Measurement of a Gaussian laser beam diameter through thedirect inversion of knife-edge data

John M. Khosrofian and Bruce A. Garetz

An inversion algorithm, involving a linear least-squares method, is developed to analyze the intensity dataobtained by application of the traveling knife-edge method. Through use of simulated intensity data, diam-eters can be measured to an accuracy of 0.05%. The method also yields good results when it is applied toHe-Ne laser beam data.

1. IntroductionThe application of lasers to investigation of many

chemical and biophysical phenomena requires knowl-edge of the diameter of the incident beam or the waistdiameter of a focused beam. Although techniques suchas ray tracing,' multiphoton ionization yields,2 andfluorescence correlation spectroscopy 3 predict accurateresults, such methods are either too complicated formost general applications or they rely on specific ex-perimental procedures not easily adaptable to otherexperiments. Other more direct methods for deter-mination of beam diameters include correlation of thesquare of the diameter of a spot burned by the laserbeam with the logarithm of the normalized power ab-sorbed from the beam by the material that incurs thedamage.4 In addition, the intensity profile of the laserbeam can be probed directly either by a photoelectricor pyroelectric detector array.5

Another class of method involves gradual eclipsingof the laser beam by a sharp knife-edge. As the knife-edge intersects the beam in a direction perpendicularto the propagation axis of the beam, a photodetectormeasures the intensity of the unmasked portion of thebeam. If the spatial profile of the laser beam is de-scribed by a Gaussian line shape, the signal measuredby the detector is represented by an integrated Gaussianfunction. In principle, the beam diameter can be de-termined from such photodetector data. One approachinvolves electronic 6 or numerical 7 differentiation of theexperimental data, thus reconstructing the originalGaussian profile.

The authors are with Polytechnic Institute of New York, ChemistryDepartment, Brooklyn, New York 11201.

Received 1 July 1983.0003-6935/83/213406-05$01.00/0.© 1983 Optical Society of America.

We present here an alternative method for analyzingknife-edge data. In the course of this paper, we willdescribe the mathematical procedure by which knife-edge data are inverted to yield accurate beam diametervalues as well as presenting an experimental demon-stration of the method using a He-Ne laser.

11. TheoryThe cross-sectional spatial profile of a Gaussian laser

beam propagating in the z direction is given by thefunction

G(xy) = Io exp[-_3 2 r2 (xy)], (1)

in which r(xy) = [(x - xO) 2 + (y - yo) 2]1/2 , where xoyodefine the center of the beam profile. I is the maxi-mum value of G (x ,y) measured at the center, and -1is the radius at which G (x ,y) falls to l/e times its max-imum value Io (37%).

Although the distribution given in Eq. (1) is radiallysymmetric, the present procedure involves a linear 1-Dtranslation of the knife-edge. If the x axis is chosen asthe translation axis, the response of the photodetectorto the transmitted portion of the beam is given by

XX f b

R(xb) = I S G(x,y)dxdy,_.1 - c

(2)

where xb is the x coordinate of the knife-edge when themeasurement is made. Evaluation of Eq. (2) yields

R(xb) = Io(ir/1 2)"/2 , exp[-f32 (x - o) 2 ]dx. (3)

This response can be normalized to the total integratedsignal of the full totally unmasked beam (xb - oa) by

FR(xb) = R(xb)/R(o) = Iir-1/2 J exp[-f32 (x - o)2]dx, (4)

where represents the normalized response func-tion.

3406 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

In principle, , which determines the ultimate valuefor the beam radius, can be deduced from the trans-formation condition which converts Eq. (4) into thenormal distribution:

N(z) = (2ir)/2 5 exp(-n2 /2)dn. (5)

The salient difference between (x) and N(z) is foundin the arguments of the exponents in the integrands.The two arguments can be made equal if the condi-tion

n2/2 = 2(X - X)2 (6)

is imposed. Solving Eq. (6) for n yield

n = /2-dx - A/2-oxo (7)

or in differential form

dn = -V-2dx. (8)

In addition, Eq. (7) provides a simple linear functionwhich can be used to determine A, if the correspondencebetween the experimentally determined translationpoints x and the equivalent points n of the normal dis-tribution is known. To separate from x0, Eq. (7) canbe rewritten as

nX = + XO,

or, evaluated for a particular measurement,

Xb = ;t=2 + XO.

Although the values for xb are determined experi-mentally, the values for z must be calculated fromtabulated normal distribution data. Such a method,however, is often tedious and cumbersome, especiallyfor a large number of data points. A better procedureis use of an analytic approximation for the normal dis-tribution to generate the values of z. Such an approachalso obviates the necessity of either storing or reenteringthe tabulated normal distribution into a limited mem-ory. As an initial approximation, the function

1 + exp(-z)(13)

is a good representation for the extrema of the normaldistribution; as z approaches + or - infinity, f(z) con-verges to unity and zero, respectively, and f(z) assumesthe value 0.5 for z = 0. However, a better representa-tion for a wider range of values can be obtained by re-placement of z by a polynomial expansion in z:

mP(z) = E aizi

i=O

as a result of Eq. (12), Eq. (13) can be modified:

f(z) =1 + exp[P(z)]

(14)

(15)

(9) The values for m, the order of the polynomial, and thecoefficients aiI can be determined by application of themethod of least squares to the logarithmic transform of

(10) Eq. (15):

Since Eq. (9) is a linear equation with two unknownquantities p-1 and x0, only two pairs of data points(Z,Xb) are necessary to uniquely determine the two un-knowns.8 9 For example, one could measure the Xb

values at which the beam intensity falls to 10 and 90%of the full value. From a normal distribution table, thevalues of z at which the normal distribution N(z) equals0.10 and 0.90 are found to be :1.28. This informationprovides two simultaneous equations:

1.28xlO = -gg2 + x0;

1.28xgo = 2- V'gB + x (lib)

which when solved yields

x0 = (xjo + xso)/2 (12a)

$- = 0.552(xio - x9o). (12b)

However, such an approach to the problem has disad-vantages. Each experimentally measured translationpoint contains an inherent random error which ispropagated to : through Eq. (10). Furthermore, themethod provides no indication of how closely the spatialintensity profile of the laser beam resembles a Gaussianline shape. The best approach is application of themethod of least squares to Eq. (10) for a large numberof data pairs. After: and x0 have been determined, Eq.(10) can be used to generate either the best fit profile orthe degree of correlation of the experimental data withthe theory.

P(z) = logV-f(z) - 1] (16)

and the use of the data from the tabulated normal dis-tribution function1 The least-squares analysis gen-erates a third-order polynomial with coefficients

ao = -6.71387 X 10-3,

a, = -1.55115,

a2 = -5.13306 X 10-2,

a3 = -5.49164 X 10-2,

(17)

which correspond to a maximum correlation of0.999891. The normal distribution data consist of everytenth listing in Table V of Ref. 10

After the values of the polynomial coefficients havebeen established, f (z) can be used to determine z fromR(xb), the experimentally measured normalized re-sponse function, by the method of successive approxi-mations; the value of z is changed by successivelysmaller increments, and f(z) is compared with R(xb)until

R(xb) = f(z) (18)

to the desired degree of accuracy.

Ill. Experimental ProcedureThe knife-edge translator which was used to probe

the beam intensity profile is illustrated in Fig. 1. Itconsists of a micrometer-driven translation stage, towhich is secured a single-edge razor blade. Beam di-ameter measurements were performed on a cw He-Ne

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3407

origin is applied as follows:

R(-xb) = -Xb G(x)dx = 5 G(x)dx

= , G(x)dx - Xb G(x)dx = 1 - R(xb).

Similarly,

f(-z) = 1 - f(z).

heliumneonlaser

t ranslator

Fig. 1. Schematic diagram of the experimental configuration.

laser with an output power of 2 mW at 632.8 nm. Theknife-edge was advanced through the beam in equalsteps of 25.4 ,tm. The integrated intensity of the une-clipsed portion of the beam was measured by a PINdiode operating in the photovoltaic mode. The voltagesignal was measured on a Tektronix 7704 oscillo-scope.

IV. Discussion

A. Computer ProgramThe computer program which analyzes the experi-

mental data was written in BASIC language for a Com-modore 8K PET microcomputer. The program is easilymodified for use with other comparable computers.

The experimental response data are entered seriallyinto the computer memory. These data are correctedfor a predetermined background and normalized to themaximum response signal, which corresponds to thesignal from a totally uneclipsed beam.

A translation axis is set up internally in the computerso that one translation unit corresponds to a single stepof the micrometer head. In our experiments, a singlestep corresponds to a distance = 25.4 m of actualtranslation.

The normalization of the response data produces aset of normalized responses R (xb) which range betweenzero and unity. To determine the value of z, the nor-malized translation distance corresponding to eachR(xb) as described in Eq. (18), a regression techniqueis used to approximate the numerical inversion of Eq.(15). The value for z is varied, initially from zero to apredetermined upper limit by increments of 0.1, untilz assumes the largest value for which

f(Z) <5 (Xb). (19)

The algorithm is repeated three times, beginning withthe current value of z, with the value for the incrementdiminished by a factor of 10 for each repetition. Thefinal value of z is expressed to four decimal places.

The procedure that has been presented is valid onlyfor R(Xb) > 0.5, because the parameters that define f (z)have been determined from tabulated normal data withpositive arguments. To extend the procedure to in-clude negative arguments of R (Xb) and f (z), the reflec-tion symmetry of the Gaussian line shape about the

(21)

In the actual execution of the program, R (-xb) isreplaced by -R(xb). The negative sign is used as a labelto indicate that the value for z that is calculated fromthe absolute value of -(xb) should be stored inmemory as -z.

The method of least squares is used to fit the pairs(Xb,Z) to a straight line, thus determining / and x0 indimensionless units. The corresponding dimensionedquantities are //6 and x0b, where is the micrometercalibration factor. The program for the analysis isavailable upon request.

B. Analysis of ResultsTo establish the accuracy of the numerical procedure

for determination of /, simulated profiles correspondingto particular values of 3 were generated from tabulatednormal distribution data through use of Eq. (10).Best-fit values of / were determined using the describedalgorithm and were compared to the exact : values.The fits were excellent, with agreement to 0.05%.

In addition, the calculated /3 values were used togenerate the best-fit profiles. Comparison of exact andcalculated profiles is shown in Figs. 2 and 3 for twodifferent values of /3.

The results from analysis of experimental data for aHe-Ne laser are presented in Fig. 4, which depicts atypical profile from a series of eleven measurements.The beam diameter d with respect to the l/e2 points ofthe Gaussian line shape is related to / through the ex-pression

d = 2V,-1. (22)

The values of the beam diameter based on eleven in-dependent measurements were 0.67 0.02 mm. Thevariations among the five measurements are probablydue to a variety of sources, including some instabilityin the mounting of the translator and to fluctuations inthe laser beam intensity. The former problem can bealleviated through a more secure mounting of thetranslator. The latter could be surmounted by splittingoff a fraction of the laser beam and normalizing allmeasurements with respect to this reference inten-sity.

The sensitivity of the translation can be increased byrotating the knife-edge in its mount by an angle 0,measured from the initial vertical position of theknife-edge. As a result, the direction of translation isalso rotated by the angle 0, in the plane perpendicularto the direction of propagation of the laser beam. If theknife-edge passes through a distance ( before the rota-tion, the corresponding translation distance for therotated knife-edge is the projection of along the new

3408 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983

photodIode

(20)

direction of translation or cosO. Such a procedureeliminates the need for a micrometer drive with a finerpitch when higher sensitivity is required.

V. ConclusionsThe results demonstrate that the knife-edge method

used in conjunction with the described algorithm for theanalysis of the detector data provides a relatively simpleprocedure for accurately measuring the diameter of aGaussian laser beam. In particular, the algorithm (Fig.5) consists of a straightforward numerical technique fordirect analysis of the experimental data rather than firstreconstructing a Gaussian distribution.

1.0

Lu0z0

0

so 0.50LuN-

0z

0.0

0 10 20 30 40 50 60 70RELATIVE TRANSLATION

Fig. 2. Integrated Gaussian profile for f: = (A/2)-l. Solid dots in-dicate experimental data points, and the solid line represents the

best-fit profile.

1.0

LU

z0

0.5LU

N _

0z

OD

0 10 20 30RELATIVE TRANSLATION

Fig. 3. Integrated Gaussian profile for f: = 0.177. Solid dots indicateexperimental data points, and solid line represents the best-fit

profile.

RELATIVE TRANSLATION

0 10 20

1.0

0Lu

Lu

N

I0.0

30

Fig. 4. Integrated Gaussian profile for the He-Ne laser beam. Asingle translation step corresponds to 25.4 tm. Solid dots indicateexperimental data points, and the solid line represents the best-fit

profile.

Fig. 5. Schematic flow chart for the computer algorithm.

Acknowledgment is made to the National ScienceFoundation (grant ECS-7908109), the Research Corp.,and the Petroleum Research Fund, administered by theAmerican Chemical Society, for support of this re-search.

1 November 1983 / Vol. 22, No. 21 / APPLIED OPTICS 3409

I I I I I I I I I I I I I I

I I I I I I I I I. .l l

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(1980).4. Y. C. Kiang and R. W. Lang, Appl. Opt. 22, 1296 (1983).5. J. G. Edwards, H. R. Gallantree, and R. M. Quilliam, J. Phys. E

10, 699 (1977).6. J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Claviere,

E. A. Franke, and J. M. Franke, Appl. Opt. 10, 2775 (1971).7. M. Mauck, Appl. Opt. 18, 599 (1979).8. Y. Suzaki and A. Tachibana, Appl. Opt. 14, 2809 (1975).9. Laser Handbook, (Metrologic Instruments, Bellmar, N.J. 1979),

p. 28.10. H. D. Brunk, An Introduction to Mathematical Statistics

(Blaisdell, Waltham, Mass., 1965), p. 389.

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3410 APPLIED OPTICS / Vol. 22, No. 21 / 1 November 1983