.

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