Ronchi ruling characterization of axially symmetriclaser beams

Robert M. O'Connell and Cheng-Hao Chen

The application of the Ronchi ruling beam characterization method to axially symmetric optical beams isanalyzed. Specific results are derived for the Airy and focused annulus diffraction patterns. Plots of theratio of minimum to maximum transmitted optical power vs the first null radius of the beam functions showthat for the Airy pattern and other focused annuli with obscuration ratios smaller than -0.30, the methodshould be as useful as with Gaussian beams.

1. IntroductionThe use of Ronchi (periodic square-wave) and other

rulings has been analyzed thoroughly and shown to bea fast, accurate, and inexpensive method for determin-ing the l/e 2 radius ro of a Gaussian laser beam.4 Themethod consists of moving a ruling (reflective or trans-missive) of known optical characteristics transverselythrough the beam and measuring the ratio K of theminimum total power Pmin reflected or transmitted tothe maximum total power Pmax reflected or transmit-ted. The beam radius can then be obtained from agraphic relationship between K and r that resultsfrom an analysis of the optical power reflected or trans-mitted when a Gaussian laser beam illuminates a spe-cific ruling (Ronchi, triangular, etc.) with specific opti-cal characteristics (period, duty cycle, and reflectanceor transmittance).

In an early analysis,' Dickson considered the case ofan ideal transmissive Ronchi ruling, i.e., one consistingof bars and spaces of equal width L (50% duty cycle)and transmittances of 0 and 100%, respectively. Hefurther simplified the analysis by restricting the rulingto one transparent space centered on the beam, twoopaque bars, and a continuous transparent space ev-erywhere else in the Pmax calculation and to its exactcomplement in the Pmin calculation. The graphic re-sult of his analysis shows that the measured K ratio

The authors are with University of Missouri-Columbia, Electrical& Computer Engineering Department, Columbia, Missouri 65211.

Received 8 June 1989.0003-6935/90/304441-06$02.00/0.© 1990 Optical Society of America.

increases smoothly and monotonically from zero forvalues of the ratio rolL greater than -0.35 and reason-ably linearly over the range of (ro/L,K) from (0.65,0.14)to (1.35,0.76).

Broockman et al.2 analyzed the case of a reflectiveRonchi ruling with the same simplified geometry as inRef. 1 but with nonideal optical properties. Theiranalysis shows that as the bar and space reflectancesvary from the ideal values of 0 and 100%, respectively,the range of K ratios available for measuring the samerange of rolL ratios as in Ref. 1 is significantly reduced.This effect reduces the accuracy of the measurement.The analysis also shows that the K vs rolL relationshipis relatively insensitive to small differences in the dutycycle of the ruling from the ideal case (50%).

Karim3 removed the geometric and optical restric-tions of Refs. 1 and 2 and derived a K vs rolL relation-ship for a Ronchi ruling having an infinite number ofbars and spaces with arbitrary values of duty cycle andreflectance or transmittance. When applied to theideal case (50% duty cycle and bar/space transmit-tances of 0%/100%), his result gives a graphic K vs r/Lrelation similar to Dickson's but with a slightly largeruseful linear region, i.e., over the range of (ro/L,K) from(0.60,0.128) to (1.45,0.80). Thus a fairly wide range ofGaussian beam radii can be accurately measured witha single Ronchi ruling with the smallest such radiusbeing set by the size of available rulings and the mini-mum measurable ro/L (0.35). For example, a 400-line/in. ruling, for which L = 31.8 Am, could be used tomeasure beam radii of 19-46 Atm in the linear rangeand a minimum radius of -10 um.

Recently, Karim et al.4 analyzed the use of transmis-sive sinusoidal and triangular rulings to measure aGaussian beam radius. Their results show that, inagreement with Broockman et al.,2 the available rangeof the measurable K ratio is significantly reduced if the

20 October 1990 / Vol. 29, No. 30 / APPLIED OPTICS 4441

maximum and minimum transmittances of the rulingare different from the ideal values of 100 and 0%,respectively. More significantly, they show that,whereas the smallest rolL obtainable with a Ronchiruling is -0.35, rolL ratios smaller than 0.05 can bemeasured with sinusoidal and triangular rulings.Thus beam radii of the order of 1,gm should be measur-able with such rulings.

While these1 4 and other studies56 have concentrat-ed on the use of rulings to characterize Gaussianbeams, it appears that the method has not yet beenapplied to non-Gaussian beams. In this paper weanalyze the use of an ideal Ronchi ruling to character-ize an axisymmetric beam, i.e., one that is circularlysymmetric about the propagation axis. Importantspecific examples considered are the Airy pattern andfocused annulus. Such beams are frequently charac-terized or profiled by approximating them as Gauss-ian.7 However, as has been shown,8 such an approxi-mation can lead to significant errors, especially whenthe beam has substantial sidelobes.

The characterization of axially symmetric beamsusing the knife-edge method was considered earlier,8

but, as pointed out in Ref. 1, the ruling method hasseveral practical advantages over the knife-edgedmethod. For example, whereas the knife-edge meth-od requires a large number of accurately positioneddata readings, the ruling method requires only two(one each for Pmin and Pmax) without concern for theactual locations of the ruling. Also, the range of mea-sured powers is much greater with the knife-edgemethod, so that the associated measurement errors aregreater.

II. Ideal Ronchi Ruling and Arbitrary AxisymmetricBeams

As with a Gaussian beam, characterizing any beamwith the ruling method requires a graphic relationshipbetween the optical power ratio K and some propertythat characterizes the beam (e.g., the 1/e2 beam radiusin the Gaussian case). A unique relationship has to bederived for each ruling-optical beam function combi-nation. For an ideal Ronchi ruling and any axiallysymmetric beam, an expression for K can be obtainedwith the aid of Fig. 1. The circles in the figure repre-sent the beam whose optical intensity function andradial extent beyond which the beam power is smallenough to be ignorable are I(r) and R, respectively.The beam is shown centered over an opaque bar in theupper portion of the figure for the calculation of Pminand over a transparent space in the lower portion forthe calculation of Pmax.

To calculate Pmin and Pmax, note first that the beampower P in the area A both to the right of any verticalline at the x-axis and above the x-axis in Fig. 1 can beobtained from the expression

(1)P= JA I(r)dA.

Using polar coordinates in which dA = rdrdO and refer-ring to Fig. 1, Eq. (1) can be rewritten as

space

(a)

(b)

-7

/ Opaque

I I LI xL LI1Fig. 1. Geometry used to analyze the characterization of an arbi-trary axially symmetric beam with an ideal Ronchi ruling: (a) for

the calculation of Pmin; (b) for the calculation of Pmax.

R -,cos(x/r) R

P= I. rI(r)dr J dO = I rI(r) cos-'(x/r)dr.

Since the last form is a function of x, we define it asq(x), i.e.,

R

0(x) = J rI(r) cos-'(xlr)dr. (2)

Referring once again to Fig. 1, it can be seen that Pminand Pmax can, respectively, be expressed in terms ofk(x) as

Pmin = 4[q5(L2) - 0(3L/2) + 0(5L/2) - (7L/2) +...]

= 4 > {q[(4m - 3)L/21 - 0[(4m - 1)L/2]1,m=1

Pmax = 4[U(0) - (L/2) + ¢,(3L2) -(5L/2) + 0(7L/2) +...]

= 4(O() -O(L/2) + E 4(4m - 1)L/2]m=l

- [(4m + 1)L/2]1).

Using these forms, the ratio K = Pmin/Pmax becomes

'0 {[(4m - 3)L/2] - 4[(4m - )L/2]1

0(0)- (L/2) + 1{0[(4m - )L/2] - 0[(4m + 1)L/2])m=1

where 0( ) is defined in Eq. (2).

4442 APPLIED OPTICS / Vol. 29, No. 30 / 20 October 1990

Equations (2) and (3) are valid for any axially sym-metric beam and an ideal Ronchi ruling. To be practi-cally useful, however, they must be applied to a specificbeam intensity function I(r) and expressed in terms ofsome parameter characteristic of that function. Thisis done specifically for the Airy diffraction pattern andthe focused annulus in the following section. Sincethe Airy pattern is just a special case of a focusedannulus, the analysis will be done for the more generalfocused annulus and the results simplified appropri-ately for the Airy pattern.

1.0

0.8

0.6-

0.4Ill. Application to the Focused Annulus and AiryDiffraction Patterns

A. AnalysisThe output of a laser based on the confocal unstable

resonator design9 is a uniform beam in the shape of anannulus.10 As illustrated in the inset of Fig. 2, theannulus consists of a circular aperture of radius a and acentral circular obstruction of radius Ea, where isdefined as the obscuration ratio. When a collimatedannular beam is focused, the intensity function of theresulting Fraunhofer diffraction pattern can be writ-ten as1l

0.2

0 l \ V0 2 4 6 8 10

rcFig. 2. Geometry of an annular aperture (inset) and normalizedcross sections of two axially symmetric beam functions: (a) the Airypattern [Eq. (6)] or focused annulus [Eq. (4)] with e = 0.0; (b) the

focused annulus [Eq. (4)] with e = 0.5.

L° ) rc ere ]J' (4)

where I0 is the intensity at the center of the beam, J1 )is the Bessel function of the first kind, first order, and cis a constant given by

c = 27ra/X. (5)

In Eq. (5) a is the radius of the diffracting aperture, f isthe lens focal length, and X is the wavelength of thediffracted light. Note that when e = 0 the diffractingaperture shown in Fig. 2 becomes a simple circle forwhich the diffraction pattern is the well known Airypattern. Accordingly, Eq. (4) reduces as expected to

I(r) = 10[2J(rc)/rcJ2. (6)

The Airy pattern is an important special case becauseit occurs frequently in most laser systems due to thecommon use of circular apertures and lenses.

Figure 2 shows normalized plots of Eq. (4) for thecases of e = 0.0 (the Airy pattern) and 0.5. Note that asE increases, an increasing fraction of the total energy inthe beam appears in the sidelobes. Thus the errorinherent in characterizing this beam function by as-suming it to be approximately Gaussian becomes moresignificant with increasing e.

Application of Eqs. (2) and (3) to the focused annu-lus begins by using Eq. (4) in Eq. (2) and defining a newvariable y = rc. Equation (2) thus becomes

O(xc) = 4o/(1 - 2)2c2 [J 1 (y) - EJl(Ey)]2

X (l/y) cos-'(xcly)dy. (7)

The upper limit of Eq. (7) is written R - because,although the Airy pattern has infinite radial extent,the integral has to be terminated at some finite pointfor purposes of computation. Since the argument ofthe function k( ) in Eq. (2) has been changed from x toxc in Eq. (7), all the arguments of 0( ) in Eq. (3) mustalso be multiplied by c. Also, it is evident that whenEq. (7) is used in Eq. (3) the factor 4o/(1 - E2)2c2 will becommon to every term and thereby cancel. It is conve-nient, therefore, to define a new function to be used inEq. (3), i.e.,

0(z) = 0(z = xc)4Id(1 -f2)2C2

IRc--= jc [J 1() - eJ1(ey)]

2(lly) cos'1(z/y)dy.

In terms of Eq. (8), Eq. (3) can be rewritten as

(8)

M-XI 10[(4m - 3)Lc/2] - 0[(4m - 1)Lc/211

m=1M-X

7r(1 - E2)/4 - 0(Lc/2) + Y 10[(4m - 1)Lc/21 - 0[(4m + 1)Lc/211

m=1

20 October 1990 / Vol. 29, No. 30 / APPLIED OPTICS 4443

K (9)

where the quantity M has been introduced in the sum-mations for computational purposes and the first termin the denominator has been replaced by its actualvalue, i.e.,

0(0) = 2 J [Jl(y) - eJ 1(ey)j2

(1/y)dy

= [f dy + e2 J dy-2e J 'Y)J'('Y) dy]

=7r ( 2 +ee2) =7r(1 E2)

where the tabulated integral

Jn,(at)Jn,(bt) t(/n.dt=o t 2n

was used to evaluate the three integrals in the calcula-tion.

The final step in applying Eq. (3) to the focusedannulus is to replace the quantity c in Eq. (9) with aparameter that characterizes the beam function. Inanalogy with the use of the 1/e2 beam radius ro in thecase of Gaussian beams, the radial distance r, to thefirst null of the focused annulus is defined here as thatcharacteristic parameter. It is evident in Fig. 2 thatthe first zero of the focused annulus depends on thevalue of e. We thus define g(e) as the value of rccorresponding to the first null of the intensity functionin Eq. (4), i.e., g(e) = ric, so that the form of c to be usedin Eq. (9) is

c = g(e)/r. (10)

From Eq. (4) it can be seen that g(E) is the smallestsolution of the transcendental equation

J1(rc) - eJj(erc) = 0, (11)

which can be solved iteratively. For example, for e =0.0 and 0.5, the solution of Eq. (11) yields g(e) = 1.2207r= 3.833 and g(e) = 1.001r = 3.145, respectively. Sub-stitution of Eq. (10) into Eq. (9) yields the requiredresult for K, i.e.,

M-Z 10[(4m - 3)g(e)LI2r,] - 0[(4m - 1)g(e)LI2r1]jm=l

M-

7(1 - 2)/4 - OOg(e)LI2rj] + E 0[(4m - 1)g(e)L/2rj] - [(4m + l)g(e)L2r]jm=1

where 0[ ] is defined in Eq. (8).

B. ResultsEquation (12) provides the desired relation between

the measured ratio K and the radial distance r1 charac-teristic of a specific focused annulus. For comparisonwith Karim's results for Gaussian beams,3 Eqs. (8) and(12) were used to obtain plots of K vs r1L. Therepeated integrations of Eq. (8) required in Eq. (12)were done numerically using Simpson's one-thirdrule.13 To ensure that the upper limit Rc of everyintegral was always larger than the lower limit, it wasnecessary to require that Rc be at least as large as the

largest argument in Eq. (12), which is (4M + 1)g(e)LI2rj. This leads to the condition

r1IL > (4M + 1)g(e)I2Rc, (13)

which reveals a computational trade-off, not present inthe case of a Gaussian beam, between the minimumr1L for which Eq. (12) can be evaluated and the num-ber of terms M used in the summations. Fortunately,the difficulty imposed by this trade-off could be essen-tially eliminated by a sufficiently large choice of theupper limit Rc (200), which also guaranteed that mostof the energy in the beam was accounted for in thecomputations.

In Ref. 11 it is shown that the fraction of the totalpower or energy of the Airy pattern (the e = 0 case)contained in the axial circle of radius R is given by theexpression

P(R) = 1 - J2(Rc) - J2f(Rc), (14)

where J0 ( ) and J1( ) are Bessel functions of the firstkind, orders zero, and one, respectively. Using largeargument formulas14 for J0( ) and J,( ) in Eq. (14), itwas found that for Rc = 200, P(R) = 99.68%. Forvalues of e other than zero, P(R) is slightly less, as canbe seen in the normalized curves of Fig. 2, where thesidelobe energy content increases with e. The fractionof total energy contained within a finite radius of afocused annulus with arbitrary e can be calculated bynumerically integrating Eq. (8) with z = 0. This hasbeen done previously,15' 7 and the results of Ref. 17show that for Rc = 200, at least 99% of the beam energyis included if e < 0.5, and at least 98% is included if E <0.8. Furthermore, with Rc = 200 in Eq. (8) and as fewas three terms (M 2 3) in the summations of Eq. (12),plots of K vs rIL were found to be virtually identicalfor values of rIL up to -2.5. At that point K c 0.99,which is well beyond the useful portion of a given plot.

Figure 3 shows plots of Kvs rIL for E = 0.0,0.25, and0.50, which were obtained with Rc = 200 in Eq. (8) andM = 5 in Eq. (12). The lower ends of the curvesterminate at rIL / 0.20 due to the limitation imposed

, (12)

by Eq. (13). Fewer terms (M < 5) in the summationswould allow the curves to extend to smaller values ofr1L. On the other hand, larger M are needed to showhow the upper ends of the curves approach 1.0 asymp-totically with increasing rIL. Therefore, the use of M= 5 represents an acceptable compromise that yieldsthe useful portion of the K vs rj/L relationship.

A K vs rj/L Ronchi ruling plot is most useful where itvaries smoothly, monotonically, and approximatelylinearly over as wide a range of (r,/L,K) values aspossible. The curves in Fig. 3 (and others not shownhere) show that, beginning with e _ 0.45, a portion ofthe K vs rjIL data exhibits nonmonotonic behaviour

4444 APPLIED OPTICS / Vol. 29, No. 30 / 20 October 1990

1.0

0.8

0.6

0.4

0.2

00 0.5 1.0 1.5 2.0 2.5

ri/ LFig. 3. Plots of K vs rlL obtained using Rc = 200 in Eq. (8) andM = 5 in Eq. (12) for various focused annuli: (a) = 0.0 (the Airy

pattern); (b) = 0.25; (c) = 0.50.

that becomes more pronounced with increasing e.Thus the method is most useful for smaller e, i.e., e <0.45. For example, curve (a) in Fig. 3 for the Airypattern (E = 0) is smooth and reasonably linear over theapproximate range of (rILK) values from (0.75,0.1) to(2.25,0.9). Also, although the lower end of the curveterminates at -rIL = 0.20 [because of Eq. (13)], it isevident that the minimum measurable rL is of theorder of 0.1. Thus, using a 400-line/in Ronchi ruling(L = 3 1 .8 m), it should be possible to measure crudelyfirst nulls of Airy patterns as small as -3 um and tomeasure accurately those ranging in size from -24 to71,um [corresponding to the linear portion of curve (a)in Fig. 3]. These results are quite similar to thoseobtained by Karim3 for Gaussian beams (see Sec. I).In fact, the useful linear portion of Fig. 3 is slightlylarger than that of Ref. 3. Since the behavior of curve(b) in Fig. 3 for e = 0.25 can be seen to be similar to thatof curve (a), similar comments can be made about itand other curves corresponding to relatively small e.Thus the Ronchi ruling method should be at least asuseful with those beams as with Gaussian beams.

Curves of Kvs rIL for larger e can also be used but ina somewhat restricted manner. For example, curve (c)in Fig. 3 for = 0.50 is monotonic and more or lesslinear over two relatively small ranges of (rILK), i.e.,from -(0.29,0.12) to (0.50,0.42) and from -(1.4,0.55) to(2.0),0.97). In terms of a 400-line/in. ruling, these tworanges would enable measurement of the first nullradii ranging from -9.2 to 15.9,um and from -44.5 to63.6,um.

Determining the r of a focused annulus from a plotofKvs rIL depends on whether the value of e is known.If is known, the procedure is the same as in the case ofGaussian beams.3 A single measurement of K with aRonchi ruling of known L allows r to be read immedi-

ately from the appropriate plot of K vs rL. If thevalue of e is unknown (the more likely situation), mea-surements of K with two different Ronchi rulings ofknown bar and space widths (Li and L2) are needed todetermine r. The first measurement of K with theknown L, will determine a set of possible r1 values, onefrom each of a set of curves of K vs rilL for different e.With the known L2, a set of rIL2 ratios can then beformed and used with the set of K vs rL curves topredict the value of K to be measured with L2. Thesecond measurement should agree with only one ofthose predictions, thereby establishing e and rl.

Note finally that a measurement of r also deter-mines the outer radius a of the diffracting aperture.This becomes evident when Eqs. (5) and (10) are com-pared, which shows that

a = g(e)fX/27rrj, (15)

where the lens focal length f and the wavelength X ofthe diffracted light are presumably known.

IV. Summary and ConclusionsWe have analyzed the use of an ideal Ronchi ruling

to characterize axially symmetric optical beams.Equations (2) and (3) express the ratio K of minimumto maximum transmitted optical powers in terms ofthe ruling parameter L and the beam function I(r) ofan arbitrary axisymmetric beam. Equations (2) and(3) are converted to Eqs. (8) and (12), respectively,when applied to the focused annulus diffraction pat-tern [Eq. (4)], a special case of which is the Airy diffrac-tion pattern [Eq. (6)].

In analogy with previous Ronchi ruling analyses ofGaussian beams,'" Eqs. (8) and (12) were used togenerate plots of K vs rL, where r is the first null ofthe specific beam function. Typical examples are giv-en in Fig. 3. The results suggest that the Ronchi rulingmethod is most useful for focused annuli with relative-ly small obscuration ratios, i.e., E < 0.30 (approximate-ly). In such cases the K vs rL plots are similar inbehavior to the results obtained for Gaussian beams,i.e., monotonic and more or less linear over a relativelywide range of (rIL,K) values.

It should be possible to characterize beams quicklyand accurately with first nulls at radii greater than orequal to -25 m. If e is known, one measurement of Kwith a ruling of known L determines both r (from a Kvs rL plot) and the outer radius a of the diffractingaperture [from Eq. (15)]. If e is unknown, a secondmeasurement with a different known ruling will deter-mine e, r, and a.

References1. L. D. Dickson, "Ronchi Ruling Method for Measuring Gaussian

Beam Diameter," Opt. Eng. 18, 70-75 (1979).2. E. C. Broockman, L. D. Dickson, and R. S. Fortenberry, "Gener-

alization of the Ronchi Ruling Method for Measuring GaussianBeam Diameter," Opt. Eng. 22, 643-647 (1983).

3. M. A. Karim, "Measurement of Gaussian Beam Diameter UsingRonchi Rulings," Electron. lett. 21, 427-429 (1985).

20 October 1990 / Vol. 29, No. 30 / APPLIED OPTICS 4445

c

4. M. A. Karim et al., "Gaussian Laser-Beam-Diameter Measure-ment Using Sinusoidal and Triangular Rulings," Opt. Lett. 12,93-95 (1987).

5. D. K. Cohen, B. Little, and F. S. Luecke, "Techniques for Mea-suring 1-um Diam Gaussian Beams," Appl. Opt. 23, 637-640(1984).

6. R. Csomor, "Techniques for Measuring 1-um Diam GaussianBeams: Comment," Appl. Opt. 24, 2295-2298 (1985).

7. J. Ebert and E. Kiesel, "Measurement of Laser-Induced Damagewith an Unstable Resonator-Type Laser," Appl. Opt. 23, 3759-3761 (1984).

8. 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).

9. W. F. Krupke and W. R. Sooy, "Properties of an UnstableConfocal Resonator CO2 Laser System," IEEE J. QuantumElectron. QE-5, 575-586 (1969).

10. J. T. Verdeyen, Laser Electronics (Prentice-Hall, EnglewoodCliffs, NJ, 1981), p. 106.

11. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford,1975), pp. 395-418.

12. G. N. Watson, A Treatise on the Theory of Bessel Functions(Cambridge, U.P. New York, 1944), p. 405.

13. R. G. Stanton,NumericalMethodsforScience and Engineering(Prentice-Hall, Englewood Cliffs, NJ, 1961), p. 116.

14. Samuel M. Selby, editor, Standard Mathematical Tables (CRCPress, Cleveland, 1965), p. 350.

15. W. T. Welford, "Use of Annular Apertures to Increase FocalDepth," J. Opt. Soc. Am. 50, 749-753 (1960).

16. B. L. Mehta, "Total Illumination in an Aberration Free AnnularAperture," Appl. Opt. 13, 736-737 (1974).

17. H. F. A. Tschunko, "Imaging Performance of Annular Apertur-es," Appl. Opt. 13, 1820-1823 (1974).

4446 APPLIED OPTICS / Vol. 29, No. 30 / 20 October 1990